The action of Kontsevich’s graph
complex on Poisson structures and
star products: an implementation
Dissertation submitted for
the award of the title Doctor of Natural Sciences
to the Faculty Physics, Mathematics, and Computer Science
of Johannes Gutenberg University Mainz
in Mainz
Ricardo Thomas Buring
Born in Groningen, the Netherlands.
Mainz, October 31st 2022.
Supervisor: Dr. hab. Arthemy V. Kiselev
University of Groningen, The Netherlands
Supervisor: Prof. Dr. Duco van Straten
Johannes Gutenberg-University of Mainz, Germany
Date of doctoral examination: October 13th, 2022
Abstract
Poisson brackets emerge whenever the pointwise product of scalar functions on an
affine manifold is deformed in such a way that it stays associative. Kontsevich
proved the converse: a universal formula assigns such an associative deformation
to every Poisson bracket. Likewise, Poisson brackets can be deformed by universal
formulae. In both constructions, the universal formulas are built by using graphs.
To handle the thousands of graphs, we develop and present the software package
gcaops (Graph Complex Action on Poisson Structures) for SageMath. Using this
package, • we expand Kontsevich’s ⋆-product up to ō(h̄4); • we assemble ⋆ mod
ō(h̄6) from external data by Banks–Panzer–Pym and we obtain the star product
⋆aff mod ō(h̄7) for affine Poisson brackets; • we verify that graph weights found
by Banks–Panzer–Pym up to ō(h̄6) satisfy many known relations; • we illustrate
the explicit proof of the associativity for the full star product modulo ō(h̄6) and
for the affine star product modulo ō(h̄7); • we find new explicit formulas of graph
cocycles and universal Poisson cocycles, and • we prove the factorization of the
Poisson cocycle condition via the Jacobi identity in each case.

Contents
Overview 1
Actual background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Local embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Research problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Scientific novelty of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Practical significance of results . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Personal contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Approbation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Publications and citation analysis . . . . . . . . . . . . . . . . . . . . . . . . . 12
Structure of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Scientific outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Star products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Poisson flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
List of publications 37
Abstract 39
I Computer demonstrations 40
0 Introduction 41
0.1 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
0.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
0.2.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
0.2.2 Differential polynomials . . . . . . . . . . . . . . . . . . . . . . . 43
0.2.3 Symbolic expressions . . . . . . . . . . . . . . . . . . . . . . . . . 44
1 Implementation of star products 45
1.1 Star products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.2 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.3 Polydifferential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2 Implementation of Poisson structures 55
2.1 Superfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3 Bi-vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.4 Poisson structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Poisson complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.6 Homogeneous polynomial Poisson complex . . . . . . . . . . . . . . . . . 59
i
ii CONTENTS
2.7 Stable Poisson cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Implementation of Formality 65
3.1 Formality graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 The bi-colored operad of Formality graphs . . . . . . . . . . . . . . . . . 68
3.3 Formality graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4 Kontsevich star product . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1 Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.2 Star product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.3 Star product from Formality morphism . . . . . . . . . . . . . . . 80
3.4.4 Star product associativity via Leibniz graphs . . . . . . . . . . . 83
3.5 Leibniz graph expansion and factorization(s) . . . . . . . . . . . . . . . . 87
3.5.1 Iterative production of Leibniz graphs . . . . . . . . . . . . . . . 87
3.5.2 Leibniz graph factorization (non)uniqueness . . . . . . . . . . . . 93
3.6 Cyclic weight relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.6.1 From a basis to relations . . . . . . . . . . . . . . . . . . . . . . . 97
3.6.2 Kontsevich graphs in f ⋆ g . . . . . . . . . . . . . . . . . . . . . . 98
3.6.3 Leibniz graphs in (f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) . . . . . . . . . . . . . . 100
3.6.4 Known weights satisfy the cyclic weight relations . . . . . . . . . 103
3.7 Kontsevich’s ⋆ product for affine Poisson structures . . . . . . . . . . . . 107
3.7.1 Affine Kontsevich star product mod ō(h̄6) . . . . . . . . . . . . . 107
3.7.2 Relations between the weights at h̄7 . . . . . . . . . . . . . . . . 108
3.7.3 From weight relations to master parameters . . . . . . . . . . . . 111
3.7.4 Substitute master parameters into ⋆aff mod ō(h̄7) and its associator 112
3.7.5 Restrict onto generic affine Poisson bivector on R2 . . . . . . . . 114
3.7.6 Restrict onto affine rescaled Nambu–Poisson bivector on R3 . . . 114
3.7.7 Direct calculation of master parameter values . . . . . . . . . . . 115
3.7.8 Certificate of associativity . . . . . . . . . . . . . . . . . . . . . . 116
3.7.9 Rationality of ⋆ 7aff mod ō(h̄ ) . . . . . . . . . . . . . . . . . . . . . 121
4 Implementation of the graph complex 125
4.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2 Graph operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.3 Full undirected graph complex . . . . . . . . . . . . . . . . . . . . . . . . 128
4.4 Graph bases: storing them in cache . . . . . . . . . . . . . . . . . . . . . 130
4.5 Undirected graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.6 Directed graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.7 Undirected graph operations . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 Examples of graph cocycles 137
5.1 The tetrahedron cocycle γ3 . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.2 The five-wheel cocycle γ5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3 The coboundary δ6 = d(β6) . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4 The heptagon-wheel cocycle γ7 . . . . . . . . . . . . . . . . . . . . . . . 144
5.5 The commutator [γ3, γ5] . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Graph complex action on Poisson structures in dimension two 151
6.1 Tetrahedral γ3 flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2 Five-wheel γ5 flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3 Graph coboundary δ6 = d(β6) and the Poisson-trivial flow . . . . . . . . 158
6.4 Heptagon-wheel γ7 flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
CONTENTS iii
6.5 Commutator [γ3, γ5] flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7 Graph complex action on rank two rescaled Nambu–Poisson structures 169
7.1 Tetrahedral flow on rescaled Nambu–Poisson structures on R3 . . . . . . 169
7.1.1 Superfunction algebra . . . . . . . . . . . . . . . . . . . . . . . . 170
7.1.2 Tetrahedral flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.1.3 The induced flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.1.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.1.5 Differential polynomial triviality . . . . . . . . . . . . . . . . . . 173
7.1.6 Total skew-symmetry of the trivializing vector field . . . . . . . . 175
7.2 Tetrahedral flow on Nambu–Poisson structures on R4 . . . . . . . . . . . 177
7.2.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.2.2 Differential polynomial (non)triviality . . . . . . . . . . . . . . . 180
7.3 Tetrahedral flow on rescaled Nambu–Poisson structures on R4 . . . . . . 181
7.3.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.3.2 Differential polynomial (non)triviality . . . . . . . . . . . . . . . 185
7.4 Five-wheel flow on rescaled Nambu–Poisson structures on R3 . . . . . . . 186
8 Graph complex action on R-matrix Poisson structures 191
8.1 The Lie algebra of 2× 2 matrices . . . . . . . . . . . . . . . . . . . . . . 193
8.2 The Lie algebra of 3× 3 matrices . . . . . . . . . . . . . . . . . . . . . . 196
8.3 The Lie algebra of traceless 2× 2 matrices . . . . . . . . . . . . . . . . . 199
8.4 The Lie algebra of traceless 3× 3 matrices . . . . . . . . . . . . . . . . . 200
9 Graph complex action on star products 203
9.1 Poisson-trivial deformations and gauge transformations . . . . . . . . . . 203
9.2 Poisson-trivial deformation and the gauge transform in terms of graphs . 207
9.3 How the tetrahedral flow deforms the star-product . . . . . . . . . . . . 208
List of references 211
II Research articles 215
10 On the Kontsevich ⋆-product associativity mechanism 217
11 The expansion ⋆ mod ō(h̄4) and computer-assisted proof schemes in the Kon-
tsevich deformation quantization 223
12 Formality morphism as the mechanism of ⋆-product associativity: how it works 303
13 The heptagon-wheel cocycle in the Kontsevich graph complex 321
14 Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph cal-
culus 345
15 The Kontsevich tetrahedral flow revisited 359
16 Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex389
17 The orientation morphism: from graph cocycles to deformations of Poisson
structures 401
iv CONTENTS
18 The Kontsevich graph orientation morphism revisited 415
19 Universal cocycles and the graph complex action on homogeneous Poisson brack-
ets by diffeomorphisms 435
20 The hidden symmetry of Kontsevich’s graph flows on the spaces of Nambu-
determinant Poisson brackets 445
Back matter 473
Curriculum Vitae 473
Acknowledgements 475
Zusammenfassung 477
Samenvatting 479
Summary for Laymen 481
Appendices 484
A Introduction to SageMath 485
B Kontsevich’s star product ⋆ mod ō(h̄6) 507
B.1 Kontsevich’s star product ⋆ mod ō(h̄4) . . . . . . . . . . . . . . . . . . . 507
B.2 Associativity of Kontsevich’s ⋆ mod ō(h̄6) . . . . . . . . . . . . . . . . . 511
C Kontsevich’s affine star product ⋆ mod ō(h̄7) 519
C.1 Original expansion ⋆ 7aff mod ō(h̄ ) . . . . . . . . . . . . . . . . . . . . . . 519
C.2 Reduced expansion ⋆red 7aff mod ō(h̄ ) . . . . . . . . . . . . . . . . . . . . . 539
C.3 Associativity of ⋆red 7aff mod ō(h̄ ) . . . . . . . . . . . . . . . . . . . . . . . 551
D Flows Qγ(P ) and factorizations [[P,Qγ(P )]] = ♢γ(P, [[P, P ]]) 565
D.1 The tetrahedral flow Qγ3(P ) . . . . . . . . . . . . . . . . . . . . . . . . . 565
D.2 The pentagon-wheel flow Qγ5(P ) . . . . . . . . . . . . . . . . . . . . . . 566
E Graph cocycles 569
E.1 The tetrahedron γ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
E.2 The pentagon-wheel cocycle γ5 . . . . . . . . . . . . . . . . . . . . . . . 569
E.3 The heptagon-wheel cocycle γ7 . . . . . . . . . . . . . . . . . . . . . . . 569
E.4 The commutator [γ3, γ5] ∈ ker d on 9 vertices and 16 edges . . . . . . . . 570
F Reference documentation for the gcaops software 573
Overview
The Formality theorem by Kontsevich introduced an extensive calculus of graphs into
both deformation quantization and universal deformations of Poisson structures. The
goal of this dissertation is to implement the Kontsevich graph calculus by algorithms
and software modules. By this, we make it possible for every scholar to do explicit
computations with Kontsevich’s star product, graph flows, and graph complex. The
software, together with the examples presented in this dissertation are available from1
https://github.com/rburing/gcaops
under the permissive MIT free software license. This new implementation allowed us
to illustrate the theory by examples, inspect and verify conjectures, reveal properties of
objects, and phrase new conjectures.
The domain of Kontsevich graph calculus not only offers theoretical concepts, but—by
its explicit nature—also suggests the design of algorithms and development of software.
Therefore, our present task is the translation of the theoretical framework into computa-
tional problems, examples, equations and solutions.
By using this software, we find the explicit expansion of the Kontsevich star prod-
uct up to order 4, we find explicit representatives of graph cohomology classes, and we
calculate the action of several graph cocycles on arbitrary 2D Poisson structures, as well
as on families of rescaled Nambu–Poisson structures in 3D and 4D, and on homogeneous
quadratic and cubic Poisson structures associated with R-matrices (those R-matrices are
associated with splittings of the Lie algebras gl2(R) and gl3(R)). Finally, we calculate
examples of the graph complex action on star products.
This dissertation is such that it can also be used in education. Indeed, the examples
and demos in this dissertation have already served as a basis for tutorials in an advanced
master’s course on deformation quantization and the graph complex (2020/21). The
dissertation lets students get acquainted with applications of the graph calculus in other
domains: how this theory is used in other parts of mathematics and physics. In this way,
through natural examples and illustrations, the dissertation opens the door to domains
such as abstract algebra, group theory, group actions, cohomology theory, deformation
theory, combinatorics, combinatorial topology, Poisson geometry, calculus of multivectors,
supergeometry, Lie groups and algebras, homotopy Lie algebras, jet bundles, geometry of
differential equations, and topological methods in physics, including the Feynman path
integral. We illustrate how these topics are united by the Kontsevich concept of graphs
in deformation quantization.
Actual background. Sophus Lie introduced the notion of a Lie algebra as a gener-
alization for linear Poisson brackets. Lie groups combine properties of manifolds and
abstract groups. They are integral objects for Lie algebras. The theory developed in
1The abbreviation gcaops stands for Graph Complex Action on Poisson Structures.
1
2 Overview
the papers by H. Poincaré, E. Cartan, H. Weyl, H.S.M. Coxeter, E. Dynkin, V.G. Kac,
and R. Moody has become fundamental in modern mathematics. Physical applications,
namely the theory of gauge fields as connections in principal fibre bundles, united ab-
stract ideas of mathematics with methods of theoretical physics. We now see that such
is the gauge model of electroweak interactions, and such is the hypothesis/description of
strong interactions. Together, they constitute the Standard Model. Independently, the
pseudogroup of diffeomorphisms of spacetime acts in the geometry of Einstein’s general
relativity, i.e. in a classical description of gravity. With the general theory of Lie groups
and algebras, we associate the problem of their representations in spaces of endomor-
phisms End( nk ) (using matrices) and realizations in the spaces X1( nk ) of vector fields,
i.e. derivations on the algebra of functions. The Cartan differential associates with a Lie
algebra a chain complex, so that there appear the corresponding Chevalley–Eilenberg
cohomology groups. Kontsevich in 1993–1994 discovered another natural class of Lie
algebras. Namely, he discovered the structure of differential graded Lie algebra (dgLa)
on the spaces of undirected graphs with external ordering of edges. Willwacher in 2010–
2015 related suitable cocycles from the Kontsevich unoriented graph complex to the Lie
algebra grt of the (infinitely generated) Grothendieck–Teichmüller group discovered by
Drinfeld in 1990. Obviously we need effective tools to operate with the calculus of graphs,
to manipulate graph expressions, to calculate cohomology, Betti numbers, and to do all
the natural operations (such as the calculation of the graph differential).
Independently from the abstract group theory (e.g. permutations) and Lie groups, the
idea of noncommutativity was a starting point for the emergence of new quantum me-
chanics by Dirac and Heisenberg, in contrast with the “old” quantum theory by Bohr and
Sommerfeld. In the course of quantization, classical dynamical variables lose permutabil-
ity. In that sense, the transition from Poisson algebra of dynamical functions to the
representations of coordinates and momenta in the spaces of endomorphisms (self-adjoint
Hermitian operators that act on typically infinite-dimensional spaces of wave functions for
quantum objects) is a leap. This leap amounts to a radical change of both the physical
sense of the geometry of the model, and of its mathematical description. Quite natu-
rally, there appears the problem of linking quantum and classical, i.e. of semiclassical
approximation, that would connect the classical picture and the quantum wave picture.
But the noncommutativity of quantum objects, by itself requiring that the objects be
ordered, is lost in the old classical purely commutative description. The theorem by H.
Groenewold (1946) and L. Van Hove (1951) establishes that a naive correspondence of
classical Poisson brackets and quantum commutators is not well-defined; counterexam-
ples are immediately produced. E. Wigner and H. Weyl posed a natural question. Can
we reinstate the noncommutativity already in the classical picture, so that we make all
the diagrams commutative? To be precise, so that we bring noncommutativity into the
classical picture in such a way that the (now, suddenly) deformed Poisson bracket is
mapped under the Wigner–Weyl transform to the commutator of Hermitian operators
on the spaces of quasiprobability distributions? Thus was created the theory of noncom-
mutative associative star products, bridging the classical Poisson geometry and quantum
mechanics by Heisenberg and Dirac. Fifty years passed, and a breakthrough result by
Kontsevich established that every finite-dimensional Poisson manifold can be deform-
quantized. To solve that problem, Kontsevich developed and applied the language of
directed graphs, by this interpreting the deformation quantization problem from analytic
into topological combinatorial. Again, by using the standard technique by Gerstenhaber–
Schlessinger–Stasheff, Kontsevich described the world of star products as a gauge theory
3
with cochain complexes. So there are equivalence classes of star products, and in turn
the products themselves depend on the Poisson classes of Poisson brackets in their own
Poisson cohomologies. Having related star products through the use of weighted graphs
to the Lobachevsky hyperbolic geometry, Kontsevich pointed out in 1999 the existence
of other classes of star products, with their weights not defined by the harmonic prop-
agators. These classes of star products were explored by Alekseev, Rossi, Torossian,
Willwacher in 2014. It is now expected that the coefficients of Kontsevich graphs in
the star product contain many hidden numbers, the properties of which constitute open
problems in number theory, e.g. ζ(3) as pointed out by Felder–Wilwacher in 2008. A
link of Kontsevich’s graph calculus with quantum field theory was explicated by Catta-
neo and Felder (1999), Kontsevich’s graphs in the star product expansion are Feynman
diagrams as those appear in the course of calculation of the correlation functions for
the Ikeda–Izawa Poisson sigma model (1993–1994) by using the path integral expansion.
The weights of Kontsevich graphs in the star product are calculated through the har-
monic propagators in the quantum field theory Poisson sigma model. Now, so much has
become understood about star products and about the relation of deformation quantiza-
tion to problems of number theory. There naturally appears the problem of calculating
Kontsevich’s star product expansion up to reasonably high order (h̄4, h̄5, h̄6, h̄7). In this
direction worked Penkava–Vanhaecke (∼ h̄3 in 1998), Buring–Kiselev (∼ h̄4 in 2017), and
Banks–Panzer–Pym (∼ h̄5, h̄6 in 2017, 2018) who based their work on that of Francis
Brown. Obviously, because there are thousands of graphs in the star product expansion,
we must have effective software to implement the directed graph calculus, in the world
of star products.
The deformed Poisson bracket becomes a derivation with respect to the new noncom-
mutative product. At the same time, it is interesting to construct symmetries of the old
space of Poisson brackets over the fixed nondeformed algebra of functions. This amounts
to the problem of constructing nontrivial second Poisson cohomology classes, such that
their construction would be universal to all Poisson manifolds of arbitrary topologies.
(If for a given Poisson manifold the second Lichnerowicz–Poisson cohomology were zero,
then if such a universal construction existed, then it would still work and produce trivial
classes.) Although a priori the formulation of this problem looks overly optimistic and
strange, Kontsevich nevertheless formulated in 1996 a solution to this problem. It again
appeals to the language of Kontsevich directed graphs, i.e. the same classes of graphs as
were used in the solution of the deformaton quantization problem. Willwacher in 2010
establishes the existence of a countable infinitely generated set of universal infinitesimal
deformations Ṗ = Q(P ) of Poisson structures, i.e. second Poisson cohomology classes.
They are differential polynomial in the coefficients of the given Poisson brackets. These
infinitesimal deformations are obtained by orienting suitable graph cocycles, namely the
wheel cocycles in the Kontsevich unoriented graph complex, and their iterated commuta-
tors. Kontsevich’s tetrahedral flow is the minimal nontrivial example. Until recently, any
study of Kontsevich’s universal flows on the spaces of Poisson structures was hopeless.
The programming of simple fast accessible computational tools would allow verification
and illustration of all previously accumulated theoretical findings, as well as further new
experiments, and prediction of new properties. A theoretical aspect of the study of such
deformations is in that, as Kontsevich in 2019 (private communication) points out, uni-
versal deformations of Poisson brackets by using the language of graphs, both in the
context of flows and noncommutative brackets, rediscover the quantization of the phase
volume in the “old” Bohr–Sommerfeld quantization.
4 Overview
The software by Dror Bar-Natan (2000) and other authors (Willwacher–Živković
2014) was designed for the Kontsevich unoriented graph complex, to serve the count
of graphs with up to 24 vertices and 36 edges, and the count of dimensions of graph
homology groups e.g. in vertex-edge bi-grading (n, 2n − 2) up to n = 8. In the context
of Kontsevich’s star products with harmonic propagators, the software to calculate the
weights of directed graphs has been applied systematically up to h̄5, h̄6 (Banks–Panzer–
Pym). Jets by Michal Marvan serves a completely different purpose of geometric analysis
of differential equations, and in that respect it is suitable for calculations in Poisson co-
homology. Nevertheless, software packages which would have been capable to correlate
the dgLa of unoriented graphs with the Lichnerowicz–Poisson complex and the calculus
of directed graphs in the theory of star products did not exist. The ad hoc use of general
purpose mathematical software was possible in principle, but the translation and transfer
of data structures or the glueing together of scripts would be laborious and slow (as expe-
rienced firsthand by the author). Both the verification of previously existing material and
further advancement required a new multi-functional package for calculation, computa-
tion, verification, and experiment. The present dissertation is devoted to the development
and use of precisely such a package. We present the first general implementation of the
action of Kontsevich’s graph complex on Poisson structures and star products.
Local embedding. The Institute of Mathematics at Johannes Gutenberg-University of
Mainz, as well as the faculty of Physics, Mathematics, and Computer Science at large,
has a strong tradition in areas of mathematics overarching the topic of this dissertation.
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The Insitute of Mathematics has been and is being part of large projects in fundamen-
tals of science: SFB 45 “Periods, Moduli Spaces, and Arithmetic of Algebraic Varieties”
consortium between Mainz, Bonn, and Essen (2007–2019) and CRC 326 GAUS “Geom-
etry and Arithmetic of Uniformized Structures”, involving Frankfurt, TU Darmstadt,
Heidelberg, Mainz, and Munich, now ongoing. Both of the grant projects are naturally
related to the subject of this dissertation.
Research problem. Problems about ⋆-products. The Kontsevich star-product is a
formal power series in h̄ defined through a weighted sum of graphs. While the weights
are perfectly well defined numbers given by integrals, their explicit closed-form values are
not easy to obtain. For any use of the formula in physics, an explicit expansion up to some
order k would be essential. The main problem is therefore to determine the (harmonic)
weights of graphs that appear in the Kontsevich star-product up to order k. An auxiliary
problem is to explicitly verify the associativity of the ⋆-product expansion up to order k.
This would reinforce the correctness of the found weights, and would also let us inspect
the extensibility of the formula e.g. to the variational setting. Here a task is to re-examine
the mechanism of associativity and the appearance of the Jacobi identity in the language
of graphs. Since the number of graphs in the Kontsevich star-product already counts in
5
the hundreds at low orders, we set ourselves the main task of implementing the Kontsevich
star product (and related objects) in computer software. As a byproduct, this will allow
us to undertake the subtask of illustrating this theory where examples are sparse. We
choose an indirect approach to the problem of determining the weights. Rather than
calculating integrals directly, we consider the weights as unknowns, and our task is to
obtain as many relations between them as we can. A subtask is how to get the most
out of the associativity equation for the star-product. In this context we ask to what
extent the rescaled Nambu–Poisson structures can be used to find relations between
weights. A completely different method is the use of the Shoikhet–Willwacher–Felder
cyclic weight relations. To what extent do they constrain the weights? Independently,
a task is to verify the output of the software by Banks–Panzer–Pym, by substituting
their weights into the relations we find. Another orthogonal task is to implement gauge
transformations of star products, and to inspect how many graphs can be gauged out
from the star-products. With a view toward the Formality morphism as the mechanism
for star-product assocativity, we also ask for a set of conventions such that left-hand
sides are equal to right-hand sides in the Formality identities, not only morally but also
arithmetically. Lastly we consider the non-uniqueness of the Formality morphism and
hence the star-product; in particular we ask how the action of the graph complex on star
products looks explicitly.
Problems about graph flows. A guiding question is the following open problem: Does the
graph complex act nontrivially on Poisson structures? Within the scientific community
(e.g. Kontsevich, Willwacher, Dolgushev et al.) it is conjectured that it does. The pos-
sibility of an abstract proof of the existence of a graph cocycle that acts nontrivially on
a Poisson structure is a priori not excluded, but it would leave the author unsatisfied.
Therefore we set ourselves the following optimistic task: Find a graph cocycle that acts
nontrivially on a Poisson structure. Our chosen method is that of experimental math-
ematics: implement everything that is required, and try examples systematically. The
calculation of the action requires explicit formulas for Poisson brackets and graph co-
cycles. While many Poisson brackets can be found in the literature, this is not true of
explicit formulas for graph cocycles. A subtask is therefore to find explicit representatives
of graph cohomology classes. For optimization purposes, it is moreover desirable to find
a representative with as few terms as possible. Next we need software to orient the graph
cocycles, and restrict to graphs with out-degree at most two. To realize the insertion of
multi-vectors into graphs, an implementation of Z2-graded math is needed.
Regarding particular classes of Poisson structures we ask: Does the graph complex
action preserve the class of (rank two) rescaled Nambu–Poisson structures? The task is
to obtain the explicit formulas for infinitesimal deformations ρ̇ and ȧ of the functional
parameters, and to verify Kontsevich’s conjecture about shape of the evolution ȧ. It
is then an exercise to express the large differential polynomials ρ̇ and ȧ as total skew-
symmetrizations of much smaller differential polynomials. Again we ask whether the
graph complex action is trivial or not. In this case we can ask more specifically if there
exists a trivializing vector field with differential polynomial coefficients. Another class of
Poisson structures to consider is that of the homogeneous quadratic and cubic Poisson
structures associated with R-matrices. In this case the homogeneity of the Poisson struc-
ture and hence of the Poisson differential reduces the problem of finding a trivializing
vector field to a problem of finite-dimensional (but often high-dimensional) linear algebra.
On top of that, using the same graph language, we ask if it is possible to produce
universal 1-cocycles (not 2-cocycles) for homogeneous Poisson structures. In this setting
6 Overview
there is also the task of illustrating this concept by (non)trivial examples.
Our final large-scale problem is: How far can the use of (known properties of) special
(classes of) Poisson structures further our quest for the (non)trivial action?
Research subject. This research is interdisciplinary within mathematics and computer
science. It refers to abstract concepts such as deformation quantization, Lie algebras, dif-
ferential graded Lie algebras (dgLas), homotopy Lie algebras and homotopy morphisms
(L∞), Gerstenhaber’s deformation theory, supergeometry and supermanifolds, graded
calculus of multivectors, Poisson geometry, Lichnerowicz–Poisson cohomology, nonlinear
partial differential equations, jet spaces, dynamical systems, combinatorics, combinatorial
topology, graph theory, but also algorithms and data structures (representing combinato-
rial data), high performance computational methods (fast computational linear algebra,
count of dimension and rank), scientific programming and computation, symbolic com-
putation, implementation of algebra of graphs and multivectors (dgLa), software design
and engineering. We keep in mind that this research is carried out in the context of
topological methods in physics.
Research object. Graphs, vector spaces spanned by graphs, algebraic structures on the
spaces of graphs (commutator, differential), multivectors (with differential polynomial co-
efficients or symbolic coefficients), algebraic—specifically dgLa—structures on the space
of multivectors (Schouten bracket [[·, ·]]), differential polynomials, equations and identities
for graphs, equations and identities involving multivectors, (ir)rational numbers as the
weights of graphs (partly known and partly unknown: undetermined variables, expressed
via multiple zeta values), multi-differential operators, graded objects (e.g. edges in graphs
in the Kontsevich graph complex), Poisson brackets (examples and classes of), nonlinear
partial differential equations and dynamical systems (in the context of the Poisson coho-
mology triviality condition for the evolution of the Nambu–Poisson brackets in 3D, 4D),
Lie algebras, R-matrices. On the computer science side of this project, there are algo-
rithms, computer programs, software libraries (linear algebra, nauty). We operate with
theoretical concepts, ideas and conjectures (primarily, by Kontsevich); the new software
helps us to convert conjectures into true statements.
Research strategy. We use a wide spectrum of analytic and computational methods
to study examples case by case (so we do experimental mathematics, on the basis of
fundamental mathematics by Kontsevich). This work combines logical reasoning (proof
of theorems), design of experiments, and running extensive computations.
Proof of statements. Within deformation theory, we first study infinitesimal deforma-
tions, and only then we consider possible obstructions to the integrability of the infinites-
imal deformations. (Instead of groups as integral objects, we study algebras, and apply
the deformation cohomology theory.) We describe cohomology classes by producing their
representatives. Overall, we operate modulo trivial objects (e.g. modulo zero graphs,
modulo graph coboundaries, modulo gauge transformations). We regularly make use of
gradings and homogeneity with respect to those gradings (e.g. numbers of vertices and
edges in graphs, degrees of polynomial coefficients of multivector fields). We deploy meth-
ods of differential geometry and supergeometry (e.g. in the calculus of multivectors), and
in the description of Formality morphism we refer to methods of homotopy Lie theory.
Experimental verification. We aim at proof or disproof of theoretical claims and conjec-
tures. We calculate the weights of Kontsevich graphs and Leibniz graphs using external
software (available from Banks–Panzer–Pym; the weights are expressed in terms of the
multiple zeta values), and verify that they satisfy relations produced by our software.
7
In the case study of examples, we borrow Poisson structures from the literature and use
them for a completely different purpose, in a new context (star products and graph flows).
By these examples, we verify and illustrate the associativity of the star product via the
Formality morphism, and similarly the Poisson cocycle condition for the Poisson graph
flows.
Digital processing of data structures. Our guiding principle is representation of large ob-
jects by small simple markers. Our programs are optimized for both size and speed,
by using the normal forms of graphs, and sparse vectors. We aim to express the graph
cocycles by using a minimal number of terms, i.e. we work with the smallest possible rep-
resentatives of cohomology classes. The graph cocycles are stored as sparse vectors with
respect to a basis consisting of graphs. Finding basis of graph cohomology is then done
using fast linear algebra methods, such as Markowitz pivoting. Let us emphasize that
large linear systems are solved using rationals, not invoking finite precision floating point
operations. When found, rational solutions are verified to be exact by direct substitution.
The software is designed in such a way that transfer of data structures between tasks
is easy. For example, the object O⃗r(γ5)—defined in terms of a graph cocycle γ5—is an
endomorphism on the space of multivectors; the software is able to evaluate it at given
Poisson structure. Since some problems are big, it is convenient to store intermediate
results; this is achieved using the pickling in the Python language, as well as encoding in
plain text files.
Identities which hold by force of the Jacobi identity can be processed in many ways.
In particular, the software is able to restrict an identity onto the 3D rescaled Nambu–
Poisson structures. Becoming a differential polynomial in the components ρ, a of the
Nambu–Poisson bracket, the identity now splits into many homogeneous components.
This idea of homogeneity and splitting is used extensively in all our software.
Within the geometry of differential equations and group theoretic methods for PDEs
(by Sophus Lie), we do symbolic calculations in differential calculus on jet bundles. For
example, we take the restriction of Kontsevich’s tetrahedral flow on the class of 3D
Nambu–Poisson brackets and by using the gcaops software, we establish the existence of
a trivializing vector field with differential polynomial coefficients.
The output of the software can be used independently of the software. We provide
both the graph encodings and the analytic formulas encoded by these graphs. Whoever
wants, can use either result.
Scientific novelty of results. The main results which reflect the scientific novelty and
which are presented for defense are these:
1. The action of Kontsevich’s graph complex on Poisson structures and star products
is implemented in the free software package https://github.com/rburing/gcaops
written in free, open source software Python and SageMath.
2. Taking into account all conventions in multivector calculus and graph calculus,
the gcaops software confirms the exact equality of left-hand sides and right-hand
sides in identities such as the Leibniz-graph factorization of associativity for star
products and the Poisson cocycle condition for graph flows (see Proposition 13
on p. 19 and Proposition 26 on p. 26, respectively). The balance of associativity
requires recalculation constants; their values cn = n/6, conjectured in Chapter 12
of Part II, are verified by computations in Chapter 3 of Part I, whereas the latter
equality for graph flows is exact and absolute (see Chapter 5).
8 Overview
3. The gcaops software is able to find explicit representatives of the non-trivial graph
cocycles γ3, γ5, γ7, and calculate commutators of graphs, in particular [γ3, γ5], in
the Kontsevich graph complex with the vertex-expanding differential (see Proposi-
tions 42 and 43 on p. 32).
4. The canonical factorization (à la Kontsevich) of the Poisson cocycle condition for
Qγ3(P ) and Qγ5(P ) via differential consequences of the Jacobi identity for (generic)
Poisson structures, using Leibniz graphs, is given in Example 27 on p. 27 and
in Table 0.4 on p. 28. The solution of this factorization problem is not unique
in either case; there exist sums of Leibniz graphs such that their expansion into
sums of Kontsevich graphs equals zero identically. The nullity of the Leibniz graph
expansion map, restricted to the bi-gradings (3, n− 1) for n = 2, . . . , 5 is reported
in Table 0.8 on p. 34.
5. The graph cocycles γ3, γ5 act on rescaled Nambu–Poisson structures. Computer
experiment shows (in 3D and 4D for γ3, and in 3D for γ5) that they preserve this
class of Poisson structures (Qγ(P [ρ, a]) = P [ρ̇, a] + P [ρ, ȧ] in 3D, and similarly
in 4D). The following conjecture by M. Kontsevich is true: the evolution of the
functional parameters has the shape reported in Propositions 32, 33, 34 on pp. 29–
29. (The huge formulas can be realized as the total skew-symmetrization of tiny
differential polynomial expressions.2) Moreover, there exists a vector field Xγ3 with
differential polynomial coefficients that trivializes the tetrahedral γ3 flow over R3,
i.e. the equation Qγ3(P ) = [[P,Xγ3 ]] has a solution Xγ3 with differential polynomial
coefficients, again realized in Proposition 32 as the total skew-symmetrization of
tiny differential polynomial expressions.
6. The graph cocycle γ3 acts on homogeneous quadratic and cubic Poisson structures
associated with R-matrices. Computer experiment establishes that this action is
Poisson-trivial in the cases of those R-matrices associated with splittings of Lie
algebras gl2(R) and gl3(R), see Proposition 36 on p. 30.
7. In 2-dimensional Poisson geometries, the Poisson cocycles Qγ(P ) defined by graph
cocycles γ ∈ {γ3, γ5, δ6, γ7} are Poisson-trivial, see Proposition 28 on p. 28. Namely,
there exist vector fields Xγ(P ), differential polynomial in P , that trivialize the flows
Qγ(P ) = [[P,Xγ(P )]]. Moreover, with respect to the standard symplectic structure
on R2, every such vector field Xγ(P ) is the Hamiltonian vector field of a Hamiltonian
Hγ(P ), again differential polynomial in P and expressed using Kontsevich graphs
(see Proposition 31 on p. 28). The case of γ3 was known to Kontsevich (1996), and
the respective Hamiltonian was found by Bouisaghouane (2016/17). The remaining
cases are established by the new software.
8. In the Kontsevich star product at order 4, at most 256 digraph isomorphism classes
of Kontsevich graphs can in principle appear. The weights of all these graphs
are completely determined by the weights of 149 basic graphs (those with positive
differential order which are nonzero and prime) and the known weights of graphs
at lower orders. By using the associativity, the cyclic weight relations, and the
known vanishing of (integrands of) some weights, the basic graphs weights are
2This was shown in collaboration with D. Lipper (2020).
9
expressed in terms of only 10 master parameters. The software by Banks–Panzer–
Pym calculates the exact (rational) values of these master parameters, giving us
the entire star product mod ō(h̄4) in Equation (13) in Chapter 11.
9. The Leibniz graphs are building blocks in the combinatorial mechanism for validity
of identities which hold by force of the Jacobi identity and its differential conse-
quences. The Leibniz graph realizations of the right-hand sides, when expanded
to sums of Kontsevich graphs with a copy of the Poisson bracket in every internal
vertex, ensure the vanishing of left-hand sides of identities, see Proposition 13 on
p. 19 (with Table 0.2 on p. 20) and Proposition 26 on p. 26 (with Table 0.4 on
p. 28). Examples of the work of this mechanism are in Example 14 on p. 20 and
Example 27 on p. 27 (see also Proposition 46 on p. 34).
10. The new software is able to ouput the Shoikhet–Felder–Willwacher (2008) cyclic
weight relations
• between Kontsevich graphs that appear in the star product at orders 3, 4, 5,
and
• between Leibniz graphs that appear in the associativity equation for the Kon-
tsevich star-product at orders 3, 4, 5 in h̄.
We used the software by Banks–Panzer–Pym to calculate the weights of Leibniz
graphs; we establish that all these values do satisfy the cyclic weight relations (see
Proposition 15 on p. 20). We discover also that the cyclic weight relations (as well
as the associativity itself, the known vanishing of some weights, and many other
relevant properties) constrain the weights of Leibniz graphs in the associator and of
Kontsevich graphs in the star product but do not completely determine them (see
Table 0.2 on p. 20 and Table 0.3 on p. 21, and Proposition 15 and 16 respectively).
Therefore, direct calculation of weights is inevitable.
11. We verify the associativity up to ō(h̄6) for the full Kontsevich star-product, known
modulo ō(h̄6) from Banks–Panzer–Pym [1] for arbitrary (non)linear Poisson brack-
ets and arbitrary arguments, by realizing (every homogeneous tri-differential com-
ponent of) the associator as a sum of Leibniz graphs from the 0th layer, that is
the Leibniz graphs produced at once by contracting edges in the Kontsevich graphs
from the associator (see Proposition 17 on p. 22).
12. The Kontsevich ⋆-product admits a restriction to the class of Poisson brackets with
affine coefficients on finite-dimensional affine manifolds (e.g., such are the Kirillov–
Kostant Poisson brackets on the duals of Lie algebras: their coefficients are strictly
linear without constant terms). In Section 3.7, see Proposition 18 on p. 23 below,
we obtain the expansion ⋆ mod ō(h̄7aff ) of affine Kontsevich’s star-product: in all
the Kontsevich graphs in it the in-degrees of aerial vertices are bounded by ⩽ 1.
The graph expansion ⋆aff mod ō(h̄7) is contained in Appendix C.1; at h̄6 and h̄7,
the coefficients of many Kontsevich graphs in ⋆aff mod ō(h̄7) contain ζ(3)2/π6 times
a nonzero rational factor, plus a rational part.
13. We discover that in both the orders h̄6 and h̄7 in ⋆aff with the harmonic graph
weights, the entire coefficient of ζ(3)2/π6, itself a Q-linear combination of Kontse-
vich graphs, assimilates into a linear combination of Leibniz graphs (doing so at
10 Overview
once, without need of the 1st layer of Leibniz graphs). In consequence, whenever
the affine star product ⋆aff mod ō(h̄7) is restricted to an arbitrary affine Poisson
structure, the constant ζ(3)2/π6 does not appear at all in the resulting analytic
expression. In fact, more terms in ⋆aff mod ō(h̄7) can be absorbed into Leibniz
graphs. Namely, in §3.7.9 we obtain and in Appendix C.2 we list the affine Kon-
tsevich graph expansion ⋆red mod ō(h̄7aff ), at once excluding the part which is now
known to vanish identically: there remain only 326 nonzero rational coefficients
of Kontsevich graphs (at all orders, up to h̄7), in contrast with the original graph
expansion of the Kontsevich star product ⋆aff mod ō(h̄7) in which the Kontsevich
integral formula yields 1423 nonzero Q-linear combinations of 1 and ζ(3)2/π6 for
the Kontsevich graph coefficients.3
14. We verify the associativity up to ō(h̄7) of the affine Kontsevich star product,
known modulo ō(h̄7) for arbitrary affine Poisson brackets and arbitrary argu-
ments, by realizing (every homogeneous tri-differential component of) the asso-
ciator as a sum of Leibniz graphs. We establish that, for the tri-differential orders
{(3, 3, 2), (2, 3, 3), (3, 2, 3), (2, 4, 2)} the 0th layer of Leibniz graphs is not enough for
any such factorization to exist, yet solutions appear in presence of the 1st layer of
Leibniz graphs in each of the four exceptional cases; see Proposition 19 on p. 23
below, and see the Proof scheme (for the reduced affine star product ⋆redaff mod ō(h̄7))
on p. 24 specifically about the properties of the associator for the reduced affine
star product ⋆redaff mod ō(h̄
7) with only rational coefficients. We finally deduce that
the indispensability of the first layer of Leibniz graphs is such that it carries on to
any factorization of the associator at h̄7 for the full star product modulo ō(h̄7); see
Example 20 on p. 24.
Practical significance of results. The PhD dissertation has both theoretical and
practical character. The software is designed for verfication of theoretical predictions,
and also for verification of other software (computational results). Independently, it can
be used to verify, extend and re-check results of computations in other works. Every
scholar can use the free open source software to solve relevant problems in this domain
of science by themselves. The implementations of multivector calculus, polydifferential
operators, and differential polynomials can be used independently from the graph complex
modules.
Both the text of the dissertation and the software implementation (available online)
can be used in education. Examples and demo sessions based on this research have
already been approbated in the advanced master’s course on deformation quantization
and the graph complex (2020/21).
Personal contribution. All results which are set for defense, the dissertant accom-
plished in person. In the works which are published jointly with other authors, the
contribution of the dissertant is as follows.
In [1] entitled On the Kontsevich star-product associativity mechanism, I did the
calculation of the star product expansion up to order 3, and the graphical calculation of
the associator up to the same order. I provided the first explicit example illustrating the
3In the above reduction of ⋆aff mod ō(h̄7) none of the Kontsevich graph weights was altered or
were redefined; the reduction of the number of terms which effectively contribute to the star-product of
functions and to its associator is due to a revealed property of those graphs and their Kontsevich weights.
11
work of Kontsevich’s Formality morphism, by expressing the ⋆-product associativity up
to order 3 as a differential consequence of the Jacobi identity.
In [2] entitled The expansion ⋆ mod ō(h̄4) and computer-assisted proof schemes in
the Kontsevich deformation quantization, I wrote all the computer implementations in
kontsevich_graph_series-cpp. Moreover I created all the examples, based on discussions
with A.V. Kiselev. The text was written in collaboration with A.V. Kiselev. Reducing
the number of parameters by using a gauge transformation was suggested by A.V. Kiselev
and performed by me. I did all the comparison with the literature in the Discussion. The
(rational) values of the 10 master parameters were received from Banks–Panzer–Pym
(2017).
In [3] entitled Formality morphism as the mechanism of ⋆-product associativity: how it
works, I calculated all of the examples in Section 4 and 5. The text was written together
with A.V. Kiselev.
In [4] entitled The heptagon-wheel cocycle in the Kontsevich graph complex, I found a
representative of the heptagon wheel cocycle, using my own software, and I counted the
dimensions of graph cohomology groups (two variants: valency-dependent). I made minor
contributions to the text, which (with extra examples and illustrations) is mainly due to
A.V. Kiselev and N. Rutten. I wrote ad hoc SageMath code for the graph differential in
Appendix B. (Similar code is now used in gcaops.) I assisted N. Rutten in generating
LATEX pictures of graphs.
In [5] entitled Infinitesimal deformations of Poisson bi-vectors using the Kontsevich
graph calculus, I wrote the code with some help from N. Rutten. Moreover I contributed
to the text describing the algorithms.
In [6] entitled The Kontsevich tetrahedral flow revisited, I did the graphical calculation
of the Poisson differential [[P,Qtetra]], including the skew-symmetrization. I assisted A.
Bouisaghouane in finding an example where a : b = 1 : 6 is necessary. I wrote the
2
computer program to find a Leibniz graph factorization of [[P,Qtetra]]. The software is
discussed further in [2].
In [7] entitled Poisson brackets symmetry from the pentagon-wheel cocycle in the graph
complex, I calculated the orientation of the five-wheel cocycle jointly with N. Rutten. My
software produced the analytic formula in Appendix A.
In [8] entitled The orientation morphism: from graph cocycles to deformations of
Poisson structures, I wrote the software graph_complex-cpp for the graph differential and
orientation of graph cocycles. The example of a coboundary δ6 = d(β6) was calculated
using that software. The text was written together with A.V. Kiselev. I calculated the
numbers of graphs in Table 2 that shows the size of the problem.
In [9] entitled The Kontsevich graph orientation morphism revisited, I provided the
encodings of graphs used in all examples; the text is due to A.V. Kiselev.
In [10] entitled Universal cocycles and the graph complex action on homogeneous
Poisson brackets by diffeomorphisms, I proposed the idea of evaluating O⃗r(γ) at tuples
other than P⊗n, and initiated the use of homogeneous P in this context. I evaluated
the flow Qγ3 on Poisson structures associated with two R-matrices for gl2(R), and found
trivializing vector fields for those flows. I calculated the action of the tetrahedral flow on
the rescaled Nambu–Poisson brackets depending on functional parameters (a, ρ), inducing
the flow (ȧ, ρ̇). The graph realization of ȧ was pointed out by M. Kontsevich immediately,
during discussion with A.V. Kiselev and myself at the IHÉS (December 2019). I verified
the claim by Kontsevich about the shape of ȧ, and also found ρ̇.
In [11] entitled The hidden symmetry of Kontsevich’s graph flows on the spaces of
12 Overview
Nambu-determinant Poisson brackets I calculated the induced evolution of the functional
parameters for the tetrahedral γ3 flow over R3 and R4 and for the pentagon-wheel γ5
flow over R3. The realization of those formulas by using Civita symbols is joint work
with D. Lipper. I established the Poisson-triviality of the γ3 flow over R3 in the Poisson
cohomology of the respective Nambu–Poisson structure: I represented the trivializing
vector field by using the Kontsevich graphs in which the vertices with ρ and Civita
symbols are resolved against the vertices with Casimirs.
Approbation of results. Prior to the start (April 2017) and in the middle (December
2019) of the project, the dissertant visited the IHÉS for one week (with A.V. Kiselev),
where the content and progress was discussed with and presented to M. Kontsevich. The
conjectures and approaches suggested there have been pursued in this work, and the
comments and feedback given there has been incorporated into the dissertation.
The work in this dissertation was presented at the following events:
International conferences. The 34th International Colloquium on Group Theoretical
Methods in Physics (Strasbourg, France, July 18–22, 2022); Poisson Geometry, Lie
Groupoids and Differentiable Stacks (Banff, Canada, June 5–10, 2022); Symmetry and In-
tegrability of Equations of Mathematical Physics (Kyiv, Ukraine, December 22–23, 2018);
Homotopy algebras, deformation theory and quantization (Będlewo, Poland, Septem-
ber 16–22, 2018); The 32nd International Colloquium on Group Theoretical Methods in
Physics (Prague, Czech Republic, July 9–13, 2018); Symposium on Advances in Semi-
Classical Methods in Mathematics and Physics (Groningen, The Netherlands, October
19–21, 2016); Group Analysis of Differential Equations and Integrable Systems (Larnaca,
Cyprus, June 12–17, 2016); Symmetries of Discrete Systems and Processes III (Děčín,
Czech Republic, August 3–7, 2015).
Colloquia. Two GQT Schools and Colloquia (Den Dolder, The Netherlands, July 3–7,
2017 and July 1–3, 2019); two Ph.D. meetings of the Sonderforschungsbereich/Transregio
45 (Physikzentrum Bad Honnef, January 26–29, 2018, and Universität Duisburg–Essen,
Febuary 1–3, 2019); Spring school Enumerative Invariants from Differential Graded Lie
Algebras and Categories (Montegufoni, Italy, March 25–31, 2018).
Seminars. Two talks at the Informal Seminar on Mathematical Aspects of Scattering
Amplitudes (JGU Mainz, Germany, January 9 and May 15, 2019); Working group on
Grothendieck-Teichmüller groups (MPIM Bonn, Germany, December 12, 2018); Floris
Takens Dynamical Systems Seminar (DSGMP, Bernoulli Institute, University of Gronin-
gen, The Netherlands, September 11, 2018); Junior Geometry and Topology seminar
(University of Oxford, United Kingdom, May 3, 2017); DIAMANT Intercity Number
Theory Seminar (University of Groningen, The Netherlands, April 7, 2017).
The feedback from seminars is incorporated into publications on which the dissertation
is based. Reciprocally, much of the work of the dissertant has served as the basis of
talks by other coauthors. Lastly, demos and examples from the dissertation have been
approbated in a master’s course on deformation quantization and the graph complex
(Autumn semester 2020/21), contributing to 15 tutorials and two PC demo sessions.
Publications and citation analysis. The dissertation is based on ten journal publica-
tions and one preprint, as well as twenty externally stored Jupyter notebooks with com-
puter demonstrations, based on free open source software packages. The main results are
contained in [2, 6, 3, 8]. All ten journal publications underwent anonymous peer review
by referees. Four papers are published in journals which are indexed by Mathematical
13
Reviews (MR), independently five works are indexed in zbMATH (formerly Zentralblatt
MATH), and three papers are indexed by the IAEA.
Citations. The paper [6] has been immediately cited by Kontsevich4 in his Séminaire
Bourbaki talk (January 2017). The concept and result of our paper [2] was used by the
Oxford group of Banks–Panzer–Pym5 to check their result. New explicit examples of
graph cocycles and Poisson bracket flows from [6, 8, 4, 5, 7, 10] are recognized in the
topical review by Morand.6
Structure of the dissertation. The dissertation consists of two major parts, as well as
this overview and appendices. Part I combines an introduction to theory, introduction to
software (computer demonstrations), and main examples which motivate further study
in Part II. By looking at Part I, the reader gets an impression of the technology of doing
this mathematics.
In Chapter 0 which is the Introduction, we inform the reader how to obtain and
install the gcaops software; we show the basics how functions (such as polynomials and
differential polynomials) are manipulated.
In Chapter 1 entitled Implementation of star products, we begin by recalling the notion
of star product; how we truncate it and the associativity equation. We recall the idea of
gauge transformation, and the notions of Gerstenhaber bracket, Hochschild differential
and Maurer–Cartan equation.
In Chapter 2 entitled Implementation of Poisson structures, we first explain the super-
calculus on the space of multivectors, endowed with the Schouten bracket. The Schouten
bracket provides the Jacobi identity for Poisson bi-vectors. Likewise, the Schouten
bracket gives us the Poisson differential and Lichnerowicz–Poisson cohomology. We illus-
trate the concept by giving examples of Poisson brackets (in particular, with homogeneous
polynomial coefficients) and Poisson cocycles. In particular Kontsevich’s tetrahedral flow
is a 2-cocycle, and when restricted to a specific Poisson bracket it is a coboundary.
In Chapter 3 entitled Implementation of Formality we first recall the construction and
properties of the graphs in Kontsevich’s Formality morphism; then we build Kontsevich’s
⋆-product modulo ō(h̄4) using Kontsevich’s graphs and we verify the associativity of ⋆ mod
ō(h̄4) by using Leibniz graphs. Moreover we investigate the (co)ranks of linear algebraic
systems of the Shoikhet–Felder–Willwacher cyclic weight relations that constrain the
graph weights at a given order h̄k (here, k ⩽ 5 for the Kontsevich graphs in the star-
product and hence n ⩽ 4 for the Leibniz graphs in the associator). In §3.5 we discuss
Leibniz graphs in detail, and we factor (every homogeneous tri-differential component
of) the associator modulo ō(h̄6) by using the 0th layer of Leibniz graphs, that is those
Leibniz produced at once from the Kontsevich graphs in the associator itself. In §3.7 we
obtain the affine star product ⋆aff mod ō(h̄7), we contrast its associativity mechanism with
the previously known mechanism that worked at orders ⩽ 6, and we reduce the affine
star product at orders h̄6, h̄7 by absorbing terms such as the Q-linear combinations of
Kontsevich graphs near ζ(3)2/π6 into linear combinations of Leibniz graphs. The graph
encoding of the reduced affine star product ⋆redaff mod ō(h̄
7) is given in Appendix C.2.
4Maxim Kontsevich. Derived Grothendieck–Teichmüller group and graph complexes [after T. Will-
wacher]. Séminaire Bourbaki. Vol 2016/2017. Exposés 1120–1135. Astérisque No. 407 (2019), Exp.
No. 1126, 183–211. ISBN: 978-2-85629-897-8.
5Peter Banks, Erik Panzer, Brent Pym. Multiple zeta values in deformation quantization. Invent.
Math. 222 (2020), no. 1, 79–159.
6Morand, Kevin. M. Kontsevich’s graph complexes and universal structures on graded symplectic
manifolds I. arXiv:1908.08253 [math.QA] – 42 pages.
14 Overview
In Chapter 4 entitled Implementation of the graph complex we demonstrate how the
definition of the graph complex is implemented in software: how graphs are encoded,
how their brackets are calculated, and how the differential acts. We give basic exam-
ples: the stick, Kontsevich’s tetrahedron, and the pentagon wheel cocycle by Kontsevich–
Willwacher; we show that they all are cocycles. After the undirected graph complex, we
study directed graphs and we introduce the directed graph complex. We give an example
of a directed graph cocycle, by illustrating how to convert from the undirected to the
directed graph complex. We recall and illustrate how the Schouten bracket comes from
the stick graph, and how the tetrahedral flow originates from the tetrahedron γ3.
Chapter 5 is entitled Examples of graph cocycles; now we start a systematic search for
new graph cocycles. We deploy methods of linear algebra and make the search automatic.
In this way, we find a coboundary δ6, and nontrivial graph cocycles γ7, and [γ3, γ5].
Further, we prove the factorization of the Poisson cocycle condition via the Jacobi identity
in each case, by providing the necessary Leibniz graphs.
In Chapter 6 entitled Graph complex action on Poisson structures in dimension two
we evaluate several graph flows at a generic Poisson structure on R2. In each case, we
obtain the formula of the flow, as well as the differential polynomial expression for the
coefficients of the trivializing vector field. Moreover, we discover its Hamiltonian with
respect to the standard symplectic structure and a graph realization of the Hamiltonian.
This is done for the tetrahedron γ3, the pentagon-wheel cocycle γ5, the coboundary δ6,
and for the heptagon wheel cocycle γ7.
Chapter 7 entitled Graph complex action on rank two rescaled Nambu–Poisson struc-
tures is about Poisson bi-vectors of the form P = ρ da/(dx∧dy∧dz) = [[ρ ∂x∧∂y ∧∂z, a]],
with, obviously, coefficients which are differential polynomial in ρ and a. We establish
that this class of Poisson brackets is preserved by the tetrahedral γ3 flow and by the
pentagon-wheel γ5 flow (as are similar brackets in dimension 4). In every case we express
the evolution ρ̇, ȧ as differential polynomials in ρ and a. Because these expressions are
highly symmetric, we collapse them by using the markers of minimal size and Civita sym-
bols. Independently, for the Kontsevich tetrahedral flow γ3 we establish the existence of
a trivializing vector field X[ρ, a] with differential polynomial coefficients, and again we
collapse it by using Civita symbols and marker monomials, which are realized by using
graphs.
Chapter 8 entitled Graph complex action on R-matrix Poisson structures is about
homogeneous quadratic and cubic Poisson brackets associated with R-matrices (they are
borrowed from Li–Parmentier); in turn those R-matrices are constructed for Lie algebras
gl2(R) and gl3(R). We use the homogeneity to establish that the Kontsevich tetrahedral
γ3 flow is trivial in each case; we give explicit formulas for the trivializing vector fields.
Chapter 9 entitled Graph complex action on star products combines star products
and gauge transformations, unoriented graph cocycles, and Poisson structures and their
Kontsevich’s flows. Finally, we give two examples of the graph complex action on ⋆-
products. We show how the Poisson (non)trivial evolution induces an evolution of star
products. For the fourth order expansion of the Kontsevich star product and for the
Kontsevich tetrahedral flow we find out whether the induced deformation of the star
product amounts to a gauge transformation.
All the software demonstrations in Part I are based on SageMath Jupyter notebooks.
Those notebooks can be retrieved from the same place as the programs themselves, i.e.
https://github.com/rburing/gcaops; the output data files (e.g. graph encodings, Kon-
tsevich graph weights and Leibniz graph weights, and the Shoikhet–Felder–Willwacher
15
cyclic weight relations) are also found there. The notebooks use the software, and the
data files can be appreciated separately.7
Based on research articles, Part II is more theoretic. Publications are clustered in
three groups: about star products, about graph calculus, and about Kontsevich flows of
Poisson structures encoded by graphs. We study relevant theory and prove new lemmas
and theorems; computational results are obtained by using the same software as in Part I,
as well as by the software modules kontsevich_graph_series-cpp in C++ also by the author
(2015–2019).
We begin with [1] entitled On the Kontsevich ⋆-product associativity mechanism, in
which we factor associativity through differential consequences of the Jacobi identity.
Here we meet Leibniz graphs for the first time, in the associator for ⋆-products. The
core publication in this group of articles is [2] entitled The expansion ⋆ mod ō(h̄4) and
computer-assisted proof schemes in the Kontsevich deformation quantization. We express
the weights of all graphs in ⋆ mod ō(h̄4) in terms of only 10 master parameters. We
import the values of these master parameters from Banks–Panzer–Pym, write down the
formula of authentic Kontsevich star product, and verify its associativity by using Leibniz
graphs. In [3] entitled Formality morphism as the mechanism of ⋆-product associativity:
how it works we study the algebraic mechanism of associativity in terms of the Formality
morphism, and we illustrate it. In hindsight, papers [1] and [2] allow us to illustrate the
concept with explicit examples at orders 3 and 4.
The paper [4] entitled The heptagon-wheel cocycle in the Kontsevich graph complex is
an introduction to the Kontsvich graph complex, in which we re-derive graph cohomology
classes related to the grt Lie algebra, namely the tetrahedron γ3, the pentagon-wheel
cocycle γ5, the heptagon-wheel cocycle γ7, and the commutator [γ3, γ5].
In [5] entitled Infinitesimal deformations of Poisson bi-vectors using the Kontsevich
graph calculus, we begin the quest for Poisson flows defined by Kontsevich directed graphs.
Here, we design algorithms and we do the full run through Kontsevich graph flows for
few-vertex graphs. The conclusion is that there are only the grt-related flows and no
others.
The Kontsevich graph flows γ3, γ5, γ7 related to grt are presented in [6], [7], [8] respec-
tively. In [6] entitled The Kontsevich tetrahedral flow revisited we find the correct balance
8 : 24 of graph coefficients in Kontsevich’s tetrahedral graph flow. In [7] entitled Poisson
brackets symmetry from the pentagon-wheel cocycle in the graph complex we obtain the
coefficients of oriented graphs in the pentagon-wheel flow, and we establish the (non-
unique) Leibniz graph factorization of the Poisson cocycle condition. In [8] entitled The
orientation morphism: from graph cocycles to deformations of Poisson structures we not
only provide the encoding of the heptagon wheel flow, but also analyze the factorization
mechanism, originally by Kontsevich, in much detail.
We revisit the graph orientation morphism in [9], where the morphism is expressed
combinatorially in terms of graphs themselves. Next, in [10] entitled Universal cocycles
and the graph complex action on homogeneous Poisson brackets by diffeomorphisms,
we construct universal Poisson 1-cocycles for homogeneous Poisson structures. We also
examine the (non)triviality of universal Poisson 2-cocycles for the brackets obtained from
R-matrices. Finally, we report on the rescaled Nambu–Poisson structures P [ρ, a]: the
Kontsevich flows preserve this class of brackets, forcing the nonlinear evolutions ρ̇, ȧ with
7© The copyright for all newly designed software modules is retained by R. Buring; provisions of the
MIT free software license apply.
16 Overview
differential polynomial r.-h.s. In [11] entitled The hidden symmetry of Kontsevich’s graph
flows on the spaces of Nambu-determinant Poisson brackets we show that the tetrahedral
flow and pentagon-wheel flow preserve the class of Nambu–Poisson bi-vectors over R3
and R4, we collapse the induced evolution of the functional parameters using the Civita
symbols, we find a further discrete symmetry of these evolution equations. For the class
of Nambu–Poisson bivectors over R3 we establish that the Poisson bracket evolution with
respect to the tetrahedral γ3 flow is trivial in the respective Poisson cohomology and we
collapse the formula of the trivializing vector field by using the Civita symbols again.
The dissertation concludes with 6 appendices, as well as with a Curriculum Vitae,
acknowledgements, three abstracts (in English, Dutch, and German), and an abstract for
laymen. In the appendices that follow, we provide in particular the encoding of ⋆ mod
ō(h̄4) of the Kontsevich authentic ⋆-product, the encoding of ⋆aff mod ō(h̄7) of the affine
Kontsevich ⋆-product, and an appendix with software reference documentation.
Scientific outline. In this overview we recall the actual background, we formulate the
research problem, set up goals which we pursue, and phrase the main results. We keep
in mind that this dissertation consists of two large parts. Part I contains ten chapters
(enumerated from 0 to 9) with computer demonstrations, each including an analysis of
the computational results. At the top of each demo chapter, we summarize its content.
Part II is based on the journal publications and one preprint. Here is a brief summary.
Star products. The phase space formulation of quantum mechanics avoids the formalism of
operators on the Hilbert spaces of functions. Here, the operator multiplication is replaced
by an associative non-commutative product defined for functions on the phase space; this
intermediate construction was proposed by Groenewold et al. around 1946, building
on earlier ideas by Weyl and Wigner. To be precise, we consider (formal) deformation
quantization in the following sense.
Definition 1. A star product on a smooth real manifold M with algebra of smooth
functi∑ons A = C
∞(M) is a R[[h̄]]-bilinear associative product ⋆ on the algebra of formal
power series A[[h̄]] that deforms the associative pointwise product on A, i.e. f ⋆ g =
f ·g+ ∞n=1 h̄
nBn(f, g) for f, g ∈ A, in such a way that the Bn are bi-differential operators.
In physics one worries about the convergence of series, perhaps restricting the domain
of the product to a subalgebra; besides, the formal parameter h̄ is replaced by ih̄ . In this
2
dissertation we only consider formal series in h̄, and we will not worry about convergence.
It is easily seen that the skew-symmetric part of B1 defined by B−1 (f, g) = 1(B1(f, g)−2
B1(g, f)) is a Poisson bracket on A, i.e. a Lie bracket (bi-linear anti-symmetric bracket
satisfying the Jacobi identity) which is also a bi-derivation with respect to the pointwise
product. Hence the natural inverse probl∑em is to construct a ⋆-product such that B−1
equals a given Poisson bracket {f, g} = 1 n iji,j=1 P · ∂i(f) · ∂j(g), where ∂ ∂` := ℓ is the2 ∂x
derivative with respect to a local coordinate x` in a chart on M .
Example 2. For a generic Poisson bracket with coefficients P ij, an analytic formula for
17
a ⋆-product modulo ō(h̄3), with B1 e(qual to the Poisson bracket, is given by
f ⋆ g = f · g + h̄P ij · ∂if · ∂ g + h̄2 1j P ij · P k` · ∂)k∂if (· ∂ ∂ g + 1∂ P ij · P k`` j ` · ∂2 3 k∂if · ∂jg
− 1∂ ij k``P ·P ·∂ f∂ 1 ij k` 3 1 ij k` mn3 i k ·∂jg− ∂`P ·∂jP ·∂if ·∂kg + h̄ P ·P ·P ·∂m∂k∂if ·∂n∂`∂6 6 jg
− 1∂ ∂ P ij · ∂ ∂ P k` · Pmn · ∂ f · ∂ g − 1P ijm ` n j i k · ∂nP k` · ∂`Pmn · ∂6 6 k∂if · ∂m∂jg
− 1∂m∂`P ij · ∂ P k` · Pmnn · ∂k∂if · ∂jg − 1∂ ∂ P ij · ∂ P k` · Pmnm ` n · ∂if · ∂k∂6 6 jg
+ 1∂ ∂ P ij · P k` · Pmn · ∂ ∂ ∂ f · ∂ g + 1∂ ∂ P ij · P k`n ` m k i j n ` · Pmn · ∂6 6 if · ∂m∂k∂jg
+ 1∂nP
ij · P k` · Pmn · ∂ ∂ ∂ f · ∂ ∂ g − 1∂ P ij · P k` · Pmnm k i ` j n · ∂k∂if · ∂3 3 m∂`∂jg
− 1∂ P ij · ∂ ∂ P k` · Pmn` n j · ∂m∂if · ∂kg + 1∂n∂ ij`P · ∂ P k`j · Pmn · ∂if · ∂6 6 )m∂kg
− 1∂ P ij · P k`∂ mn 1 ij k` mn 3
6 n `
· P · ∂k∂if · ∂m∂jg − ∂`P · ∂nP · P · ∂6 k∂if · ∂m∂jg + ō(h̄ ),
where the sum over all indices—each index running from 1 to dim(M)—is implicit. This
formula illustrates a major result by Kontsevich (1997) stating that there always exists
a solution to the inverse problem of constructing ⋆ = ⋆(P ) on finite-dimensional affine
Poisson manifolds (M,P ).
Inspired by the technique of Feynman diagrams, Kontsevich assigned formulas to the
following class of graphs.
Definition 3. A Formality graph is a simple directed graph (that is, without double
edges and without tadpoles) on m+n vertices {0, . . . ,m−1,m, . . . ,m+n−1}, such that
the m ground vertices 0, . . . ,m− 1 are sinks (with no outgoing edges) and the n vertices
m, . . . ,m+ n− 1 are called aerial. The set of edges of the graph is endowed with a total
ordering.
In the graphs that we will meet in practice, the ground vertices will be drawn on
the page along R = ∂H2 unlabeled from left to right. The aerial vertices inside H2 will
typically have two or three outgoing edges, which will be labeled L ≺ R or L ≺ M ≺ R
respectively. The total ordering on the set of edges is then inherited from the ordering
of aerial vertices and the ordering of edges at each aerial vertex. In pictures of Formality
graphs without edge labels, the edge ordering is by convention lexicographic.
Notation 4. The set of all Formality graphs with m ground vertices and n aerial vertices
will be denoted by Gnm. The subset of Kontsevich graphs built of wedges (with each aerial
vertex having exactly two outgoing edges) will be denoted by Ĝnmr ⊂ G
n
m.
r 3R2 @Rr2
Example 5. Kontsevich graphs built of wedges: L	r R@Rr ∈ Ĝ1 L Rr r2, r?	rL@Rr ∈ Ĝ
2
2.
3 3  L 4
Formality graphs containing a tripod: rLr@R	 ? 1 L@R 
R 2M@Rr ∈ G3, 	r r?M@Rr
 ∈ G3.
Formulas are associated with Formality graphs as follows.
2r
Convention 6. To the edges L and R of the wedge graph L RΛ = r	@Rr we ascribe inde-
pendent indices i and j respectively, and with this graph Λ we associate the operator
Λ(P )(f, g) = P ij · ∂if · ∂jg which is the Poisson bracket. More generally, for a Formal-
ity graph Γ ∈ Gnm we ascribe independent indices to all edges; the multi-linear multi-
differential operator Γ(J0, . . . , Jn−1)(f0, . . . , fm−1) associated with the graph Γ is then a
18 Overview
sum over those indices, with each summand being a product over the (differentiated)
contents of vertices, the ground vertex k containing the argument fk of the operator, and
the aerial vertex m + ℓ containing the component of the multi-vector field J` specified
by indices of the ordered outgoing edges; here the content of each vertex is differentiated
with respect to the local coordinates specified by the incoming edges (if any).
3r 3R
@Rr r`2 @Rr2
Example 7. To the Kontsevich graph L RΓ = ?r	@Rr ∈ Ĝ2L 2 we ascribe the indices k r j?	i@Rr
and hence with Γ we associate ther operator Γ(P, P )(f, g) = P k` · ∂ (P ij` )r· ∂k∂i(f) · ∂j(g).3 3
To the Formality graph L@RT := 	r r?MR@r ∈ G13 we ascribe the indices i @k	r r?j @Rr and hence
with it we associate the operator T (B)(f, g, hr ) = Bijkr · ∂i(rf)r · ∂j(g) · ∂k(L rh).L	@ 	
 @RRR r
To the sum of Konrtsevich graphs rJ :=
L H	r@RRrr @R
Rr − rL 
r
 HjRr − L	r L	r@RRr we ascribe
r `r `	@ r	
 @R` rthe indices 	i@jRr @Rkr − r ji j H
r
 Hjk r − 	ri 	r@Rkr and hence with it we associate the
operator
J(P, P )(f, g, h) = (∂ (P ij) · P `k − ∂ (P ik) · P `j − ∂ (P jk` ` ` ) · P i`) · ∂i(f) · ∂j(g) · ∂k(h)
= {{f, g}, h}+ {{h, f}, g}+ {{g, h}, f},
which is the Jacobiator for P . We have the identity T (1 [[P, P ]])(f, g, h) = J(P, P )(f, g, h),
2
where [[·, ·]] denotes the Schouten bracket.
Theorem-Definition 8 (Kontsevich, 1997). For every Poisson bi-vector P on a finite-
dimensional affine real manifold M and an infinitesimal deformation × 7→ ×+ h̄{·, ·}P +
ō(h̄) towards the respective Poisson bracket, there exists a system of weights w(Γ), uni-
formly given by an integral form∑ula, suc∑h that the R[[h̄]]-bilinear star-product
× h̄
n
⋆ = + w(Γ) · Γ(P, . . . , P )(·, ·) (1)
⩾ n!n 1 Γ∈Ĝn2
is associative.
Elementary properties of the graph weights w(Γ) are summarized in [2, Lemma 1–5
and Remark 8], and the Shoikhet–Felder–Willwacher cyclic weight relations are recalled in
[2, Proposition 7]. These relations are not enough to determine the weights completely.
Another ample source of relations between weights is the associativity of ⋆; this can
be exploited as in [2, Method 1–3]. To make these methods effective, the evaluation
of operators associated with graphs (having Poisson structures implanted into them) is
implemented in software. In this way, a system of equations is formed with the weights
w(Γ) as unknowns. Up to ō(h̄4), the above methods express the weights of all graphs in
terms of just 10 (and up to gauge transformations, just 6) parameters. The weights of
those remaining 10 graphs are imported from Banks–Panzer–Pym (2017).
Proposition 9. The analytic formula for the Kontsevich star product modulo ō(h̄4) is
displayed in Chapter 11, Eq (11) in Conclusion thereof. The encodings of all the graphs
in ⋆ mod ō(h̄4) together with their coefficients are given in Encoding 1 in Appendix B.1.
19
Out of the 247 graphs showing up in the Kontsevich ⋆-product at order four, as many
as 138 contain two-cycles [2, Appendix A.2]. We are now in a position to inspect whether
this ⋆-product will be associative for a generic Poisson structure, in arbitary dimension.
In this direction we make use of the following lemma. ∑
Lemma 10 (Lemma 1 in [1]). A tri-differential operator IJK|I|,|J |,|K|⩾0 c ∂I ⊗ ∂J ⊗ ∂K
vanish∑es identically iff all its coefficients vanish: c
IJK = 0 for every triple (I, J,K) of
multi-indices; here ∂ = ∂α1L 1 ◦ · · · ◦ ∂αnn for a multi-index L = (α1, . . . , αn). Moreover, the
sums IJK|I|=i,|J |=j,|K|=k c ∂I ⊗ ∂J ⊗ ∂K are then zero for all homogeneity orders (i, j, k).
Definition 11. A Leibniz graph is a Formality graph containing at least one aerial vertex
with three outgoing edges, such that those three edges have three distinct targets, and
none of those three edges are tadpoles. The other aerial vertices (if any) have two outgoing
edges, and the ground vertices are as usual. These graphs will be evaluated with the
Jacobiator 1 [[P, P ]] of the Poisson structure P in the vertex with three outgoing edges,
2
hence representing a differential operator that is identically zero whenever P is Poisson.
3 r 3 r L r4
Example 12. Leibniz graphs: rLr@R L@R 
	 ? 1 R 2M@Rr ∈ G3, 	r r?M@Rr
 ∈ G3.
We recall from Kontsevich (1997) the guaranteed existence of a factorization of the
star-product associator via Leibniz graphs. To the best of our knowledge, nobody checked
the general mechanism of associativity explicitly before. In [3] we analyze the mechanism
and illustrate it in detail (see section 5 in [3]). An (earlier) explicit example of the
factorization of the associator up to order 3 is in [1].
Proposition 13 (Corollary 4 and Conjecture ending §4 in [3]). The operator ♢ that
solves the factorization problem ( )
Assoc(⋆)(P )(f, g, h) = ♢ P, [[P, P ]] (f, g, h), f, g, h ∈ A[[h̄]], (2)
is given by ∑ n ( )
♢ h̄= 2 · · cn · Fn−1 [[P, P ]], P, . . . , P . (3)
n⩾1 n!
where Fk is the k-ary component of the Formality L∞-morphism, and where we claim
that the constants cn are equal to n/6.
The number of graphs which actually show up at order h̄k in the left and right-hand
sides of factorization problem (2) is reported in Table 0.1.
Table 0.1: Number of graphs in either side of the associator’s factorization.
k 2 3 4 5 6 7
LHS: # Kontsevich graphs, 3 (Jac) 39 740 12464 290305 ?
coeff 6= 0
RHS: # Leibniz graphs, 1 (Jac) 13 241 4609 ? ?
coeff 6= 0
In [3, Section 5] we inspect many graphs of different orders, and establish the equality
of sums of Kontsevich graphs in the associator and and sums of Leibniz graphs—in the
factorizing operator ♢—after they are expanded into the Kontsevich graphs.
20 Overview
Example 14 r(Example 8 in [3] and Cells 56–58 in Section 3.4). The Leibniz graphr5L@RL331 := 	+r)qsr r?MR@r of differential orders (3, 3, 1) has the weight 1/24 according to
Panz(er’s kontsevint. Multiplied by a universal (for all graphs at h̄
4) factor 24 = 16
and the factor 1/(# Aut(L331)) = 1/2 due to this g)raph’s symmetry (3 ⇄ 4), it expands
to 1 [01; 01; 01; 52]+ [01; 01; 12; 50]+ [01; 01; 20; 51] by the definition of Jacobi’s identity.
3
This sum of three weighted Kontsevich oriented graphs reproduces exactly the compo-
nent A(4)331 of homogeneity order (3, 3, 1) in the associator at h̄4, which is known from [2,
Table 8 in App. D].
In the right-hand side of the associator for ⋆, there are Leibniz graphs: at h̄k⩾2, such
Leibniz graphs have 3 sinks, k−1 aerial vertices (of which one vertex, the Jacobiator, has
three outgoing edges, and the remaining k − 2 vertices (if any) each have two outgoing
edges), and, by the above, 3+ (k− 2) · 2 = 2k− 1 edges; tadpoles are not allowed, graphs
with multiple edges are discarded. For each k = 2, 3, 4, 5 we generate all such admissible
Leibniz graphs (those can be zero graphs with a parity-reversing automorphism); the
respective number of such Leibniz graphs at each order h̄k is in Table 0.2. At every order
k, we generate the entire set of the cyclic weight relations (Willwacher–Felder (2008),
Shoikhet (2000)); every cyclic weight relation is a linear constraint upon the weights of
several Leibniz graphs (all those weights are given by the Kontsevich integral formula
(1997)). The number of these linear relations and the (co)rank of this linear algebraic
system follow in Table 0.2.
Table 0.2: The count of admissible Leibniz graphs in the associator for Kontsevich’s ⋆.
k 2 3 4 5 6
# Leibniz graphs, generated 1 24 520 11680 293748
# Leibniz graphs generated, nonzero 1 24 490 11260 285684
# Leibniz graphs generated, nonzero, diff. order > 0 1 15 301 6741 171528
• of them, with in-degree(aerial vertices) ⩽ 1 1 15 177 1573 12045
# Leibniz graphs (coeff 6= 0 in associator) 1 13 241 4609 ?
# Cyclic weight relations 1 15 301 6741 171528
Corank of linear algebraic system 0 3 66 1469 ?
Banks–Panzer–Pym do not list the weights of Leibniz graphs (as in Table 0.2 above),
for these graphs do not show up in the ⋆-product itself where the vertex-edge valency is
different for the Kontsevich graphs. We use the software by Banks–Panzer–Pym (2018)
to calculate the Kontsevich weights of all the Leibniz graphs which are admissible for
the right-hand side of star-product’s associator. (Some weights can—and actually do—
vanish because either the graph is zero, or the weight integrand is identically zero, or
the weight formula integrates to a zero number.) The count of admissible Leibniz graphs
with nonzero weights is in the fourth line of Table 0.2: the corresponding line in Table 0.1
is reproduced verbatim. (The Leibniz graphs with zero weights do nominally show up in
the cyclic weight relations for Leibniz graphs, but in fact stay invisible in the formulas.)
Proposition 15 (Cells 56–70 in Chapter 3). The numeric values of the Kontsevich
weights of Leibniz graphs with k aerial vertices on 3 sinks, which we calculated using
21
Panzer’s kontsevint, do satisfy the system of linear algebraic equations given by the cyclic
weight relations, for k = 1, 2, 3, 4.8
For the Kontsevich graphs admissible for the ⋆-product at h̄n, that is on two sinks,
on n aerial vertices and 2n edges (from n wedges), all of the above is repeated for n =
1, 2, 3, 4, 5, 6, 7. The various counts of Kontsevich graphs and (co)ranks of the cyclic
weight relation systems are in Table 0.3.
Table 0.3: The count of admissible Kontsevich graphs in the ⋆-product.
n 1 2 3 4 5 6 7
# Kontsevich graphs, generated 1 6 44 475 6874 126750 2814225
# Kontsevich generated, nonzero 1 6 38 445 6488 122521 2744336
# Kontsevich generated, nonzero, diff. order > 0 1 4 30 331 4907 91694 2053511
# Kontsevich generated, nonzero, diff. order > 0, 1 4 30 330 4893 91489 2049704
connected
• of them, with in-degree(aerial vertices) ⩽ 2 1 4 30 265 2801 33690 451927
• of them, prime 1 3 24 215 2327 28649 391958
• of them, with in-degree(aerial vertices) ⩽ 1 1 4 14 51 161 542 1723
• of them, prime 1 3 8 23 59 171 477
# Kontsevich graphs (coeff 6= 0 in ⋆ at h̄n) 1 4 13 247 2356 66041 ?
• of them, with in-degree(aerial vertices) ⩽ 1 1 4 6 35 84 334 958
# Cyclic weight relations 1 4 30 331 4907 91694 2053511
Corank of linear algebraic system 0 1 11 103 1561 ? ?
Now for such admissible Kontsevich graphs, the values of their weights can be di-
rectly imported from the on-line kontsevint repository of Panzer.9 We compose the linear
algebraic system of cyclic weight relations, but now, we merge these systems with many
other linear equations (upon the weights) that stem from the associativity of ⋆, as well as
from the elementary properties of graph weights such as mirror reflections. In April 2017,
we submitted the agglomerated system of linear algebraic constraints upon the weights
of Kontsevich graphs in ⋆ mod ō(h̄5) to the developers of kontsevint; Banks–Panzer–Pym
confirmed that all of the relations are satisfied by the weight values found by using their
own software. We verified this independently by using our software:
Proposition 16 (Cells 37–55 in Chapter 3 and Chapter 11). The numeric values of the
weights of Kontsevich graphs with ⩽ 5 aerial vertices on 2 sinks, generated by Banks–
Panzer–Pym (2018), do satisfy the entire system of constraints given by the basic proper-
ties [2, Lemma 1–5], the cyclic weight relations up to order 5, and the system of equations
obtained by restricting the associativity of the Kontsevich ⋆-product mod ō(h̄5) to the 3D
rescaled Nambu–Poisson structure.
In Chapter 3, further links to externally stored plain text files are available: the files
contain graphs, weights, and relations.
8The relations are satisfied exactly, without involvement of any conventional recalculating constants
and normalizations (in contrast with the mandatory use of auxiliary constants cn = n/6 in Proposition 13,
see above). But let us remember that the multiplicativity of Kontsevich weights is more subtle for graphs
on three ground vertices than for Kontsevich’s graphs on two sinks.
9https://bitbucket.org/PanzerErik/kontsevint/
22 Overview
Proposition 17 (Cells in Appendix B.2). The Kontsevich ⋆-product with the harmonic
graph weights, known up to ō(h̄6) from Banks–Panzer–Pym (2018), is associative mod-
ulo ō(h̄6): every tri-differential homogeneous component of the associator admits some
realization by Leibniz graphs; to find every such solution, the 0th layer of Leibniz graphs
suffices for each of the tri-differential orders.
Proof scheme. The associativity of Kontsevich’s ⋆-product up to ō(h̄4), that is,
Assoc(⋆(P ))(f, g, h) mod ō(h̄4) = ♢(P, [[P, P ]])(f, g, h) mod ō(h̄4), is the core of paper [2],
which is Chapter 11 in Part II below; see also §3.5.1 in Part I. Next, in §3.5.1 we provide
a realization of the component ∼ h̄5 in the associator Assoc(⋆) mod ō(h̄5) in terms of the
Leibniz graphs from the 0th layer, that is, by using the Leibniz graphs obtained at once
by contracting edges between aerial vertices in the Kontsevich graphs from the associa-
tor. (We keep in mind that the representability of the associator by using the 0th layer
Leibniz graphs is previewed in the proof of Kontsevich’s Formality theorem, and we seek
to illustrate this.)
There are 105 homogeneous tri-differential order components at h̄6 in the associator
Assoc(⋆) mod ō(h̄6). We import the harmonic graph weights at h̄5 and h̄6 in ⋆ mod
ō(h̄6) from the kontsevint repository of E. Panzer (Oxford). At order h̄6, the weights of
Kontsevich graphs in ⋆ are expressed as Q-linear combinations of 1 and ζ(3)2/π6. In
consequence, the coefficients of Kontsevich graphs in the associator at order h̄6 are also
Q-linear combinations of that kind. Every tri-differential homogeneous component of
the associator is thus split into the rational- and ζ(3)2/π6-slice: either of the slices is a
linear combination of Kontsevich’s graphs with rational coefficients. The rational slices
are met in all of the 105 tri-differential orders; we detect that in every such slice the
Kontsevich graphs provide the 0th layer of Leibniz graphs, which suffices to realize that
sum of Kontsevich graphs as a linear combination of these Leibniz graphs. The ζ(3)2/π6-
slice is nontrivial in 28 tri-differential orders of the associator at h̄6; here the Formality
mechanism works as follows. For all but 6 tri-differential orders, the Kontsevich graphs
from the linear combination near ζ(3)2/π6 suffice to provide the set of 0th layer Leibniz
graphs which are enough for a solution of the factorization problem. The tri-differential
orders {(1, 1, 3), (3, 1, 1), (2, 1, 2), (1, 2, 2), (2, 2, 1), (1, 3, 1)} are special: for a solution to
appear, the sets of Kontsevich graphs from the rational and ζ(3)2/π6-slices within that
tri-differential order must be merged and then the union set is enough to provide a
factorization of the ζ(3)2/π6-slice by the 0th layer of Leibniz graphs. The corresponding
computations are presented in Appendix B.2. We conclude that at order 6 for the full
Kontsevich star product, Kontsevich’s Formality mechanism works as expected.
All of the above was true for arbitrary Poisson structures (on affine finite-dimensional
real manifolds). For the class of Poisson brackets with affine coefficients (whose higher
derivatives vanish identically), e.g. the Kirillov–Kostant linear brackets, we advance fur-
ther in the expansion of the Kontsevich ⋆-product.
Indeed, the restriction of Kontsevich’s ⋆-product to the spaces of affine Poisson brack-
ets is well-defined: all the Kontsevich graphs in ⋆aff mod ō(h̄n) and in its associator up
to ō(h̄n) only have aerial vertices with in-degree ⩽ 1. The linear algebraic system of the
Shoikhet–Felder–Willwacher cyclic weight relations is by construction triangular with re-
spect to the weights of Kontsevich graphs (in ⋆) with an overall bound for the in-degrees
of aerial vertices. (The linear system of cyclic weight relations is also triangular with
respect to the in-degrees of aerial vertices in the Leibniz graphs which can be used to
express the associator via differential consequences of the Jacobi identity.)
23
Proposition 18 (see Section 3.7). The encoding and analytic formula of Kontsevich’s
affine star product ⋆aff mod ō(h̄7)—in particular, for all the Kirillov–Kostant Poisson
brackets, linear on the duals g∗ of finite-dimensional Lie algebras—is given in Appendix C.
There are 1423 nonzero Kontsevich weights of affine Kontsevich graphs in ⋆aff mod ō(h̄7)
terms overall at all orders ⩽ 7. The multiple zeta value ζ(3)2/π6 starts appearing in the
weights at n ⩾ 6 vertices.10
Proof scheme. The ansatz for ⋆aff mod ō(h̄n) contains, at h̄7, 1731 affine Kontsevich
graphs with in-degree ⩽ 1 of aerial vertices; their Kontsevich weights are constrained
by the elementary properties (such as mirror reflections, whence basic graphs), by the
weights multiplicativity (whence the prime graphs), by the vanishing statements for the
Kontsevich graphs which are disconnected over the sinks, and for the Kontsevich graphs
which contain a triangle subgraph standing on a sink (as in Example 23 on p. 25 below),
and by the cyclic weight relations: the corank of the merged linear algebraic system upon
the 1731 unkowns equals 76. The restriction of the associator for ⋆aff mod ō(h̄7) to a
generic affine Poisson bracket P = (ax+ by+ c)∂x ∧ ∂y on R2 decreases the corank down
to 74. The values of the 76 master parameters (themselves the weights of certain affine
Kontsevich graphs on n = 7 aerial vertices in the affine star-product ⋆aff mod ō(h̄7)) have
been computed using the kontsevint program by Banks–Panzer–Pym; these values are
listed in Cell 54 in Section 3.7 below.
This affine star product expansion is associative up to ō(h̄7):
Proposition 19 (Cells 58–71 in Section 3.7). The affine Kontsevich star product expan-
sion ⋆aff mod ō(h̄7) found in Proposition 18 above is associative modulo ō(h̄7). Namely,
(every homogeneous tri-differential component of ) the associator (f ⋆aff g) ⋆aff h − f ⋆aff
(g ⋆aff h) mod ō(h̄7) is realized as some sum of Leibniz graphs.
Proof scheme. With not yet specified undetermined coefficients of Kontsevich graphs at
h̄7 in the affine star product ⋆aff mod ō(h̄7), its associator’s part at h̄7 expands to 203 tri-
differential order components. As soon as the weights of all the new Kontsevich graphs
on n = 7 aerial vertices are fixed by Proposition 18, the number of tri-differential or-
ders (d0, d1, d2) actually showing up at h̄7 in the associator A for ⋆aff mod ō(h̄7) drops
to 161. For all but four tri-differential order components Ad0d1d2 in the associator A,
the 0th layer of Leibniz graphs, which are obtained by contracting11 one edge between
aerial vertices in the Kontsevich graphs of every such tri-differential component Ad0d1d2 ,
is enough to provide a solution for the factorization problem, Ad0d1d2 = ♢d0d1d2(P, [[P, P ]]),
expressing that component by using differential consequences of the Jacobi identity (en-
coded by Leibniz graphs). We detect that for the tri-differential orders (d0, d1, d2) in
the set {(3, 3, 2), (2, 3, 3), (3, 2, 3), (2, 4, 2)}, the Leibniz graphs from the 0th layer are not
enough to reach a solution ♢d0d1d2 ; still a solution ♢d0d1d2 appears in each of these four
exceptional cases after we add the Leibniz graphs from the 1st layer (i.e. those graphs
obtained by contraction of edges in the Kontsevich graph expansion of Leibniz graphs
from the previous layer; see [10]). (There are 2294 Kontsevich graphs in A2,3,3, producing
3584 Leibniz graphs in the respective 0th layer immediately after the edge contractions;
the component A3,3,2 contains equally many Kontsevich graphs and the same number of
10The Kontsevich weight of the Felder–Willwacher affine graph (2008) equals 13 1 2 62903040 − 256ζ(3) /π ,
thus now correcting a typo in the kontsevint program description by Banks–Panzer–Pym.
11Note that the Leibniz graphs in the 0th layer have vertices of in-degree ⩽ 2 because they are
obtained by the contraction of a single edge in the Kontsevich graphs with vertices of in-degree ⩽ 1.
24 Overview
Leibniz graphs in the 0th layer; the largest component A3,2,3 contains 2331 Kontsevich
graphs and gives 3603 Leibniz graphs in the 0th layer; and finally A2,4,2 contains 1246
Kontsevich graphs and produces 2041 Leibniz graphs in the 0th layer.) In Section 3.7.8 of
Part I below we generate a Leibniz graph factorization of all tri-differential components
in the associator for ⋆aff mod ō(h̄7) and we provide the data files of Leibniz graphs and
their coefficients.
Proof scheme (for the reduced affine star product ⋆redaff mod ō(h̄7)). The reduced affine
star product ⋆redaff mod ō(h̄
7) is obtained from the affine star product ⋆aff mod ō(h̄7) by
realizing the coefficient of ζ(3)2/π6 as the Kontsevich graph expansion of a linear combi-
nation of Leibniz graphs with rational coefficients and, now that this combination does
not contribute to either the star product or its associator when restricted to any affine
Poisson structure, by discarding this part of ⋆ mod ō(h̄7aff ) proportional to ζ(3)2/π6. In
the reduced affine star product ⋆redaff mod ō(h̄
7) there remain only 326 nonzero rational
coefficients of Kontsevich graphs at h̄k for k = 0, . . . , 7 (in contrast with 1423 nonzero
(ir)rational coefficients at orders up to h̄7 in ⋆aff mod ō(h̄7)).
The associator for ⋆redaff contains 95 tri-differential orders at h̄
6 and 161 tri-differential
orders at h̄7. We see that the associator Assoc(⋆redaff ) mod ō(h̄
7) becomes much smaller
than Assoc(⋆aff) mod ō(h̄7), now containing only 29371 Kontsevich graphs instead of
59905. But the work of the associativity mechanism for ⋆redaff requires the use of the 1st
and higher layer(s) of Leibniz graphs much more often than it already was for the affine
star product ⋆aff mod ō(h̄7) before the reduction. Now, at orders ⩽ 7 in h̄, new Leibniz
graphs from the layer(s) beyond the 0th are indispensable for the factorization of 114 out
of 336 homogeneous tri-differential order components of the associator, see Appendix C.3
where we list all these exceptional orders.
We observe that the number ζ(3)2/π6, not showing up in any restriction of the
affine star product f ⋆aff g mod ō(h̄7) to an affine Poisson structure and any arguments
f, g ∈ C∞(M)[[h̄]], acts in effect as a placeholder of the Kontsevich graphs which, by
contributing to the associator and then creating the Leibniz graphs by edge contraction,
provide almost all of the Leibniz graphs needed for a factorization of the associator for
⋆aff mod ō(h̄7) via the Jacobi identity. When the ζ(3)2/π6-part of ⋆aff mod ō(h̄7) itself is
eliminated by using the Jacobi identity for affine Poisson structures, the remaining ⋆redaff
mod ō(h̄7) and its associator rely heavily on the use of higher layers of Leibniz graphs for
a factorization solution to be achieved.
Example 20. Consider the Leibniz graph
t
tJ6H JHHjt J
L1 = t C t Jt ) tiPtC QJ?	@R@CW?

 QĴst
on three sinks 0, 1, 2 with ñ = 6 aerial vertices, and with edges [(3, 2), (3, 7), (4, 1),
(4, 8), (5, 1), (5, 3), (6, 1), (6, 2), (6, 4), (7, 0), (7, 5), (8, 0), (8, 1)]. This Leibniz
graph is needed for the factorization of the tri-differential component of order (2, 4, 2)
at h̄7 in the associator for ⋆ 7aff mod ō(h̄ ). This graph appears only in the 1st layer of
Leibniz graphs, not in the 0th layer, as we contract edges of Kontsevich’s graphs on n = 7
aerial vertices in the associator for ⋆ 7aff mod ō(h̄ ), and as we expand the resulting Leibniz
25
graphs to the old and possibly new Kontsevich graphs.12 This Leibniz graph created in
the 1st layer appears with coefficient 2/135 in an iteratively found factorization of the
associator. The genuine Kontsevich weight of this Leibniz graph calculated by using the
program kontsevint 2by E. Panzer is also nonzero: w(L1) = −3/128 · ζ(3)6 + 31/725760.π
The actual coefficient of L1 in the canonical factorization of the associator, as guaranteed
by the Formality Theorem, equals w(L1) multiplied by some nonzero constant. The
discrepancy between the found rational value in some solution and the (ir)rational value
in Kontsevich’s canonical solution is likely due to an identity between Leibniz graphs
which expand to a zero sum of Kontsevich graphs (see §3.5.2 on p. 93 below). But
anyway, based on this empiric evidence we conclude that the 0th layer of Leibniz graphs
is not enough to provide a factorization of the associator for the (either affine or full)
Kontsevich star product at order h̄7, whereas, according to Proposition 17 above, the 0th
layer of Leibniz graphs was enough at order h̄6 to factor the associator for the full star
product.
Remark 21. The above iterative scheme gives us a solution to the weak factorization
problem: each tri-differential component Ad0d1d2 is factorized independently from the
others, so that the coefficients of the Leibniz graphs are not yet constrained overall—
over different components—by the Shoikhet–Felder–Willwacher cyclic weight relations
and other relations. In particular, the above scheme does not guarantee that the found
coefficients of Leibniz graphs are equal (up to the multiplicity and recalculation con-
stants) to the genuine Kontsevich weights of those Leibniz graphs. The above scheme
provides the necessary minimum number of layers of Leibniz graphs, whereas the calcula-
tion of Kontsevich’s genuine weights of Leibniz graphs is sufficient to build a solution (the
canonical one) for the associator’s factorization problem. We remember that there exist
identities, i.e. sums of Leibniz graphs which expand to zero sums of Kontsevich graphs
(here, in the associator); such identities could make unnecessary the use of a Leibniz
graph with nonzero genuine weight from a (high number — in particular the last) layer.
Hypothetically it might be that any solution needs the 0th and 1st layers, hence they
are “necessary”, but Kontsevich’s canonical solution stretches over the 0th, 1st and 2nd
layers, thus they are “sufficient”. In conclusion, the above scheme does not guarantee that
the genuine Kontsevich weight of a Leibniz graph in the known associator’s factorization
at order h̄7 will definitely be equal (up to the multiplicity and recalculation constants)
to this Leibniz graph’s coefficient in the last necessary layer.
In section 2.5 of [2] and in Chapters 1 and 9 below, we study gauge transformations
of star products.
Definition 22. Let ⋆ : A[[h̄]] × A[[h̄]] → A[[h̄]] be a star-product, where A = C∞(M).
A gauge transformation is an R[[h̄]]-linear map T : A[[h̄]] → A[[h̄]] of the form f 7→ f +
h̄D1(f)+h̄
2D2(f)+. . . for f ∈ A, where the Di are differential operators; by construction,
T is formally invertible. The star-product ⋆′ defined by f ⋆′ g = T (T−1(f) ⋆ T−1(g)) is
called gauge equivalent to ⋆. q+ q
Example 23 (Examples 23 and 24 in [2]). The map defined by • →7 • + h̄2 AU3r
 or12
f 7→ h̄2f + ∂ P ij∂ P k`k j ∂i∂`f for f ∈ C∞(M) is a gauge transformation. When applied to12
12This Leibniz graph cannot originate from any Kontsevich graph in the associator itself — even with
aerial vertex in-degree ⩾ 2. Namely, all candidate Kontsevich graphs are composite with one of the
factors having zero weight.
26 Overview
the Kontsevich ⋆-product, the respective gauge-equivalent product ⋆′ contains no graph
with loop at h̄2.
Gauge transformations enable us to bring down the number of unknown parameters
in ⋆ mod ō(h̄4) from 10 down to 6 in [2, Theorem 14]. Independently, gauge transforma-
tions allow us to verify claims about (non)equivalence of star products. In section 4 of [2]
we compare some earlier calculations of ⋆-product expansions up to ō(h̄3) with the au-
thentic Kontsevich ⋆-product expansion. For more details about gauge transformations,
see Chapter 9.
Poisson flows. We want to deform Poisson structures in such a way that they stay
Poisson.
Definition 24. Let P be a Poisson bi-vector on the manifold M at hand and consider
its deformation P + εQ + ō(ε) where Q is a bi-vector and ε is a formal parameter. We
say that after such deformation the bi-vector stays infinitesimally Poisson if [[P + εQ +
ō(ε), P + εQ+ ō(ε)]] = ō(ε), that is if [[P,Q]] = 0. The deformation P + εQ+ ō(ε) is called
trivial if Q is a coboundary in the Poisson complex w.r.t. the differential ∂P = [[P,−]],
i.e. if there exists a vector field X such that Q = [[P,X]].
The existence and classification of (non)trivial deformations of a Poisson structure P
naturally depends strongly on the manifold M and the Poisson structure P . Nevertheless
we can ask if there exist universal deformations, in the sense of a general formula or recipe
P 7→ P+εQ(P )+ō(ε), which is defined for all (affine) manifolds in terms of the coefficients
P ij of the Poisson bi-vector. The existence of such a formula would then also require a
universal proof of the Poisson cocycle condition [[P,Q(P )]] = 0.
Example 25. As a byproduc(t of Kontsevich’s Formality Con)jecture (1996), the formula′
∂3P ij ∂P kk ∂P ``
′
∂Pmm
′
∂ ∂
Qtetra(P ) = 1 · ∧
∂(xk∂x`∂xm ∂x`′ ∂xm′ ∂xk′ )∂xi ∂xj
2 ij 2 km k′` m′`′
+ 6 · ∂ P ∂ P ∂P ∂P ∂ ∧ ∂
∂xk∂x` ∂xk′∂x`′ ∂xm′ ∂xj ∂xi ∂xm
defines a universal deformation of Poisson structures. Here the balance 1 : 6 is neces-
sary, as shown by experiment with 3D rescaled Nambu–Poisson structures in [6]. A very
detailed illustration of the pictorial proof of the Poisson cocycle condition’s factoriza-
tion [[P,Qtetra(P )]] = ♢(P, Jac(P )), with the Kontsevich graphs in the left-hand being
expressed as the expansion of sums of Leibniz graphs in the right-hand side, is given
in [6].
In fact there are more such universal flows, originating from cocycles γ ∈ ker d in the
Kontsevich graph complex (which will be discussed soon, see p. 31 below).
Proposition 26 (Theorem 1 and Corollary 3 in [8]). Whenever P is a Poisson bi-vector so
that the Schouten bracket πS(P, P ) vanishes ([[P, P ]] = 0), and whenever γ ∈ ker d is a co-
cycle on k vertices and 2k−2 edges (so that d(γ) = [••, γ] = 0 in Kontsevich’s unoriented
graph complex), then O⃗r(γ)(P⊗k) is a Poisson 2-cocycle (so that [[P, O⃗r .(γ)(P⊗k)]] = 0
modulo the Jacobi identity 1 [[P, P ]] = 0 for the Poisson structure). The operator ♢ in the
2
factorization problem
∂P (O⃗r(γ)(P⊗k)) = ♢(P, [[P, P ]]), γ ∈ ker d,
27
is the sum of Leibniz graphs obtained from the graph cocycle γ by inserting the Jacobiator
1 [[P, P ]] into one of its vertices (by the Leibniz rule) and skew-symmetrizing w.r.t. the
2
sinks.
Proposition 26 is proved and illustrated explicitly in [8] and references therein: for
the tetrahedron γ3 (see also Example 27 below), the five-wheel cocycle γ5, an example of
a coboundary δ6 = d(β6), and the heptagon-wheel cocycle γ .137 In particular the formula
for ♢ is exact, giving the factorizing operator for the Poisson cocycle condition in each
case (see Table 0.4). Each of these canonical factorizations is also given in Chapter 5
below.
Example 27 (see Equation (11) in [6] and Cells 16–22 in Chapter 5). For the tetrahedral
flow we haver
@ r r@R • • • •
? ? ? ∑  r? B ∑  B♢ = + 3 (−)τ  B  ?r+ 3 B
• • τ∈S r  @ @B  @2 ⟳ Br  @B
 @RBN r
	 R@BH BNr


 ? JĴ ? ? ? ?HHj ?
( ) ( ) ( ) [( ) ( )] ( ) ( ) ( ) ( )
{ ? ? ?∑ • • ? ? • • }
+ 3 + • • A + 
 @
⟳  r AU r *r 
 @Rr 	@@Rr  ? 
@  @YH
r 
 ?
H r YrH
 ? @R 
 ? @R H? H

 = ? H?r
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∑ {
? ?
• • • • }
+ 3 (−)σ 
 @R@r + 
 @R@r .
σ∈S3 
 r  
? *

H
 
 YHr?

 H

  Hj?r 
 C HHr?? 
 CW ?
( ) ( ) ( ) ( ) ( ) ( )
The flow associated with a graph d-coboundary δ = d(β) in the graph complex is
universally Poisson-trivial; the vector field associated with β being the trivializing vector
field [10, Corollary 4]. Flows associated with graph d-cocycles which are not coboundaries
(e.g., γ3, γ5, γ7, [γ3, γ5], γ9) are not universally Poisson-trivial in that particular way. Still
the question remains whether there exists a Poisson structure on an affine manifold on
which the graph complex truly acts nontrivially.
In this direction we can say more when we restrict ourselves to particular (classes
of) Poisson structures. This is the subject of Chapters 6, 7, and 8 below. The classes
will be arbitrary bi-vectors on R2, as well as rescaled Nambu–Poisson structures P =
ρ(x, y, z) da/(dx ∧ dy ∧ dz) on R3 (and similar brackets on R4), and finally, Poisson
brackets constructed by using R-matrices.
13Pictures of these graph cocycles are drawn in Example 41 on p. 32 below.
28 Overview
Table 0.4: The number of graphs in the problem [[P, O⃗r(γ)(P )]] = ♢(P, [[P, P ]]).
Cocycle: γ3 γ5 δ6 = d(β6) γ7 [γ3,γ5]
#vertices: 4 6 7 8 9
#edges: 6 10 12 14 16
#graphs: 1 2 4 46 68
#or.graphs in Q(P ) = O⃗r(γ)(P, . . . , P ): 3 167 1,500 37,185 ?
#or.graphs in [[P,Q(P )]]: 39 3,495 35,949 1,003,611 ?
#Leibniz graphs in ♢(P, [[P, P ]]): 27 3,876 45,965 ? ?
#skew Leibniz graphs in ♢(P, [[P, P ]]): 8 843 9,556 293,654 ?
In dimension two the cocycle condition for Poisson 2-cocycles is satisfied for every
bivector, but the condition to be a 2-coboundary is generally still nontrivial. Nevertheless,
all the universal flows which we have tried are trivial.
Proposition 28 (Cells 17, 31, 45, 67 in Chapter 6). In 2-dimensional Poisson geometries,
the Poisson cocycles Qγ(P ) defined by graph cocycles γ ∈ {γ3, γ5, δ6, γ7} are Poisson-
trivial. Namely, there exist vector fields Xγ(P ), differential polynomial in P , that trivialize
the flows Qγ(P ) = [[P,Xγ(P )]]. Moreover, with respect to the standard symplectic structure
on R2, every such vector field Xγ(P ) is the Hamiltonian vector field of a Hamiltonian
Hγ(P ), again differential polynomial in P .
The case of γ3 was known to Kontsevich (1996), and the respective Hamiltonian was
found by our colleague Bouisaghouane (2016/17). The remaining cases are established
by the new software.
Example 29. Letting P = u ∂x ∧ ∂y be the generic Poisson bi-vector on R2, we have
H 2γ3 = 8uyuxx − 16uxuyuxy + 8u2xuyy, Hγ5 = 6u2u u2 − 12u u u3 − 6u2y xx xy x y xy yu2xxuyy + 12uxuyuxxuxyuyy +
6u2 2 2 2xuxyuyy−6uxuxxuyy−2u3yu 2 3xyuxxx+2uxuyuyyuxxx+2uyuxxuxxy+2uxu2yuxyu 2 2xxy−4uxuyuyyuxxy−4uxuyuxxuxyy+
2u2xu
3
yuxyuxyy + 2uxuyyuxyy + 2u
2 3 4 3 2 2 3
xuyuxxuyyy − 2uxuxyuyyy − 2uyuxxxx + 8uxuyuxxxy − 12uxuyuxxyy + 8uxuyuxyyy −
2u4xuyyyy, and H = 199γ7 u2yuxxu4 − 199u 5xy xuyuxy − . . .. The full formula for Hγ7 is given in4 2
cell 68 on p. 163 in Chapter 6.
Remark 30. In [6, Appendix F, Remark 13] we establish that the formula for the triv-
ializing vector field Xγ3 can be realized as a sum of Kontsevich graphs (although this
representation generally does not provide a trivializing vector field in dimensions greater
than two). Moreover it was established by Bouisaghouane (2016) that the Hamiltonian
Hγ3 itself is a sum of Kontsevich graphs.
The fact that the Hamiltonian associated with γ3 can be represented as a sum of
Kontsevich graphs is not an isolated incident:
Proposition 31 (Cells 23, 37, 54, 72 in Chapter 6). For γ ∈ {γ3, γ5, δ6, γ7} and an
arbitrary Poisson structure P on R2, the differential polynomial Hamiltonians Hγ(P )
from Proposition 28 are sums of Kontsevich graphs; their shapes are on pp. 154, 157,
161, 164 in Chapter 6. The respective trivializing vector fields are obtained from these
Hamiltonians by using the standard symplectic structure ω = dx ∧ dy on R2.
Next we consider the rescaled Nambu–Poisson structures in three and four dimensions.
29
Proposition 32 (see Proposition 18 in [10] and Proposition 1, Corollary 2, Example 5,
and Theorem 7 in [11]). The tetrahedral flow Ṗ = O⃗r(γ3)(P⊗4) restricts to the class of
rescaled Nambu–Poisson brackets P = ρ da/(dx dy dz) = [[ρ ∂ 3x ∧ ∂y ∧ ∂z, a]] on R with
coordinates x, y, z, that is there exist (ρ̇, ȧ) such that Qγ3(P [ρ, a]) = P [ρ̇, a] + P [ρ, ȧ].
• The velocity ȧ is given by Kontsevich’s graphs: ȧ = Qγ3(P, P, P, a).
• The velocity ρ̇ is expressed by ρ̇ = (Qγ3(P [ρ, a])− P [ρ, ȧ])/P [1, a].
• The velocities ρ̇, ȧ are obtained by total skew-symmetrization:14
∑
ȧ = σσ,τ,ζ∈S (−) (−)τ (−)ζ(2au1au2au3ρw1ρw2ρw3av1v2v3 − 6ρau1v2au2au3ρw1ρw3av1v3w23
∑− 6ρ2au1au2u3av1v2ρw3av3w1w2)
ρ̇ = σ τσ,τ,ζ∈S (−) (−) (−)ζ(−2au1au2au3ρv1ρv2ρv3ρw1w2w3 + 6au1v2au2au3ρv1ρv3ρw2ρw1w33
− 12ρau1au2u3av1v2ρv3ρw1ρw2w3− 6ρau1v2au2au3ρv1ρv3ρ 2w1w2w3 + 6ρ au1au2u3av1v2ρv3ρw1w2w3),
where each sum runs over three permutations σ, τ, ζ ∈ S3, giving three triples (u1, v1, w1) =
(σ(x), σ(y), σ(z)), (u2, v2, w2) = (τ(x), τ(y), τ(z)), and (u3, v3, w3) = (ζ(x), ζ(y), ζ(z)).
• The cocycle Qγ3(P [ρ, a]) is the coboundary of a vector field X[ρ, a] with differential
polynomial coefficients (cubic in both ρ and a, of total differential order eight).
• The vector field X[ρ, a] is again realized—by using Civita symbols—as the total skew-
symmetrization of tiny differential polynomial expressions, themselves encoded by graphs.
• The induced velocities are such that ȧ = −[[X, a]] and ρ̇ ∂x∧∂y∧∂z = −[[X, ρ ∂x∧∂y∧∂y]].
This is demonstrated in detail in §7.1, and established in [11].
Proposition 33 (see Example 6 in [11]). The tetrahedral flow restricts to the class of
rescaled Nambu–Poisson brackets P = ρ da db/(dx dy dz dw) = [[[[ρ ∂x ∧ ∂y ∧ ∂z ∧ ∂w, a]], b]]
on R4 with coordinates x, y, z, w, that is there exist (ρ̇, ȧ, ḃ) such that Qγ3(P [ρ, a, b]) =
P [ρ̇, a, b] + P [ρ, ȧ, b] + P [ρ, a, ḃ].
• The velocities ȧ, ḃ are given by Kontsevich graphs: ȧ = Qγ3(P, P, P, a) and ḃ =
Qγ3(P, P, P, b).
• The velocity ρ̇ is expressed by ρ̇ = (Qγ3(P [ρ, a, b])− P [ρ, ȧ, b]− P [ρ, a, ḃ])/P [1, a, b].
• The velocities ρ̇, ȧ, ḃ are obtained by total skew-symmetrization:14 in particular, we have
that
∑
ȧ = σ τσ,τ,ζ∈S (−) (−) (−)ζ4
+3a 3 3 2s1u2u3at1t2bs2bs3v1bt3u1av2av3ρ +6as1u2at1at2v3au3v1v2bt3u1bs2bs3ρ +3av2at1u2v3bv1ρs1au1au3bs2t3bs3t2ρ
−6as1v3b
2
s2t1ρv1as3u1v2au2au3bt2bt3ρ −6as1v2v3at1a
2 2
u2au3v1bs2bs3u1bt3ρt2ρ +6as1as2v3as3u1at1t2t3bv1ρv2bu2bu3ρ
−6as1at1u2u3bu1v2at2at3bs2bs3ρv1ρv3ρ+6av2as1as2s3t1at2t3ρv1ρv3bu1bu2bu3ρ−2as1at2at3u1u2au3bt1bs2bs3ρv1ρv2ρv3 ,
where (s1, t1, u1, v1) = (σ(x), σ(y), σ(z), σ(w)), (s2, t2, u2, v2) = (τ(x), τ(y), τ(z), τ(w)),
and (s3, t3, u3, v3) = (ζ(x), ζ(y), ζ(z), ζ(w)). The other formulas, for ḃ and ρ̇, are on
p. 185 in Chapter 7.
This is discussed in §7.3; the case with scale ρ ≡ 1 is discussed first in §7.2.
Proposition 34 (see Example 7 in [11]). The pentagon-wheel flow Ṗ = O⃗r(γ )(P⊗65 )
restricts to the class of rescaled Nambu–Poisson brackets on R3, with the same mechanism
ȧ = O⃗r(γ5)(P, P, P, P, P, a) and the same subtract-and-divide mechanism for ρ̇ as above;
again ρ̇ and ȧ are obtained by total skew-symmetrization using 5 permutations of the
independent coordinates x, y, z on R3.
14Their expressions as total skew-symmetrizations were obtained in collaboration with D. Lip-
per (2020).
30 Overview
Remark 35. For the tetrahedral γ3 flow on (the space of) Nambu–Poisson brackets on R3,
the above totally skew-symmetric expressions for ρ̇ and ȧ enjoy a further hypersymmetry
property. Namely, choosing any differential monomial (with a nonzero coefficient) in
ρ̇ (resp. ȧ), and choosing any way to skew-symmetrize it15 still not producing zero
identically, the skew-symmetrization then reproduces the entire homogeneous component
in which the respective monomial is contained. There are three homogeneous components
in ȧ, and five in ρ̇.
Finally we consider the class of R-matrix Poisson brackets. With a Lie algebra g
equipped with a non-degenerate bilinear form (such as A,B →7 tr(AB) on some ma-
trix Lie algebras), we associate R-matrices (e.g., from a direct sum decomposition of
g one obtains a difference of projections), as in Chapter 10 of the book by Gengoux–
Pichereau–Vanhaecke (2013). These ingredients then allow us to cook homogeneous lin-
ear, quadratic, and cubic Poisson brackets. We consider the tetrahedral γ3 flow for these
R-matrix Poisson brackets.
Proposition 36 (Cells 22–27, 31–35 and the rest in Chapter 8). The graph cocycle γ3
acts on homogeneous quadratic and cubic Poisson structures associated (with R) -matrices.
• On gl2(R) with coordinates x, y, z, v there is an R-matrix ( x yz v ) 7→ 0 y−z 0 . The cu-
bic polynomial Poisson structure associated with this R-matrix is P = (x2y + y2z) ∂x ∧
∂ + (x2y z + yz
2) ∂x ∧ ∂z + (2 xyz + 2 yzv) ∂x ∧ ∂v + (y2z + yv2) ∂y ∧ ∂v + (yz2 + zv2) ∂z ∧
∂v. The tetrahedral flow of this Poisson structure is Ṗ = Qγ3(P ) = (−48x5y −
288x3y2z−240xy3z2+192 y3z2v−384xy2zv2−192 y2zv3)∂x∧∂y+(−48x5z−288x3yz2−
240xy2z3 + 192 y2z3v − 384xyz2v2 − 192 yz2v3)∂x ∧ ∂z + (−336x4yz − 480x2y2z2 −
576x3yzv+480 y2z2v2+576 xyzv3+336 yzv4)∂x∧∂v+(192 x3y2z−192xy3z2+288 y2zv3+
48 yv5 + 48 (8 x2y2z + 5 y3z2)v)∂y ∧ ∂v + (192 x3yz2 − 192xy2z3 + 288 yz2v3 + 48 zv5 +
48 (8 x2yz2 + 5 y2z3)v)∂z ∧ ∂v. We detect that this bi-vector is a coboundary, Qγ3(P ) =
[[Y⃗ , P ]] with the vector Y⃗ = (−24x4+120 y2z2−96 yzv2)∂x+(96x3y−96 yv3)∂y+(96x3z−
96 zv3)∂z + (96x
2yz − 120 y2z2 + 24 v4)∂v.
• Similarly also on gl2(R) there is the R-matrix ( x yz v )→7 (
x y
−z v ) and the cubic polynomial
P(oisson s)truct(ure associa)ted wit(h it; its t)etrahedral flow is also Poisson-trivial.
16
• On gl3(R) there are quadratic and cubic Pois(son bracket)s associated with the R-matrices
x0 x1 x2 0 x1 x2 x x x xand 0 1 2 0
x1 x2
x3 x4 x5 →7 −x3 0 x5 x3 x4 x5 7→ −x3 x4 x5 . In those cases the tetrahedral
x6 x7 x8 −x6 −x7 0 x6 x7 x8 −x6 −x7 x8
flow is also Poisson-trivial.16
Staying in the same context of homogeneous Poisson brackets, we provide a construc-
tion of universal Poisson 1-cocycles in [10]; this is Proposition 48 on p. 35 below. We
consider two examples: in the above cases of R-matrices associated with the splittings of
gl2(R), the 1-cocycles are identically zero.
The list of classes of Poisson brackets discussed so far is not exhaustive. There are
more methods for constructing Poisson brackets. One such further method is available
from Gengoux–Pichereau–Vanhaecke: they construct symplectic brackets with polyno-
mial coefficients. Those Poisson structures can also be tried by using our algorithms and
software. Another approach to the (non)triviality of Kontsevich’s universal flows in the
context of algebraic varieties is studied by Dolgushev–Rogers–Willwacher (2015).
15That is, prescribing which of the letters xxxyyyzzz in that monomial belong to which of the three
triples xyz.
16The full formulas are given in Cells 31–35, 41–43, 44–48, 49–51, 54–56 in Chapter 8.
31
Remark 37. We have studied infinitesimal deformations P →7 P + εQ + ō(ε) defined
via graph cocycles. These infinitesimal deformations can be formally integrated; higher
order terms in the series 2P + εQ + ε R + . . . are obtained recusively by inserting the
2
graphs from Q into vertices of themselves. The study of convergence of the series P (ε) is
a different aspect, which we do not consider in this dissertation.
Graphs. Just as the Kontsevich ⋆-product is a byproduct of the respective Formality
L∞-morphism F : T d dpoly(R ) → Dpoly(R ), the universal flows on the spaces of Poisson
structures are a byproduct of universal L∞-automorphisms Tpoly(Rd)→ T dpoly(R ), which
themselves are defined via graph cocycles in the Kontsevich graph complex. We presently
recall the construction of this cochain complex, in which the elements are sums of graphs
(with extra structure, namely an ordering of edges) modulo relations. As the next step,
we will (re)compute the dimensions of graded parts of the respective graph cohomology,
and we find (new) explicit representatives of cohomology classes, which are necessary to
evaluate the flows on spaces of Poisson structures.
Notation 38. For n ∈ N⩾1 let Gra(n) denote the N⩾0-graded vector space with the kth
component spanned by simple17 undirected graphs on the vertex set {0, . . . , n− 1} with
an ordered set of k edges labeled18 from 0 to k − 1, modulo the relations γσ = (−)σγ,
where the graphs γ and γσ differ only by the permutation σ of edge labels. A graph with
an automorphism that induces an odd permutation on edges is called a zero graph.
Definition 39. The insertion γ1 ◦⃗i γ2 of a graph γ1 on n1 vertices into the ith vertex of
another graph γ2 on n2 vertices is a sum of graphs on n1 + n2 − 1 vertices. Each graph
in the sum consists of the graph γ2 with its ith vertex replaced19 by the entire graph γ1;
the edges which were incident to the ith vertex in γ2 are re-attached to the vertices of γ1
in all possible ways (each possible way to re-attach edges provides one term in the sum
of graphs).
The operati∑on ◦⃗i is extended to linear combinations of graphs in Gra(n) by bilinearity.The insertion ◦⃗ : Gra(n) ⊗ Gra(m) → Gra(n + m − 1) is defined for graphs by the
sum γ1 ◦⃗ γ2 = m−1i=0 γ1 ◦⃗i γ2 of insertions into all vertices of γ2, and extended to linear
combinations of graphs by bilinearity.
With the partial composition operations ◦⃗i and the action of the group Sn that per-
mutes the labels of vertices, the collection of graded vector spaces Gra(n) forms an operad.
Definition 40. The full Kontsevich graph complex fGC is the collection of graded20
vector spaces fGC(n) defined as the quotient of Gra(n): namely, graphs that differ only
by their vertex labeling are identified. That collection of vector spaces is equipped with
the vertex-expanding differential d defined by d = [••,−], where the Lie bracket is
defined for graphs γ1 on e1 edges and γ2 on e2 edges by the commutator of insertions
[γ1, γ2] = γ1 ◦⃗ γ − (−)e1e22 γ2 ◦⃗ γ1, and the Lie bracket is extended to fGC by bilinearity.
17Without double edges and without tadpoles, not necessarily (strongly) connected, not necessarily
with each vertex at least trivalent.
18In the papers [4, 11, 8] the edges are labeled by using roman numbers I, II, . . . so that the ordered
set of edges E(γ) is I ∧ II ∧ . . ..
19The labels of the vertices of γ1 are shifted up by i, and the labels of the vertices > i in γ2 are shifted
up by n1 − 1. The labels of the edges of γ1 are shifted up by the number of edges in γ2.
20The grading initially inherited from Notation 38 can be shifted, as seen in the literature.
32 Overview
See also [4] for an elementary introduction to the graph complex. Detailed proofs of
the defining properties of this graph complex are found in Rutpten–Kiselev (2018).
Example 41. rThree examples of graph cocycles are γ3 = p ,r r r r
 r p p
5
γ = + r r r
r r
5 r r  r and δ6 = d(β6) where β =
r r r2 6 r .
The tetrahedral cocycle γ3 was found by Kontsevich (1996); the pentagon-wheel cocycle
γ5 was known to Kontsevich and to Willwacher; the coboundary δ6 is an example of a
trivial cocycle. Further examples and pictures of graph cocycles are found in Chapter 5.
Proposition 42 (see Chapter 5). The new software is able to find explicit representa-
tives of the non-trivial graph cocycles γ3, γ5, γ7, and calculate commutators of graphs,
in particular [γ3, γ5] ∈ ker d, in the Kontsevich graph complex with the vertex-expanding
differential. The encodings of these four graph cocycles are given in Appendix E.
More details about and various calculations in the Kontsevich graph complex are
found in Chapter 5. There are variants (in fact, subcomplexes) of the full graph complex
spanned e.g. by connected graphs in which each vertex has degree at least three. The full
graph complex is a symmetric product of the subcomplex spanned by connected graphs,
and Willwacher (2010) proved that the cohomology of the connected graph complex is
expressed as the direct sum of the cohomology of the degree-restricted graph complex
and some known classes, namely (4n+1)-gons for n ⩾ 1 (with 4n+1 two-valent vertices
and 4n+ 1 edges).
Proposition 43. The new software is capable of finding the dimensions of the graded
parts of the (connected) graph cohomology spaces. In Table 0.5 we give a precise count,
up to 9 vertices; the meaning of Nδ, Nker, N0, Nim is explained in Chapter 13.
Table 0.5: Dimensions of connected graph spaces and cohomology groups.
n #E #(graphs) #(= 0) #(=6 0), Nδ N ∗ker, N0 Nim dimH (n)
4 6 1 0 1 1 1
3 5 0 – – – – –
5 8 2 2 0 – 0
4 7 0 – – – – –
6 10 14 8 6 1 1
5 9 1 1 – 0 – –
7 12 126 78 48 1 0
6 11 9 8 – 1 0 1
8 14 1579 605 974 36 1
7 13 95 60 – 35 0 35
9 16 26631 7557 19074 883 1
8 15 1515 602 – 913 31 882
33
The analogous counts for graph cohomology spanned by connected graphs having at
least trivalent vertices are given in [4, Table 3]. The same numbers of nonzero graphs were
previously calculated by Willwacher–Živković (2014); their methods partly depended on
floating-point arithmetic. We provide explicit representatives of graph cohomology classes
in Chapter 5.
We now recall the origin of the graph orientation morphism γ 7→ O⃗r(γ), which
maps graph cocycles γ on n vertices and 2n − 2 edges to infinitesimal symmetries
Ṗ = O⃗r(γ)(P⊗n) of Poisson bi-vectors P on affine manifolds. The reasoning in [8] is
based on that of Jost (2013), which in turn follows an outline by Willwacher (2010), itself
referring to the seminal paper by Kontsevich (1996). A combinatorial interpretation of
the orientation morphism is in [9]. Examples of graph cocycles suitable as input to the
orientation morphism can be borrowed from [4]. An extension of the above technique,
that now yields universal 1-cocycles in the case of homogeneous Poisson bi-vectors, is
contained in [10].
The inspiration for the orientation morphism comes from a very precise analogy be-
tween two worlds: that of graphs on the one hand, and that of endomorphisms on the
spaces of multi-vectors on affine manifolds on the other hand. In fact, graphs describe
endomorphisms defined by natural formulas. The world of endomorphisms on the spaces
of multi-vectors is recalled in [8, Section 1]. The first natural example of such an endo-
morphism is the Schouten bracket πS = ±[[·, ·]]. Besides, endomorphisms can be inserted
one into the other, so that there is the Nijenhuis–Richardson bracket [·, ·]NR which is the
commutator of insertions, and there is the differential [πS,−]NR which is the bracket with
the Maurer–Cartan element πS. Here is the dictionary that we explore: see Table 0.6.
Table 0.6: From graphs to endomorphisms: the respective objects or structures.
World of graphs World of endomorphisms
Graphs (γ,E(γ)) Endomorphisms
Insertion ~◦ of graph into ithi vertex Insertion of endomorphism into ith argument
Insertion ~◦ of graph into graph Insertion ~◦
Bracket [a, b] = a~◦ b− (−)|E(a)|·|E(b)|b ~◦ a Bracket [a, b] = a~◦ b− (−)|a|·|b|b ~◦ a
Lie bracket ([a, b], E([a, b]) := E(a) ∧ E(b)) Nijenhuis-Richardson bracket [a, b]NR on the
space of skew endomorphisms
The stick • • The Schouten bracket πS = ± [[·, ·]]
Master equation [• •, • •] = 0 Master equation [πS , πS ]NR = 0
Graded Jacobi identity for [·, ·] Graded Jacobi identity for [·, ·]NR
Differential d = [• •, ·] Differential ∂ = [πS , ·]NR
We analyze this correspondence in more detail in [8]. By using this dictionary, it
becomes easy to provide the canonical factorizing operator ♢ for the Poisson cocycle
condition [[P, O⃗r(γ)(P⊗n)]] = ♢(P, [[P, P ]]). This is a corollary of⊕the(follo∧wing resu)lt:
Proposition 44 (Proposition 2 in [8]). The mapping O⃗r : n Gra i
edgei
#Vert=:n⩾1 →S
End∗
n
,∗
skew(Tpoly(M)[1]) is a Lie algebra morphism: O⃗r([β, γ]) = [O⃗r(β), O⃗r(γ)]NR.
In the case when β = •• and γ is a graph cocycle, this fact is a key to the factorization
of the Poisson cocycle condition. Namely in that case the left-hand side O⃗r(d(γ)) evaluates
to zero and the right-hand side, [πS, O⃗r(γ)]NR, evaluated at n + 1 copies of Poisson bi-
vector P , is a linear combination of [[P, O⃗r(γ)(P⊗n)]] and O⃗r(γ)(P⊗(n−1)⊗ [[P, P ]]), where
the former has a nonzero coefficient and the latter is a sum of Leibniz graphs which
evaluates to zero at every Poisson bi-vector P . We refer again to Table 0.4 for statistics.
34 Overview
Proposition 44 is not the only mechanism which provides solutions ♢ to the factor-
ization problem [[P, O⃗r(γ)(P⊗n)]] = ♢(P, [[P, P ]]) of the Poisson cocycle condition. The
above was the original construction by Kontsevich. We can consider more generally the
problem of constructing the (minimal) set of Leibniz graphs needed to express [[P,Qγ]] as
the expansion of a linear combination of Leibniz graphs from that set.
To build all the potentially needed Leibniz graphs, we take the Kontsevich graph
expansion of the left-hand side [[P,Qγ]] of the Poisson cocycle condition, and find all the
Leibniz graphs in whose expansion (again into Kontsevich graphs) we reproduce at least
one previously known graph. Now having expanded these Leibniz graphs into Kontsevich
graphs built of wedges, we have clearly reproduced the original set, but we can also
obtain more (new) Kontsevich graphs — previously not contained in the Poisson cocycle
condition’s left-hand side, or after the previous iterations, now to start. Repeating this
step iteratively, until saturation, we get a large set of Leibniz graphs, see [5, §1.2].
Example 45. For the pentagon-wheel cocycle γ5 and the flow Ṗ = Qγ5(P ), an example
of such saturation is reported in Table 0.7 (Table 1 in [5]).
Table 0.7: The number of skew Leibniz graphs produced iteratively for [[P,Qγ5(P )]].
No. iteration i 1 2 3 4 5 6 7 8
# of graphs 1518 14846 41031 54188 56318 56503 56509 56509
of them new all +13328 +26185 +13157 +2130 +185 +6 none
Lastly, we equate the Kontsevich graph expansions of all these Leibniz graphs, taken
with undetermined coefficients, to the Poisson cocycle condition left-hand side. Solving
the arising linear algebraic system upon the coefficients of Leibniz graphs, we discover
multiple solutions, naturally including Kontsevich’s canonical solution. The properties
of the canonical solution are listed in Table 0.4. Let Ln3 denote the set of Leibniz graphs
with positive differential order over 3 ground vertices and n aerial vertices (of which n−1
are wedges and one is a tripod). The non-uniqueness of Leibniz graph factorizations is
expressed in the following proposition.
Proposition 46 (see Section 3.5.2). The solution of a Leibniz graph factorization problem
is not unique as soon as the number of aerial vertices in the Leibniz graphs exceeds 2;
then there exist sums of Leibniz graphs such that their expansion into sums of Kontsevich
graphs equals zero identically. The nullity of the Leibniz graph expansion map, restricted
to the bi-gradings (3, n− 1) for n = 2, . . . , 5 is reported in Table 0.8.
Table 0.8: The nullity of the Leibniz graph expansion map restricted to Ln3
n 1 2 3 4
#Leibniz graphs in Ln3 1 15 301 6741
Nullity of Leibniz graph expansion map restricted to Ln3 0 0 12 538
Remark 47. Viewing Proposition 44 in the case where γ is a d-coboundary, there is
a trivializing vector field for O⃗r(γ)(P⊗n) defined by graphs; see [8, Corollary 4]. We
illustrate this in Chapter 6 with the coboundary δ6 = d(β6).
35
Let us now look at Proposition 44 from another perspective. Let both β and γ be
graph cocycles. Then on the one hand one can commute the graph cocycles, and on the
other hand one can commute the respective flows. The relation in Proposition 44 can
be verified using the new software. The graph commutator [γ3, γ5] offers the minimal
nontrivial illustration (cf. Section 6.5).
Another use for Proposition 44 is found by evaluating the endomorphism O⃗r(γ) at
tuples different from P⊗n:
Proposition 48 (Theorem 4 in [10]). Let (M,P ) be an a∑ffine finite-dimensional real
Poisson manifold with P = [[V⃗ , P ]] homogeneous. Let γ = a ca · γa be a graph cocycle
consisting of unoriented graphs γa over n vertices and 2n−2 edges (with a fixed ordering of
edges in each γa). Then the 1-vector X⃗(γ, V⃗ , P ) = O⃗r(γ)(V⃗ ⊗ ⊗
n−1
P ), which is obtained
by representing each edge i−−j with the operator21 ∆⃗ij and by (graded-)symmetrizing over
all the ways σ ∈ Sn to send the n-tuple ⊗ ⊗
n−1
V⃗ P into the n vertices in each γa, is a
Poisson cocycle: X⃗ ∈ ker[[P, ·]]. The vector field X⃗ is defined up to adding arbitrary
Poisson 1-cocycles Z⃗ ∈ ker[[P, ·]].
As said, Proposition 48 is illustrated by using two examples of Poisson structures
obtained from R-matrices, see Proposition 36.
Conclusion. Star products. The new software package gcaops developed in this disserta-
tion allows independent verification of results obtained by other software (within different
implementations, written in different languages). The software illustrates the theory and
verifies theoretical predictions. In particular, we know the cyclic weight relations for all
Kontsevich- and Leibniz graphs up to order 5 in h̄, and we know these relations specifi-
cally for those Kontsevich graphs on n = 7 aerial vertices with their in-degrees bounded
by ⩽ 1, that is for the affine Kontsevich graphs in the affine star product ⋆aff mod ō(h̄7).
We verified that the values of weights obtained by Banks–Panzer–Pym satisfy all the
relations found by our software (e.g. the cyclic weight relations for Kontsevich graphs at
h̄5, and the relations from the associativity of ⋆-product at h̄6).
• To advance in the determination of closed-form expressions for the weights, we need
to find more relations between them (in the case of linear relations, we must increase the
rank). Kontsevich claims that the weights have many links to number theory: interesting
numbers are hidden in the weights of graphs. We expect that there are probably also
many more hidden relations between the weights. In this context, it is natural to wonder
if the question of the (ir)rationality of ζ(3)/π3 can be resolved in this way, because
by knowing star products’ high order expansions, we obtain (known and possibly new)
algebraic relations between the Riemann zeta values (with ζ(3) in particular).
• In another direction (as Gutt et al. (2008), for ⋆-product modulo ō(h̄3)), for universal
star products on manifolds with a torsion-free linear connection (the coefficients of the
star product depending in a differential polynomial way on not only the Poisson structure
but also the curvature tensor), an explicit expansion up to orders higher than three—and
its uniqueness—is an open problem. We propose to use a variation of Kontsevich’s graphs,
now with two colors of aerial vertices: some vertices containing copies of the Poisson bi-
vector, and other vertices containing the curvature tensor — in such a way that the graph
encodes a tensorial expression over the curved Poisson manifold. The gcaops software can
be extended to handle different content of aerial vertices in the graphs.
21These superoperators ∆⃗ij , acting from left to right, were introduced by Kontsvich in Ascona ‘96.
36 Overview
• In the proof of the associativity of the Kontsevich star product via the Formality mor-
phism, we have verified the identity Assoc(⋆(P ))(f, g, h) = ♢(P, Jac(P )(·, ·, ·))(f, g, h) up
to some recalculation constants (we give their values). Understanding all the conventions
from which these recalculation constants originate will deepen our understanding of the
work of Kontsevich’s Formality mechanism in deformation quantization.
Poisson flows. In this dissertation we extended the collection of explicit examples of
flows and the respective factorizations, all realized in software. Our initial software im-
plementation in [6] immediately detected the correct balance for the components of the
tetrahedral flow which was invented in 1996. Moreover, we verified the mechanism of
validity of the Poisson cocycle condition which was predicted by Kontsevich in ‘96. For
all the flows under consideration we establish a factorization of the Poisson cocycle con-
dition via the Jacobi identity exactly, taking into account all the normalizations and
conventions.
• We found that the class of rescaled Nambu–Poisson structures P [ρ, a] is closed un-
der the Kontsevich graph flows. We discover that the explicit infinitesimal evolution
ρ̇, ȧ of the functional parameters ρ, a is hypersymmetric (expressed as the total skew-
symmetrization of differential polynomials, with arbitrarily chosen markers). We found
the reason for the tetrahedral flow on rescaled 3D Nambu–Poisson structures to be Pois-
son cohomology trivial, by producing a graph realization of the trivializing vector field
X[ρ, a]; its coefficients are differential polynomials in the functional parameters. It re-
mains to be seen how the evolution works for the five-wheel flow and for Nambu–Poisson
structures in higher dimensions.
• Although the flows are designed to be not universally trivial, all the examples of
flows evaluated at concrete Poisson structures in this dissertation are Poisson cohomology
trivial, without exception. While we have verified the Poisson-triviality of the Kontsevich
graph flows in 2-dimensional Poisson geometries (for γ3, γ5, δ6, γ7) and we have proved the
Poisson-triviality of the tetrahedral flow for the class of 3D Nambu–Poisson brackets, it
is still an open problem why it happens in other cases. This needs further exploration:
in particular, how universal is the (non)triviality of Kontsevich’s graph construction with
respect to the dimension of Poisson manifolds? (We refer to Chapter 6 of this dissertation
where we obtain a graph realization of the Hamiltonians for the trivializing vector fields
in dimension two.)
Graph complex. The Poisson flows discussed above originate from cocycles in the Kon-
tsevich undirected graph complex with the vertex-expanding differential. In this domain
we counted the dimensions of bi-graded parts, verifying the calculations by Willwacher–
Živković. A natural open problem is to predict a universal formula for the coefficients of
graph cocycles. In this direction Willwacher–Rossi (2014) give integral formulas. It would
be interesting to illustrate these formulas by examples, and to find whether alternative
formulas are possible.
Programming. On an average desktop computer, one can study the Kontsevich graph
theory up to around 7 or 8 vertices, dealing with on the order of one million graphs.
Namely, one can verify the associativity of the full star product up to order 6 (and
up to orders 7 and 8 for the restricted case of the affine star product for affine Poisson
structures), find explicit factorizations via Leibniz graphs, calculate relations between the
weights of graphs, do gauge transformations of star products, insert a Poisson structure
into a given flow, factor the Poisson cocycle condition via Leibniz graphs, and more.
The initial implementations kontsevich_graph_series-cpp and graph_complex-cpp
(written in C++) were fast, but their design (as separate programs, communicating by
37
file input/output) was not very convenient for fast experimentation. Adding that level
of interactivity to C++ programs would have been a lot of extra parsing work. The new
software gcaops is based on Python and SageMath, which have interactivity built into
their interpreters; for our purposes the Jupyter notebook interface is very convenient, as
it allows calling the interpreters and receiving not only plain text output but also graph
pictures. High performance is achieved by calling into C/C++ libraries such as nauty for
graphs, libSingular for polynomials, and whatever SageMath uses for linear algebra.
Educational. The software developed for this dissertation could be suitable as a whole or
in part as a new package for SageMath. It has been designed to be consistent with the
conventions of SageMath, it is documented with examples (and its formal description in
Appendix F), and it is intended to be easy to use. Hence it can be used by students to
learn the subjects of deformation quantization and the graph complex. We hope that by
this we may renew the interest in the topic.
Publications. The results of this dissertation are contained in the following publications
and one preprint. All the texts which are published are freely open access from arXiv.
[1] R. Buring and A.V. Kiselev. On the Kontsevich ⋆-product associativity mecha-
nism. Physics of Particles and Nuclei Letters, 14(2), 403–407, 2017. (Preprint
arXiv:1602.09036 [q-alg] – 4 p.) IAEA 48098722
[2] R. Buring and A.V. Kiselev. The expansion ⋆ mod ō(h̄4) and computer-assisted proof
schemes in the Kontsevich deformation quantization. Experimental Math. 31(3)
or (4), 54 p., 2022 (in press). (doi:10.1080/10586458.2019.168046) (Preprint
arXiv:1702.00681 [math.CO] – 77 p.) MR (pending), Zbl. (pending)
[3] R. Buring and A.V. Kiselev. Formality morphism as the mechanism of ⋆-product
associativity: how it works. Collected works Inst. Math., Kyiv 16:1 Symmetry
& Integrability of Equations of Mathematical Physics, 22–43, 2019. (Preprint
arXiv:1907.00639 [q-alg] – 16 p.) Zbl. 143853125
[4] R. Buring, A. V. Kiselev, and N. J. Rutten. The heptagon-wheel cocycle in the
Kontsevich graph complex. J. Nonlin. Math. Phys., 24: Suppl. 1 ‘Local & Nonlocal
Symmetries in Mathematical Physics’, 157–173, 2017. (Preprint arXiv:1710.00658
[math.CO] – 17 p.) MR3750843; Zbl. 142053084
[5] R. Buring, A. V. Kiselev, and N. J. Rutten. Infinitesimal deformations of Pois-
son bi-vectors using the Kontsevich graph calculus. J. Phys.: Conf. Ser., 965:
Proc. XXV Int.conf. ‘Integrable Systems & Quantum Symmetries’ (6–10 June 2017,
CVUT Prague, Czech Republic), Paper 012010, 2018. (Preprint arXiv:1710.02405
[math.CO] – 12 p.)
[6] A. Bouisaghouane, R. Buring, and A.V. Kiselev. The Kontsevich tetrahedral flow
revisited. J. Geom. Phys., 119, 272–285, 2017. (Preprint arXiv:1608.01710 [q-alg]
– 29 p.) MR3661536; Zbl. 137153092
[7] R. Buring, A. V. Kiselev, and N. J. Rutten. Poisson brackets symmetry from the
pentagon-wheel cocycle in the graph complex. Physics of Particles and Nuclei,
49(5): Supersymmetry and Quantum Symmetries 2017, 924–928, 2018. (Preprint
arXiv:1712.05259 [math-ph] – 4 p.) IAEA 52022014
38 Overview
[8] R. Buring and A. V. Kiselev. The orientation morphism: from graph cocycles to
deformations of Poisson structures. J. Phys.: Conf. Ser. 1194: Proc. 32nd Int.
colloquium on Group-theoretical methods in Physics: Group32 (9–13 July 2018,
CVUT Prague, Czech Republic), Paper 012017, 2019. (Preprint arXiv:1811.07878
[math.CO] – 10 p.)
[9] A.V. Kiselev and R. Buring. The Kontsevich graph orientation morphism revisited.
Banach Center Publ. 123 Homotopy algebras, deformation theory & quantization,
123–139, 2021. (Preprint arXiv:1904.13293 [math.CO] – 18 p.) MR4276046; Zbl.
07350411
[10] R. Buring and A.V. Kiselev. Universal cocycles and the graph complex action on
homogeneous Poisson brackets by diffeomorphisms. Physics of Particles and Nu-
clei Letters 17:5 Supersymmetry and Quantum Symmetries 2019, 707–713, 2020.
(Preprint arXiv:1912.12664 [math.SG] – 8 p.) IAEA (pending)
[11] R. Buring, D. Lipper and A. V. Kiselev. The hidden symmetry of Kontsevich’s graph
flows on the spaces of Nambu-determinant Poisson brackets. (submitted) (Preprint
arXiv:2112.03897 [math.SG] – 23+iv p.)
39
Buring R.T. The action of Kontsevich’s graph complex on Poisson structures
and star products: an implementation.
Dissertation submitted for the award of the title Doctor of Natural Sciences to the Faculty
of Physics, Mathematics, and Computer Science at Johannes Gutenberg-University Mainz
in Mainz, Germany.
Abstract. Whenever the associative pointwise product of scalar functions on an affine
manifold is deformed by adding a bi-differential term proportional to a formal deforma-
tion parameter h̄, the deformed product remains associative modulo ō(h̄2) if and only
if the skew-symmetric part of the leading deformation term is a Poisson bracket. The
problem of deformation quantization of Poisson manifolds is to extend such a first-order
deformation to a formal power series in h̄, preserving associativity, with a bi-differential
operator at each order in h̄. In 1997, M. Kontsevich proved the Formality Theorem and
thus solved the problem of deformation quantization for all (affine) Poisson manifolds at
once, by introducing graphs which represent universal differential polynomial expressions
and integrals over configuration spaces of points in the Lobachevsky plane. If one wishes
to work with Kontsevich’s ⋆-product and illustrate this theory explicitly, by expanding
formal power series in h̄, one runs into millions of graphs already at the order h̄7.
In a similar style, Kontsevich’s universal flows on the spaces of Poisson structures
(1996), given by Poisson cocycles in the Lichnerowicz–Poisson cohomology, are built from
graph cocycles in Kontsevich’s graph complex with the vertex-expanding differential. In
this setting the graph cocycles with nine vertices already contain thousands of graphs.
To operate with all the graphs and evaluate the universal formulas at particular Poisson
structures, efficient software is needed.
We develop and present the software package gcaops (Graph Complex Action on Pois-
son Structures) for SageMath, to deal with both the ⋆-products and universal flows on
spaces of Poisson structures. Using this package, we achieve the following: • we expand
the entire Kontsevich ⋆-product, i.e. for generic Poisson structures and with harmonic
weights, up to ō(h̄4); • we assemble ⋆ mod ō(h̄6) from external data by Banks–Panzer–
Pym, and we obtain the star product ⋆aff mod ō(h̄7) for affine Poisson brackets; • we
illustrate the explicit proof of the associativity for the full star product modulo ō(h̄6)
and for the affine star product modulo ō(h̄7); • we verify that the weights found by
Banks–Panzer–Pym (2018) up to ō(h̄6) satisfy many known relations; • we find new
explicit formulas of graph cocycles and universal Poisson cocycles, and • we prove the
factorization of the Poisson cocycle condition via the Jacobi identity in each case.
Although Kontsevich’s universal Poisson flows associated with nonzero graph coho-
mology classes are designed to be not universally Poisson-trivial, we establish that the
flows associated with the grt-related graph cocycles γi are Poisson trivial in these cases:
γ3, γ5, γ7 in two-dimensional Poisson geometries, γ3 for 3D Nambu–Poisson structures,
and γ3 for several (quadratic and cubic) homogeneous polynomial Poisson brackets as-
sociated with some R-matrices for Lie algebras gl2(R) and gl3(R). Finally, we illustrate
explicitly how the tetrahedron γ3 in the graph complex and the resulting flow on the
spaces of Poisson structures act upon the Kontsevich ⋆-product, and we find out whether
this action of the graph cocycle γ3 on Kontsevich’s ⋆-product is gauge trivial or not.
Keywords. Poisson bracket, star-product, deformation quantization, Kontsevich graph
complex, Poisson cohomology, differential graded Lie algebras, software design.
Mathematics Subject Classification (2020). 17B63, 17B70, 18M30, 18M60, 53-04,
53-08, 53D17, 53D55, 68W30, 97-02.
Part I
Computer demonstrations
Abstract
This part provides a tutorial for the gcaops (Graph Complex Action on Pois-
son Structures) software, using the SageMath notebook interface. This software
grew out of the course Deformation Quantization, Graph Complex, and Number
Theory, which was taught in the Dutch national Mastermath program in Fall
2020, with lectures by Arthemy Kiselev and exercise classes led by the author.
Tutorials on November 5th and December 10th (both in 2020) were dedicated to
the demonstration of this software. The present tutorial has been annotated and
extended, with further examples added.
Contents of Part I
0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1 Implementation of star products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2 Implementation of Poisson structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Implementation of Formality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Implementation of the graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5 Examples of graph cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6 Graph complex action on Poisson structures in dimension two . . . . . . . . . . . . . . . . . 151
7 Graph complex action on rank two rescaled Nambu–Poisson structures . . . . . . . . 169
8 Graph complex action on R-matrix Poisson structures . . . . . . . . . . . . . . . . . . . . . . . . 191
9 Graph complex action on star products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Chapter 0
Introduction
The gcaops (Graph Complex Action on Poisson Structures) software is a package written
in Python 3, designed to be used with SageMath version 9.2 or later [15]. It is released
under the MIT free software license. It is available from https://github.com/rburing/
gcaops. An Introduction to SageMath is included as Appendix A in the dissertation.
0.1 Installation
This software can be obtained from https://github.com/rburing/gcaops. Up-to-date
installation instructions are also listed there.
How to install the package:
1. Navigate to https://github.com/rburing/gcaops in a web browser; press the
Code button and click the Download ZIP link.
2. Extract the ZIP file to a directory such as /path/to/gcaops-master.
3. In a terminal (e.g. the SageMath Shell on Windows), run the following:
sage -pip install --upgrade /path/to/gcaops/master
This completes the installation.
4. It is optional but highly recommended to configure a default directory where data
(such as lists of graphs) can be stored, so it doesn’t have to be re-computed each
time.
This can be done by setting the environment variable GCAOPS_DATA_DIR to the path
you desire, before starting SageMath. A convenient way to achieve this is by adding
a line such as the following to SageMath’s shell script sagerc:
export GCAOPS_DATA_DIR='/home/sage/Documents/gcaops_data/'
Be warned that this directory can grow large. If no directory is configured, then
graphs are only stored in memory (which may be limiting).
41
42 CHAPTER 0. INTRODUCTION
5. It is optional but convenient to enable the importing of all names from the
gcaops package (e.g. UndirectedGraphComplex) into the global namespace of every
SageMath session, so that the functionality can be used immediately.
This can be done by adding the following line to SageMath’s startup script
init.sage:
from gcaops.all import *
Now the functionality can be imported, e.g.:
[1]: from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
from gcaops.algebra.differential_polynomial_ring import DifferentialPolynomialRing
from gcaops.graph.undirected_graph_complex import UndirectedGraphComplex
from gcaops.graph.directed_graph_complex import DirectedGraphComplex
Or import all the relevant functionality at once:
[2]: from gcaops.all import *
0.2 Functions
The meaning of the word “function” depends on the context. We shall presently consider
three types of functions: those given (as coordinate expressions) by polynomials, by
differential polynomials, and by symbolic expressions. We now describe how to work
with each of these in SageMath.
0.2.1 Polynomials
We can restrict ourselves to functions defined by polynomials. For our purposes (that
is, to work with formulas), polynomial functions (over a field of characteristic zero) are
identified with polynomials in a polynomial ring.
[3]: R.<y1,y2,y3> = PolynomialRing(QQ); R
[3]: Multivariate Polynomial Ring in y1, y2, y3 over Rational Field
This simultaneously defines the variables y1, y2, y3 and the ring R which acts as a parent
for all the polynomials in these variables. All objects f in SageMath which are “elements”
of some sort have a parent object f.parent() that they belong to. This is helpful for
bookkeeping purposes, and in particular for conversion from one type of object to another,
as will be seen later.
We can now define polynomials:
[4]: f = y1^2 + y2^2 - 1
[5]: f.parent() is R
[5]: True
0.2. FUNCTIONS 43
The arithmetic with polynomials works as usual, and we can also differentiate polynomials
with respect to the variables:
[6]: f.derivative(y1)
[6]: 2*y1
[7]: diff(f, y1)
[7]: 2*y1
For the notion of “parameters” in this context, one can define a polynomial ring over (the
fraction field of) another polynomial ring.
[8]: C.<c1,c2> = PolynomialRing(QQ)
S.<w1,w2,w3> = PolynomialRing(C)
c1^2*w1 + (c1+c2)*w2^2 + w3^3
[8]: w3^3 + (c1 + c2)*w2^2 + c1^2*w1
0.2.2 Differential polynomials
We will often work with expressions involving sums of products of derivatives. The gcaops
package provides differential polynomials, which are polynomials in jet variables.
Define a differential polynomial ring with fibre variable u and base variables z1, z2:
[9]: D = DifferentialPolynomialRing(QQ, ('u',), ('z1','z2'), max_differential_orders=[3]);␣
↪→D
[9]: Differential Polynomial Ring in z1, z2, u, u_z1, u_z2, u_z1z1, u_z1z2, u_z2z2,
u_z1z1z1, u_z1z1z2, u_z1z2z2, u_z2z2z2 over Rational Field
Retrieve the fibre and base variables:
[10]: u, = D.fibre_variables()
z1, z2 = D.base_variables()
Arithmetic with differential polynomials works as usual. The (default) derivative in this
context is the total derivative:
[11]: (u^2).derivative(z1)
[11]: 2*u*u_z1
[12]: diff(u^2, z1)
[12]: 2*u*u_z1
It is also possible to take partial derivatives:
[13]: (u^2).partial_derivative(u)
[13]: 2*u
44 CHAPTER 0. INTRODUCTION
0.2.3 Symbolic expressions
Finally we can consider functions defined by symbolic expressions (possibly more general
than polynomials).
For this we first define symbolic variables, which we interpret as coordinates in a chart:
[14]: var('x1,x2,x3')
[14]: (x1, x2, x3)
The commutative arithmetic with these variables works as usual. An example of a func-
tion of (x1, x2, x3) is:
[15]: g = x1^2*sin(x2^2)*exp(1/x3); g
[15]: x1^2*e^(1/x3)*sin(x2^2)
We can differentiate symbolic expressions:
[16]: diff(g, x3)
[16]: -x1^2*e^(1/x3)*sin(x2^2)/x3^2
The identification of functions with symbolic expressions does not limit us to elemen-
tary functions, because we can use symbolic (indeterminate) functions, evaluated at the
coordinates:
[17]: h = function('h')(x1,x2,x3); h
[17]: h(x1, x2, x3)
[18]: diff(h, x1)
[18]: diff(h(x1, x2, x3), x1)
Symbolic expressions in SageMath are elements of a “ring”: the Symbolic Ring SR.
[19]: SR
[19]: Symbolic Ring
Of course this is not implemented as a set containing all symbolic expressions, but it acts
as a parent for all symbolic expressions; that is, for any particular symbolic expression
which actually occurs.
Chapter 1
Implementation of star products
This chapter is an introduction to the usage of star products, polydifferential opera-
tors, the Hochschild complex, and the associativity as a Maurer–Cartan equation in the
gcaops software package. The theory is standard, see e.g. Chapter 11 and references
therein, as well as the foundational papers by Gerstenhaber [22] [23], Hochschild [25],
Groenewold [24], Weyl [34], DeWilde–Lecomte [14], Fedosov [19]; the expository works
by Esposito [18], Cattaneo–Indelicato [12], and Gengoux–Pichereau–Vanhaecke [30]; from
a historic perspective Bayen–Flato–Frønsdal–Lichnerowicz–Sternheimer [2], and also the
MSc thesis of Willem de Kok at the University of Groningen [13] for interpretations and
more references.
1.1 Star products
Let f, g, h be scalar functions on R2:
[1]: var('x,y')
f = function('f')(x,y)
g = function('g')(x,y)
h = function('h')(x,y)
The ordinary pointwise product of functions is associative:
[2]: bool(f*(g*h) == (f*g)*h)
[2]: True
The pointwise product is also commutative:
[3]: g*f
[3]: f(x, y)*g(x, y)
[4]: bool(f*g == g*f)
[4]: True
We want to deform the pointwise product f · g to a product f ⋆ g = f · g + h̄B1(f, g) +
h̄2B2(f, g) + . . . where h̄ is a formal parameter, Bk is a bi-linear bi-differential operator
on C∞(Rd) for each k ⩾ 1, and ⋆ remains associative (but not necessarily commutative).
45
46 CHAPTER 1. IMPLEMENTATION OF STAR PRODUCTS
Let us keep in mind that in physical applications, the formal parameter equals ih̄ where
2
this h̄ = h/2π is the Planck constant.
Example 1. Here is an example ⋆1 of such a star-product modulo ō(h̄2):
[5]: var('hbar')
star1 = lambda f,g: f*g + hbar*x*(diff(f,x)*diff(g,y) - diff(f,y)*diff(g,x)) + \
hbar^2*x^2*(diff(f,x,x)*diff(g,y,y) - 2*diff(f,x,y)*diff(g,x,y) +␣
↪→diff(f,y,y)*diff(g,x,x))/2 + \
hbar^2*x*(diff(f,y,y)*diff(g,x) - diff(f,x,y)*diff(g,y) + diff(f,x)*diff(g,y,y) -␣
↪→diff(f,y)*diff(g,x,y))/3 + \
hbar^2*diff(f,y)*diff(g,y)/6
The product is noncommutative:
[6]: (star1(f,g) - star1(g,f)).coefficient(hbar)
[6]: -2*(diff(f(x, y), y)*diff(g(x, y), x) - diff(f(x, y), x)*diff(g(x, y), y))*x
[7]: star1(x,y) - star1(y,x)
[7]: 2*hbar*x
The product is associative modulo ō(h̄2):
[8]: assoc1 = (star1(star1(f,g),h) - star1(f,star1(g,h))).expand()
[9]: assoc1.subs(hbar==0), assoc1.coefficient(hbar), assoc1.coefficient(hbar^2)
[9]: (0, 0, 0)
The above 3-tuple contains the free term in the associator and the terms at h̄1 and h̄2.
Example 2. Here is another choice ⋆2 (cf. M. Kontsevich’s [29]):
[10]: star2 = lambda f,g: f*g + hbar*x*y*(diff(f,x)*diff(g,y) - diff(f,y)*diff(g,x))/2 + \
hbar^2*x^2*y^2*(diff(f,x,x)*diff(g,y,y) + diff(f,y,y)*diff(g,x,x) -␣
↪→2*diff(f,x,y)*diff(g,x,y))/8 + \
hbar^2*x*y*(x*diff(f,x,x)*diff(g,y) + y*diff(f,y,y)*diff(g,x) -␣
↪→x*diff(f,x,y)*diff(g,x) - y*diff(f,x,y)*diff(g,y))/12 + \
hbar^2*x*y*(y*diff(f,x)*diff(g,y,y) + x*diff(f,y)*diff(g,x,x) -␣
↪→y*diff(f,y)*diff(g,x,y) - x*diff(f,x)*diff(g,x,y))/12 + \
hbar^2*(x*y*diff(f,x)*diff(g,y) + x*y*diff(f,y)*diff(g,x) - y^2*diff(f,y)*diff(g,y) -␣
↪→x^2*diff(f,x)*diff(g,x))/24
It is also associative modulo ō(h̄2):
[11]: assoc2 = (star2(star2(f,g),h) - star2(f,star2(g,h))).expand()
[12]: assoc2.subs(hbar==0), assoc2.coefficient(hbar), assoc2.coefficient(hbar^2)
[12]: (0, 0, 0)
We have the property that x ⋆2 y = exp(h̄)(y ⋆2 x), see §1.30 in [29].
[13]: star2_symm = (star2(x,y) - (1 + hbar + hbar^2/2)*star2(y,x)).expand()
1.2. GAUGE TRANSFORMATIONS 47
[14]: star2_symm.subs(hbar==0), star2_symm.coefficient(hbar), star2_symm.coefficient(hbar^2)
[14]: (0, 0, 0)
Example 3. Here is an example ⋆3 in which the first-order term B1 is a symmetric
operator:
[15]: star3 = lambda f,g: f*g + hbar*(diff(f,x)*diff(g,y) + diff(f,y)*diff(g,x)) + \
hbar^2*(diff(f,x,x)*diff(g,y,y) + 2*diff(f,x,y)*diff(g,x,y) +␣
↪→diff(f,y,y)*diff(g,x,x))/2
[16]: assoc3 = (star3(star3(f,g),h) - star3(f,star3(g,h))).expand()
[17]: assoc3.subs(hbar==0), assoc3.coefficient(hbar), assoc3.coefficient(hbar^2)
[17]: (0, 0, 0)
1.2 Gauge transformations
We can transfo∑rm a star-product using a gauge transformation T : C
∞(M)[[h̄]] →
C∞(M)[[h̄]], namely a formally invertible unary R[[h̄]]-linear differential operator of the
form T = id+ ∞ kk=1 h̄ Dk where each Dk is a finite order differential operator. Gauge
transformations T yield the gauged star-products f ⋆′ g = T−1(T (f) ⋆ T (g)).
Example 4. We transform the star-product ⋆1 from the previous section using a gauge
transformation 2T1(f) = f + h̄2 ∂ f∂y2 :
[18]: T1 = lambda f: f + hbar^2*diff(f,y,y)
[19]: T1_inverse = lambda f: f - hbar^2*diff(f,y,y)
The gauged star product is f ⋆′1 g = T−11 (T1(f) ⋆1 T1(g)):
[20]: star1_gauged = lambda f,g: T1_inverse(star1(T1(f),T1(g))).expand()
This gauge transformation leaves the first order term untouched:
[21]: bool(star1_gauged(f,g).coefficient(hbar) == star1(f,g).coefficient(hbar))
[21]: True
The second order term of ⋆′1 has one summand fewer than ⋆1:
[22]: (star1_gauged(f,g).coefficient(hbar^2) - star1(f,g).coefficient(hbar^2)).expand()
[22]: -2*diff(f(x, y), y)*diff(g(x, y), y)
The gauged star product is also associative modulo ō(h̄2):
[23]: gauged_assoc = star1_gauged(f,star1_gauged(g,h)) - star1_gauged(star1_gauged(f,g),h)
[24]: gauged_assoc.subs(hbar==0), gauged_assoc.coefficient(hbar), gauged_assoc.
↪→coefficient(hbar^2)
48 CHAPTER 1. IMPLEMENTATION OF STAR PRODUCTS
[24]: (0, 0, 0)
Example 5. We transform the symmetric star-product ⋆3 from the previous section
using a gauge transformation 2T2(f) = f + h̄ ∂ f :∂x∂y
[25]: T2 = lambda f: f + hbar*diff(f,x,y)
[26]: T2_inverse = lambda f: f - hbar*diff(f,x,y) + hbar^2*diff(f,x,x,y,y)
[27]: star3_gauged = lambda f,g: T2_inverse(star3(T2(f),T2(g))).expand()
In the transformed product ⋆′3 the first-order term is zero:
[28]: star3_gauged(f,g).coefficient(hbar)
[28]: 0
The product ⋆′3 is also associative modulo ō(h̄2):
[29]: gauged_assoc3 = star3_gauged(f,star3_gauged(g,h)) - star3_gauged(star3_gauged(f,g),h)
[30]: gauged_assoc3.subs(hbar==0), gauged_assoc3.coefficient(hbar), gauged_assoc3.
↪→coefficient(hbar^2)
[30]: (0, 0, 0)
We inspect the vanishing of the associator for ⋆′3 only modulo ō(h̄) because the inverse
T−12 is taken only up to ō(h̄). By expanding T−12 mod ō(h̄2) we could inspect the vanishing
of the associator already up to ō(h̄2).
1.3 Polydifferential operators
We now rewrite the associativity equation using more algebraic structures; this will be
helpful later.
Namely, we introduce the algebra of polydifferential operators; it is endowed with the
structure of differential graded Lie algebra by using the Gerstenhaber bracket (see below).
[31]: from gcaops.algebra.polydifferential_operator import PolyDifferentialOperatorAlgebra
D.<ddx,ddy> = PolyDifferentialOperatorAlgebra(SR, var('x,y'),␣
↪→simplify='expand',is_zero='is_trivial_zero'); D
[31]: Algebra of multi-linear polydifferential operators over Symbolic Ring with coordinates
(x, y) and derivatives (ddx, ddy)
In this algebraic setting we can define a ⋆-product as an operator as follows (cf. ⋆1 in
Example 1):
[32]: m1 = D.multiplication_operator() + hbar*x*(D(ddx,ddy) - D(ddy,ddx)) + \
hbar^2*x^2*(D(ddx^2, ddy^2) - 2*D(ddx*ddy, ddx*ddy) + D(ddy^2, ddx^2))/2 + \
hbar^2*x*(D(ddy^2, ddx) - D(ddx*ddy, ddy) + D(ddx, ddy^2) - D(ddy,ddx*ddy))/3 + \
hbar^2*D(ddy,ddy)/6; m1
1.3. POLYDIFFERENTIAL OPERATORS 49
[32]: (id ⊗ id) + (hbar*x)*(ddx ⊗ ddy) + (-hbar*x)*(ddy ⊗ ddx) + (1/2*hbar^2*x^2)*(ddx^2 ⊗
ddy^2) + (-hbar^2*x^2)*(ddx*ddy ⊗ ddx*ddy) + (1/2*hbar^2*x^2)*(ddy^2 ⊗ ddx^2) +
(1/3*hbar^2*x)*(ddy^2 ⊗ ddx) + (-1/3*hbar^2*x)*(ddx*ddy ⊗ ddy) + (1/3*hbar^2*x)*(ddx ⊗
ddy^2) + (-1/3*hbar^2*x)*(ddy ⊗ ddx*ddy) + (1/6*hbar^2)*(ddy ⊗ ddy)
The natural extension of composition to multi-linear operators is the following.
Definition. The pre-Lie product of (k1 + 1)-linear polydifferential operator Φ1 and of
(k2+1)-linear polydifferential operator Φ2 is the sum of insertions of Φ2 into an argument
slot of Φ1,
∑k1
(Φ1◦Φ2)(a ik20⊗. . .⊗ak1+k2) = (−) Φ1(a0⊗· · ·⊗Φ2(ai⊗· · ·⊗ai+k2)⊗ai+k2+1⊗· · ·⊗ak1+k2).
i=0
The Gerstenhaber bracket is the shifted-graded commutator of pre-Lie products,
[Φ1,Φ2]G = Φ1 ◦ Φ k1·k22 − (−) Φ2 ◦ Φ1.
The Gerstenhaber bracket is thus shifted-graded skew-symmetric.
In the new algebraic language, a bi-linear bi-differential operator m is associative if and
only if it satisfies the master equation [m,m]G = 0.
(In what follows, we shall take the tensor products of algebras with the algebra of formal
power series k[[h̄]], so that the bi-linear bi-differential operator m can be a finite order
bi-differential operator at every finite order of h̄, but unbounded-order overall.)
[33]: m1_assoc = (1/2)*m1.bracket(m1)
[34]: m1_assoc == m1.insertion(0,m1) - m1.insertion(1,m1)
[34]: True
[35]: m1_assoc.subs(hbar==0), m1_assoc.coefficient(hbar), m1_assoc.coefficient(hbar^2)
[35]: (0, 0, 0)
Similarly, we can define ⋆2 from Example 2:
[36]: m2 = D.multiplication_operator() + hbar*x*y*(D(ddx,ddy) - D(ddy,ddx))/2 + \
hbar^2*x^2*y^2*(D(ddx^2,ddy^2) + D(ddy^2,ddx^2) - 2*D(ddx*ddy, ddx*ddy))/8 + \
hbar^2*x*y*(x*D(ddx^2,ddy) + y*D(ddy^2,ddx) - x*D(ddx*ddy,ddx) - y*D(ddx*ddy,ddy))/12␣
↪→+ \
hbar^2*x*y*(y*D(ddx,ddy^2) + x*D(ddy,ddx^2) - y*D(ddy,ddx*ddy) - x*D(ddx,ddx*ddy))/12␣
↪→+ \
hbar^2*(x*y*D(ddx,ddy) + x*y*D(ddy,ddx) - y^2*D(ddy,ddy) - x^2*D(ddx,ddx))/24
[37]: m2_assoc = (1/2)*m2.bracket(m2)
[38]: m2_assoc.subs(hbar==0), m2_assoc.coefficient(hbar), m2_assoc.coefficient(hbar^2)
[38]: (0, 0, 0)
Likewise, ⋆3 from Example 3:
50 CHAPTER 1. IMPLEMENTATION OF STAR PRODUCTS
[39]: m3 = D.multiplication_operator() + hbar*(D(ddx,ddy) + D(ddy,ddx)) + \
hbar^2*(D(ddx^2,ddy^2) + 2*D(ddx*ddy,ddx*ddy) + D(ddy^2,ddx^2))/2; m3
[39]: (id ⊗ id) + (hbar)*(ddx ⊗ ddy) + (hbar)*(ddy ⊗ ddx) + (1/2*hbar^2)*(ddx^2 ⊗ ddy^2) +
(hbar^2)*(ddx*ddy ⊗ ddx*ddy) + (1/2*hbar^2)*(ddy^2 ⊗ ddx^2)
[40]: m3_assoc = (1/2)*m3.bracket(m3)
[41]: m3_assoc.subs(hbar==0), m3_assoc.coefficient(hbar), m3_assoc.coefficient(hbar^2)
[41]: (0, 0, 0)
The Gerstenhaber bracket satisfies its own shifted-graded Jacobi identity,
(−)(|a|−1)(|c|−1)[a, [b, c]G]G + (−)(|b|−1)(|a|−1)[b, [c, a]G]G + (−)(|c|−1)(|b|−1)[c, [a, b]G]G = 0,
where |a| is the arity (number of arguments) of the polydifferential operator a.
Here is an example (for brevity, we calculate the arity signs by hand):
[42]: jac_m123 = (-1)*m1.bracket(m2.bracket(m3)) + (-1)*m2.bracket(m3.bracket(m1)) +␣
↪→(-1)*m3.bracket(m1.bracket(m2))
[43]: jac_m123
[43]: 0
Taking the Gerstenhaber bracket with an associative product m is called the Hochschild
differential, dH = [m,−]G.
Let µ be the ordinary product in the algebra of scalar functions C∞(Rd); the respec-
tive Hochschild differential is implemented as the hochschild_differential method.
Indeed, it is a differential:
[44]: m1.hochschild_differential().hochschild_differential()
[44]: 0
Moreover, taking the bracket with a product which is associative modulo ō(h̄n) is also a
differential modulo ō(h̄n):
[45]: m2_diff_m1 = m1.bracket(m1.bracket(m2))
m2_diff_m1.subs(hbar==0), m2_diff_m1.coefficient(hbar), m2_diff_m1.coefficient(hbar^2)
[45]: (0, 0, 0)
We recall from G∑erstenhaber [23] that the problem of extending an associative product
f ⋆ g = f · g + nk=1∑h̄
kBk(f, g) + ō(h̄
n), already given modulo ō(h̄n) for n ⩾ 1, to a
next-order associative product modulo ō(h̄n+1) is equivalent to expressing the Hochschild
3-cocycle C 1n+1 = − i+j=n+1[Bi, Bj] as a 3-coboundary d2 H(Bn+1) with respect to the
i ̸=0,j ̸=0
Hochschild differential dH = [µ,−] containing the leading term associative structure µ at
h̄0.
Let us define the object C2 for the product ⋆1 and show that it is a Hochschild 3-cocycle:
1.3. POLYDIFFERENTIAL OPERATORS 51
[46]: C2 = -(1/2)*(m1.coefficient(hbar).bracket(m1.coefficient(hbar)))
C2.hochschild_differential()
[46]: 0
Indeed, it equals the 3-coboundary dH(B2):
[47]: m1.coefficient(hbar^2).hochschild_differential() == C2
[47]: True
Also, C3 is a Hochschild 3-cocycle:
[48]: C3 = -(1/2)*(m1.coefficient(hbar).bracket(m1.coefficient(hbar^2)) + m1.
↪→coefficient(hbar^2).bracket(m1.coefficient(hbar)))
C3.hochschild_differential()
[48]: 0
The associativity of a binary operator ⋆ = µ+B can be expressed as the Maurer–Cartan
equation for B, which reads dH(B) + 1 [B,B] = 0.2
Let us give an example based on ⋆1:
[49]: B = m1 - D.multiplication_operator()
mc = B.hochschild_differential() + (1/2)*B.bracket(B)
mc.subs(hbar==0), mc.coefficient(hbar), mc.coefficient(hbar^2)
[49]: (0, 0, 0)
This is relevant for the Formality morphism which sends the Maurer–Cartan elements
P in the Poisson world to the Maurer–Cartan elements B in the associative world (see
Chapter 3).
Gauge transformations T (which we discussed in Section 1.2) mod ō(h̄k) which are con-
centrated in degree k (that is, without terms at h̄` for ℓ =6 0 and ℓ =6 k), act on associative
star-product expansions modulo ō(h̄k) in a particular way, namely by adding Hochschild
coboundaries: ∑
Proposition. If ⋆′ = h̄nB′n mod ō(h̄k) is a star-product expansion o∑btained by a gauge
transformation of the form T = id+h̄kTk mod ō(h̄k) acting on ⋆ = h̄nBn mod ō(h̄k)
via f ⋆′ g = T−1(T (f) ⋆ T (g)), then the difference B′k − Bk is a Hochschild coboundary;
specifically it is none other than dH(Tk).
Example 6. We revisit the gauge transformation 2T1(f) = f + h̄2 ∂ f2 of ⋆1 mod ō(h̄2)∂y
from Example 4.
[50]: t1 = D.identity_operator() + hbar^2*ddy^2; t1
[50]: id + (hbar^2)*ddy^2
We now have k = 2, so that the gauge transformation acts nontrivially at h̄2 and its
inverse is taken modulo ō(h̄2).
[51]: t1_inverse = D.identity_operator() - hbar^2*ddy^2; t1_inverse
52 CHAPTER 1. IMPLEMENTATION OF STAR PRODUCTS
[51]: id + (-hbar^2)*ddy^2
We (re)calculate the gauged star-product expansion ⋆′ mod ō(h̄21 ):
[52]: m1_gauged = t1_inverse.insertion(0, m1.insertion(0, t1).insertion(1, t1))
At h̄2, the difference ⋆′1 − ⋆1 consists of a single term:
[53]: m1_gauged.coefficient(hbar^2) - m1.coefficient(hbar^2)
[53]: (-2)*(ddy ⊗ ddy)
That term is indeed a Hochschild coboundary 2dH( ∂ 2 ), i.e. the Hochschild differential∂y
applied to the second-order part in the gauge transformation:
[54]: (ddy^2).hochschild_differential()
[54]: (-2)*(ddy ⊗ ddy)
This line of reasoning about gauge transformations is continued in Chapter 9.
Each ⋆-product gives rise to a bracket on∣ functions defined by
{ } f ⋆ g − g ⋆ ff, g ? = ∣∣∣ = 1(B (f, g)−B2 1 1(g, f)).2h̄ h̄=0
Example 7. Here are the brackets associated with three star products in this section:
{−,−}?1 :
[55]: m1.coefficient(hbar).skew_symmetrization()/2
[55]: (x)*(ddx ⊗ ddy) + (-x)*(ddy ⊗ ddx)
{−,−}?2 :
[56]: m2.coefficient(hbar).skew_symmetrization()/2
[56]: (1/2*x*y)*(ddx ⊗ ddy) + (-1/2*x*y)*(ddy ⊗ ddx)
{−,−}?3 :
[57]: m3.coefficient(hbar).skew_symmetrization()/2
[57]: 0
By construction the bracket {f, g}? associated with a star product ⋆ is bi-linear and
skew-symmetric. Moreover the associativity of ⋆ mod ō(h̄1) implies that {f, g}? is a bi-
derivation, and associativity mod ō(h̄2) implies that {f, g}? satisfies the Jacobi identity
{f, {g, h}?}? + {g, {h, f}?}? + {h, {f, g}?}? = 0.
These properties of {f, g}? are the defining properties of a Poisson bracket. We will study
Poisson brackets in the next chapter, and in Chapter 3 we explore Kontsevich’s solution
to the natural inverse problem of constructing a star-product ⋆ such that {−,−}? equals
a given Poisson bracket.
1.3. POLYDIFFERENTIAL OPERATORS 53
See [12] and references therein for more details about the algebraic structures in this
section.

Chapter 2
Implementation of Poisson
structures
This chapter is an introduction to the usage of Poisson structures and (stable) Poisson
cohomology in the gcaops software package; the theory is standard (see e.g. [30]).
2.1 Superfunctions
Superfunctions in a chart are polynomials in odd variables ξk satisfying the anticommu-
tation relations ξjξi = −ξiξj (in particular ξ2i = 0) with functions as coefficients of those
polynomials.
Define an algebra of superfunctions on R3 with variables x1, x2, x3 as even coordinates
(the symbolic ring SR is considered to be the base ring):
[1]: from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
SA.<xi1,xi2,xi3> = SuperfunctionAlgebra(SR, var('x1,x2,x3'))
print(SA)
Superfunction algebra over Symbolic Ring with even coordinates (x1, x2, x3) and odd
coordinates (xi1, xi2, xi3)
This simultaneously defines the odd coordinates xi1, xi2, xi3 and the object SA which
acts as a parent for all the superfunctions depending on these variables. We can create
superfunctions by entering them as polynomials in the odd coordinates.
Do some basic arithmetic:
[2]: xi2*xi1
[2]: (-1)*xi1*xi2
[3]: xi1^2
[3]: 0
[4]: x1*xi1 + x2*xi2
55
56 CHAPTER 2. IMPLEMENTATION OF POISSON STRUCTURES
[4]: (x1)*xi1 + (x2)*xi2
Retrieve the coefficients of a superfunction (as a polynomial in the odd variables ξk):
[5]: V = x1*xi1 + x2*xi2
V[0], V[1]
[5]: (x1, x2)
[6]: B = x1*xi2*xi3 + x2*xi3*xi1 + x3*xi1*xi2
B[0,2]
[6]: -x2
[7]: B[2,0]
[7]: x2
Calculate even and odd derivatives (note e.g. ∂ (ξ ∂ ∂
∂ξ 1
ξ2) = (−ξ2ξ1) = − (ξ ξ ) = −ξ ):
2 ∂ξ2 ∂ξ 2 1 12
[8]: (x2^2*xi1*xi2).diff(x2)
[8]: (2*x2)*xi1*xi2
[9]: (xi1*xi2).diff(xi1)
[9]: xi2
[10]: (xi1*xi2).diff(xi2)
[10]: (-1)*xi1
By default, the coefficients of results are not automatically simplified or expanded (on
the other hand, the products of supervariables ξi1 · ξi2 · . . . are normalized by the ordering
i1 < i2 < · · ·):
[11]: Z = (x1+x2)^2*xi1*xi2 + (x1^2 + 2*x1*x2 + x2^2)*xi2*xi1; Z
[11]: ((x1 + x2)^2 - x1^2 - 2*x1*x2 - x2^2)*xi1*xi2
But the expression can be simplified manually, by calling the expand method on each
coefficient:
[12]: Z.map_coefficients(lambda z: z.expand())
[12]: 0
For convenience, in the definition of the superfunction algebra we can pass a simplify
method to be applied to the coefficients after each operation:
[13]: SA.<xi1,xi2,xi3> = SuperfunctionAlgebra(SR, var('x1,x2,x3'), simplify='expand',␣
↪→is_zero='is_trivial_zero')
print(SA)
Superfunction algebra over Symbolic Ring with even coordinates (x1, x2, x3) and odd
coordinates (xi1, xi2, xi3)
2.2. VECTOR FIELDS 57
Now the tool simplfies all coefficients before displaying them:
[14]: W = (x1+x2)^2*xi1 - (x1^2 + 2*x1*x2 + x2^2)*xi1; W
[14]: 0
2.2 Vector fields
Vector fields X i ∂ i can be identified with superfunctions X iξi which are linear in the odd∂x
coordinates ξi:
[15]: X = x1*xi1
Y = x2*xi1 + x3*xi2
[16]: X + Y
[16]: (x1 + x2)*xi1 + (x3)*xi2 ∑ ( )
The Lie bracket of vector fields is [X,Y ] = X i ∂j iY j − Y i ∂ Xj ∂ :∂x ∂xi ∂xj
[17]: X.bracket(Y)
[17]: (-x2)*xi1
2.3 Bi-vector fields
Bi-vector fields Bij ∂ ∧ ∂i j can be identified with superfunctions Bijξiξj quadratic in ξk.∂x ∂x
An example of such an object is a wedge product of vector fields:
[18]: X*Y
[18]: (x1*x3)*xi1*xi2
The Schouten bracket (or odd Poisson bracket on a finite-dimensional (super)manifold of
finite (super)dimension (d|d)),
←− −→ ←− −→
∂ ∂ ∂ ∂
[[A,B]] = A B − A B,
∂ξk ∂xk ∂xk ∂ξk
is the natural extension of the Lie bracket of vector fields on the underlying manifold of
dimension d.
[19]: Y.bracket(X*Y)
[19]: (x2*x3)*xi1*xi2
[20]: Y*X.bracket(Y)
[20]: (x2*x3)*xi1*xi2
Let B be a generic bi-vector field:
58 CHAPTER 2. IMPLEMENTATION OF POISSON STRUCTURES
[21]: B = function('B12')(x1,x2,x3)*xi1*xi2 + function('B13')(x1,x2,x3)*xi1*xi3 +␣
↪→function('B23')(x1,x2,x3)*xi2*xi3; B
[21]: (B12(x1, x2, x3))*xi1*xi2 + (B13(x1, x2, x3))*xi1*xi3 + (B23(x1, x2, x3))*xi2*xi3
Then [[B,X]] is the following bi-vector field:
[22]: B.bracket(X)
[22]: (-x1*diff(B12(x1, x2, x3), x1) + B12(x1, x2, x3))*xi1*xi2 + (-x1*diff(B13(x1, x2, x3),
x1) + B13(x1, x2, x3))*xi1*xi3 + (-x1*diff(B23(x1, x2, x3), x1))*xi2*xi3
More generally an m-vector field is represented by a homogeneous polynomial of degree
m in the odd coordinates ξk.
2.4 Poisson structures
Poisson structures are bi-vector fields P satisfying the Jacobi identity 1 [[P, P ]] = 0.
2
Here is an example of a rescaled Nambu–Poisson structure:
[23]: rho = x1^2 - x2*x3
a = (x1^2 + x2^2 + x3^2)/2
P = rho*(diff(a,x1)*xi2*xi3 + diff(a,x2)*xi3*xi1 + diff(a,x3)*xi1*xi2); P
[23]: (x1^3 - x1*x2*x3)*xi2*xi3 + (-x1^2*x2 + x2^2*x3)*xi1*xi3 + (x1^2*x3 - x2*x3^2)*xi1*xi2
[24]: P.bracket(P)
[24]: 0
Another natural class of Poisson structures consists of those of the form P = E ∧ V ,
where E = xi ∂ i is the Euler vector field and V is a vector field such that each of its∂x
coefficients is a homogeneous polynomial of the same degree.
[25]: E = x1*xi1 + x2*xi2 + x3*xi3
V = (x1^2-x2*x3)*xi1 + (x2^2-x1*x3)*xi2 + (x3^2-x1*x2)*xi3
P_wedge = E*V; P_wedge
[25]: (-x1^2*x2 + x1*x2^2 - x1^2*x3 + x2^2*x3)*xi1*xi2 + (-x1^2*x2 - x1^2*x3 + x1*x3^2 +
x2*x3^2)*xi1*xi3 + (-x1*x2^2 - x2^2*x3 + x1*x3^2 + x2*x3^2)*xi2*xi3
[26]: P_wedge.bracket(P_wedge)
[26]: 0
2.5 Poisson complex
The graded Jacobi identity for the Schouten bracket implies that ∂P = [[P,−]] for a
Poisson structure P defines a differential on the space of multivector fields (∂2P = 0).
We illustrate this for the Poisson structure P from the previous section:
2.6. HOMOGENEOUS POLYNOMIAL POISSON COMPLEX 59
[27]: P.bracket(P.bracket(x1^2 + x2^3))
[27]: 0
[28]: P.bracket(P.bracket(x1*xi1 + x2*xi2))
[28]: 0
Hence, there is the Poisson complex of P , where the cochains are multivector fields
(starting with functions as 0-vector fields).
An m-vector field in the kernel of the Poisson differential is called a Poisson m-cocycle:
[29]: H = x1^2 + x2^2 + x3^2
P.bracket(H)
[29]: 0
[30]: V0 = (x1^2*x3 - x2*x3^2)*xi2 + (-x1^2*x2 + x2^2*x3)*xi3
P.bracket(V0)
[30]: 0
So, H is a Poisson 0-cocycle for the bi-vector P, and an example of a Poisson 1-cocycle is
V0.
An m-vector field in the image of the Poisson differential is called a Poisson m-
coboundary:
[31]: E = x1*xi1 + x2*xi2 + x3*xi3
PbracketE = P.bracket(E); PbracketE
[31]: (x1^2*x2 - x2^2*x3)*xi1*xi3 + (-x1^2*x3 + x2*x3^2)*xi1*xi2 + (-x1^3 +
x1*x2*x3)*xi2*xi3
The object E is the Euler vector field, and the Poisson structure P is homogeneous:
[32]: PbracketE == -P
[32]: True
Every Poisson coboundary is a Poisson cocycle:
[33]: P.bracket(PbracketE)
[33]: 0
A Poisson cohomology class is the equivalence class of a Poisson cocycle modulo arbitrary
Poisson coboundaries for that Poisson differential.
2.6 Homogeneous polynomial Poisson complex
Construct the algebra of polynomial superfunctions over a polynomial ring Q[x1, x2, x3]:
60 CHAPTER 2. IMPLEMENTATION OF POISSON STRUCTURES
[34]: PSA = SuperfunctionAlgebra(PolynomialRing(QQ, names='x1,x2,x3'), names='xi1,xi2,xi3')
print(PSA)
Superfunction algebra over Multivariate Polynomial Ring in x1, x2, x3 over Rational
Field with even coordinates (x1, x2, x3) and odd coordinates (xi1, xi2, xi3)
Consider P as a homogeneous polynomial Poisson structure, by converting it into an
element of PSA:
[35]: P_poly = PSA(P); P_poly
[35]: (x1^3 - x1*x2*x3)*xi2*xi3 + (-x1^2*x2 + x2^2*x3)*xi1*xi3 + (x1^2*x3 - x2*x3^2)*xi1*xi2
[36]: P_poly.bracket(P_poly)
[36]: 0
For a Poisson structure P with homogeneous polynomial coefficients, the Poisson
(sub)complex of homogeneous polynomial Poisson cochains has finite-dimensional bi-
graded components, and if a homogeneous polynomial Poisson cochain is a coboundary
(in the smooth Poisson complex), then it is in particular the coboundary of a homoge-
neous polynomial Poisson cochain.
Proof : The map j∞0 that takes a smooth function to its infinite jet at the origin ex-
tends to a map from the smooth Poisson complex (with differential dP ) to the formal
Poisson complex (with differential dP0 where P = j∞0 0 (P )) which respects the differen-
tials: d ◦ j∞ = j∞ ◦ d . Suppose Q = d (X) is a d -coboundary. Then j∞P0 0 0 P P P 0 (Q) =
j∞ ∞0 (dP (X)) = dP0(j0 (X)) is a dP0-coboundary. Now taking Poisson cochains P,Q with
homogeneous polynomial coefficients of degrees p, q respectively implies Q = j∞0 (Q) and
P = j∞0 (P ), hence Q = d (j∞P 0 (X)), where j∞0 (X) has formal power series coefficients.
Finally, compare degrees: since Q has homogeneous polynomial coefficients of degree q
and dP is homogeneous of degree p − 1, it follows that j∞0 (X) = X0 +X ′ where X0 has
homogeneous polynomial coefficients of degree q − (p − 1) and dP (X0) = Q, as desired.
(The remaining terms in X ′ can be ignored as dP (X ′) = 0.)
This proven possibility to work only with polynomials splits the problem of determining
Poisson cohomology for homogeneous Poisson structures into subproblems that can be
solved using finite-dimensional linear algebra, as will be seen below.
[37]: from gcaops.algebra.homogeneous_polynomial_poisson_complex import PoissonComplex
PC = PoissonComplex(P_poly)
print(PC)
Poisson complex of (x1^3 - x1*x2*x3)*xi2*xi3 + (-x1^2*x2 + x2^2*x3)*xi1*xi3 + (x1^2*x3
- x2*x3^2)*xi1*xi2
Consider P_poly as a cochain in its own Poisson complex:
[38]: PC(P_poly)
2.7. STABLE POISSON COCYCLES 61
[38]: Poisson cochain (x1^2*x3 - x2*x3^2)*xi1*xi2 + (-x1^2*x2 + x2^2*x3)*xi1*xi3 + (x1^3 -
x1*x2*x3)*xi2*xi3
[39]: PC(P_poly).differential()
[39]: Poisson cochain 0
Test whether this cocycle is a coboundary:
[40]: PC(P_poly).is_coboundary()
[40]: True
Find a trivializing vector field V such that [[P, V ]] = P :
[41]: V = PC(P_poly).is_coboundary(certificate=True)[1]; V
[41]: Poisson cochain (-x1)*xi1 + (-x2)*xi2 + (-x3)*xi3
We obtain V = −E where E is the Euler vector field.
Determine a basis of the cohomology in several bi-graded components (with respect to
arity and degree):
[42]: PC.cohomology_basis(2,3)
[42]: [Poisson cochain (x1^2*x2)*xi1*xi2 + (-x2^3 - x1^2*x3 + x2*x3^2)*xi1*xi3 +
(x1*x2^2)*xi2*xi3,
Poisson cochain (x1^2*x3)*xi1*xi2 + (-x2^2*x3)*xi1*xi3 + (x1*x2*x3)*xi2*xi3,
Poisson cochain (x2^3)*xi1*xi2 + (x2^3 - x2*x3^2 - x3^3)*xi1*xi3 + (-3*x1*x2^2 -
3*x1*x2*x3 - 2*x1*x3^2)*xi2*xi3]
[43]: PC.cohomology_basis(1,2)
[43]: [Poisson cochain (x2^2 - x3^2)*xi1 + (-x1*x2 - 2*x1*x3)*xi2 + (2*x1*x2 + x1*x3)*xi3]
In bi-grading (2, 3) we have obtained all the second Poisson cohomology classes which
are represented by bi-vector cocycles with polynomial coefficients of degree 3. Likewise,
we learn that there is a unique cohomology class of Poisson 1-cocycles with degree 2
polynomial coefficients.
2.7 Stable Poisson cocycles
The condition for a multivector to be a Poisson cocycle is generally non-trivial:
[44]: P.bracket(x1)
[44]: (x1^2*x2 - x2^2*x3)*xi3 + (-x1^2*x3 + x2*x3^2)*xi2
[45]: P.bracket(xi1)
[45]: (-3*x1^2 + x2*x3)*xi2*xi3 + (2*x1*x2)*xi1*xi3 + (-2*x1*x3)*xi1*xi2
Nevertheless, there exist formulas — depending on the Poisson structure coefficients P ij
— that always define a Poisson cocycle, such as [27]:
62 CHAPTER 2. IMPLEMENTATION OF POISSON STRUCTURES
( )
∂3P ij
′ ′ ′
· ∂P
kk ∂P `` ∂Pmm ∂ ∂
Qtetra(P ) = 1
∂(xk∂x`∂xm ∂x`′ ∂xm′ ∧∂xk′ )∂xi ∂xj
∂2P ij 2
′ ′ ′
· ∂ P
km ∂P k ` ∂Pm ` ∂ ∂
+ 6 ∧ .
∂xk∂x` ∂xk′∂x`′ ∂xm′ ∂xj ∂xi ∂xm
It can be implemented as follows (see also the tetrahedron graph γ3 below):
[46]: x = SA.even_coordinates()
xi = SA.odd_coordinates()
dimension = len(x)
import itertools
Gamma1 = sum(sum(diff(P[i,j],x[k],x[l],x[m]) * diff(P[k,ka],x[la]) *␣
↪→diff(P[l,la],x[ma]) * diff(P[m,ma],x[ka]) for (k,l,m,ka,la,ma) in itertools.
↪→product(range(0,dimension),repeat=6))*xi[i]*xi[j] for (i,j) in itertools.
↪→combinations(range(0,dimension),2))
Gamma2 = sum(sum((diff(P[i,j],x[k],x[l]) * diff(P[k,m],x[ka],x[la]) -␣
↪→diff(P[m,j],x[k],x[l]) * diff(P[k,i],x[ka],x[la]))/2 * diff(P[ka,l],x[ma]) *␣
↪→diff(P[ma,la],x[j]) for (j,k,l,ka,la,ma) in itertools.
↪→product(range(0,dimension),repeat=6))*xi[i]*xi[m] for (i,m) in itertools.
↪→combinations(range(0,dimension),2))
Q_tetra = 1*Gamma1 + 6*Gamma2
Q_tetra
[46]: (36*x1^3*x2^2*x3 - 36*x1*x2^3*x3^2 - 36*x1^3*x3^3 + 36*x1*x2*x3^4)*xi1*xi2 +
(-36*x1^3*x2^3 + 36*x1*x2^4*x3 + 36*x1^3*x2*x3^2 - 36*x1*x2^2*x3^3)*xi1*xi3 +
(36*x1^4*x2^2 - 36*x1^2*x2^3*x3 - 36*x1^4*x3^2 + 36*x1^2*x2*x3^3)*xi2*xi3
Indeed, Qtetra(P ) is a Poisson cocycle for this particular Poisson bi-vector P:
[47]: P.bracket(Q_tetra)
[47]: 0
A good question is whether this Poisson 2-cocycle is or is not a Poisson coboundary.
Define a vector field V with undetermined coefficients:
[48]: V = function('V1')(x1,x2,x3)*xi1 + function('V2')(x1,x2,x3)*xi2 +␣
↪→function('V3')(x1,x2,x3)*xi3; V
[48]: (V1(x1, x2, x3))*xi1 + (V2(x1, x2, x3))*xi2 + (V3(x1, x2, x3))*xi3
The equation to solve is [[P, V ]] = Qtetra, or equivalently the vanishing of the components
of the following bi-vector field:
[49]: P.bracket(V) - Q_tetra
[49]: (36*x1^3*x2^3 - 36*x1*x2^4*x3 - 36*x1^3*x2*x3^2 + 36*x1*x2^2*x3^3 -
x1^2*x2*diff(V1(x1, x2, x3), x1) + x2^2*x3*diff(V1(x1, x2, x3), x1) + x1^3*diff(V1(x1,
x2, x3), x2) - x1*x2*x3*diff(V1(x1, x2, x3), x2) + x1^2*x3*diff(V3(x1, x2, x3), x2) -
x2*x3^2*diff(V3(x1, x2, x3), x2) - x1^2*x2*diff(V3(x1, x2, x3), x3) +
x2^2*x3*diff(V3(x1, x2, x3), x3) + 2*x1*x2*V1(x1, x2, x3) + x1^2*V2(x1, x2, x3) -
2*x2*x3*V2(x1, x2, x3) - x2^2*V3(x1, x2, x3))*xi1*xi3 + (-36*x1^4*x2^2 +
36*x1^2*x2^3*x3 + 36*x1^4*x3^2 - 36*x1^2*x2*x3^3 - x1^2*x2*diff(V2(x1, x2, x3), x1) +
2.7. STABLE POISSON COCYCLES 63
x2^2*x3*diff(V2(x1, x2, x3), x1) + x1^3*diff(V2(x1, x2, x3), x2) -
x1*x2*x3*diff(V2(x1, x2, x3), x2) - x1^2*x3*diff(V3(x1, x2, x3), x1) +
x2*x3^2*diff(V3(x1, x2, x3), x1) + x1^3*diff(V3(x1, x2, x3), x3) -
x1*x2*x3*diff(V3(x1, x2, x3), x3) - 3*x1^2*V1(x1, x2, x3) + x2*x3*V1(x1, x2, x3) +
x1*x3*V2(x1, x2, x3) + x1*x2*V3(x1, x2, x3))*xi2*xi3 + (-36*x1^3*x2^2*x3 +
36*x1*x2^3*x3^2 + 36*x1^3*x3^3 - 36*x1*x2*x3^4 + x1^2*x3*diff(V1(x1, x2, x3), x1) -
x2*x3^2*diff(V1(x1, x2, x3), x1) - x1^3*diff(V1(x1, x2, x3), x3) +
x1*x2*x3*diff(V1(x1, x2, x3), x3) + x1^2*x3*diff(V2(x1, x2, x3), x2) -
x2*x3^2*diff(V2(x1, x2, x3), x2) - x1^2*x2*diff(V2(x1, x2, x3), x3) +
x2^2*x3*diff(V2(x1, x2, x3), x3) - 2*x1*x3*V1(x1, x2, x3) + x3^2*V2(x1, x2, x3) -
x1^2*V3(x1, x2, x3) + 2*x2*x3*V3(x1, x2, x3))*xi1*xi2
In general [[P, V ]] = Qtetra is a difficult-to-solve system of partial differential equations.
If P and Q have homogeneous polynomial coefficients, then the existence of smooth
V satisfying Q = [[P, V ]] implies that there is such a V with homogeneous polynomial
coefficients, so it suffices to search for V of that type.
In our case P has degree d = 3 and Q has degree 4d− 6 = 6, so deg(Q) = deg([[P, V ]]) =
deg(P ) + deg(V )− 1 implies V must be of degree 4.
Convert Q_tetra from a superfunction with symbolic coefficients to a superfunction with
polynomial coefficients:
[50]: Q_tetra_poly = PSA(Q_tetra); Q_tetra_poly
[50]: (36*x1^3*x2^2*x3 - 36*x1*x2^3*x3^2 - 36*x1^3*x3^3 + 36*x1*x2*x3^4)*xi1*xi2 +
(-36*x1^3*x2^3 + 36*x1*x2^4*x3 + 36*x1^3*x2*x3^2 - 36*x1*x2^2*x3^3)*xi1*xi3 +
(36*x1^4*x2^2 - 36*x1^2*x2^3*x3 - 36*x1^4*x3^2 + 36*x1^2*x2*x3^3)*xi2*xi3
Test whether it is a coboundary in the Poisson complex of P:
[51]: PC(Q_tetra_poly).is_coboundary()
[51]: True
Find a trivializing vector field V such that [[P, V ]] = Qtetra:
[52]: V = PC(Q_tetra_poly).is_coboundary(certificate=True)[1]; V
[52]: Poisson cochain (18*x1^2*x2^2 - 18*x1^2*x3^2)*xi1 + (-18*x1^3*x2 - 36*x1^3*x3)*xi2 +
(36*x1^3*x2 + 18*x1^3*x3)*xi3
[53]: V.differential()
[53]: Poisson cochain (36*x1^3*x2^2*x3 - 36*x1*x2^3*x3^2 - 36*x1^3*x3^3 +
36*x1*x2*x3^4)*xi1*xi2 + (-36*x1^3*x2^3 + 36*x1*x2^4*x3 + 36*x1^3*x2*x3^2 -
36*x1*x2^2*x3^3)*xi1*xi3 + (36*x1^4*x2^2 - 36*x1^2*x2^3*x3 - 36*x1^4*x3^2 +
36*x1^2*x2*x3^3)*xi2*xi3
[54]: P.bracket(V.lift()) == Q_tetra
[54]: True
For this particular Poisson bi-vector P, the tetrahedral flow Q_tetra is Poisson-trivial.
We shall explain the origin of the formula for Qtetra(P ) in §4.7, we will recall why it works
64 CHAPTER 2. IMPLEMENTATION OF POISSON STRUCTURES
in §5.1, and we will inspect the (non)triviality of the Poisson 2-cocycle in other cases in
Chapters 6, 7, and 8.
In the next chapter, we first return to the subject of star products as we had it in
Chapter 1.
Chapter 3
Implementation of Formality
First, we recall the construction and properties of the graphs in Kontsevich’s Formality
morphism and the respective ⋆-product: e.g. Kontsevich graphs and Leibniz graphs.
Taken modulo the equivalence relation provided by the graph automorphisms acting on
the edges, the directed Formality graphs span vector spaces, so that sums of graphs are
well-defined and bases of graphs can be introduced. We recall how these vector spaces
–of directed Formality graphs with two types of vertices, aerial ones and sinks– assemble
to a bi-colored operad with respect to the graph insertion. When we take the quotient
of each vector space modulo the labeling of aerial vertices, we obtain the Kontsevich
Formality graph complex FGC; the differential is the graph realization of the Hochschild
differential with respect to the usual multiplication. We implement all of the above in
software. In §3.4 we write present a high-order expansion ⋆ mod ō(h̄4) of the Kontsevich
star product, using Kontsevich’s graphs, and verify the associativity of ⋆ mod ō(h̄4)
modulo the Jacobi identity by using Leibniz graphs. To constrain the graph weights even
further, we generate the cyclic weight relations [21, Appendix E]. We confirm that all
the previously known weights –in particular from the work of Banks–Panzer–Pym [1]–
do satisfy all the relations which we produce/generate here explicitly for the first time.
All our presentation in this chapter is accompanied by large data files with graphs, weights
and relations. These plain text files are stored at https://rburing.nl/gcaops/.
3.1 Formality graphs
We recall the construction of graphs which show up in Kontsevich’s proof of the Formality
Theorem in [28].
[1]: from gcaops.graph.formality_graph import FormalityGraph
from gcaops.graph.formality_graph_operad import FormalityGraphOperad
from gcaops.graph.formality_graph_complex import FormalityGraphComplex
Let us describe the relevant graphs.
Definition 1. A Formality graph is a directed graph on m + n vertices {0, . . . ,m −
1,m, . . . ,m + n − 1} such that the m ground vertices 0, . . . ,m− 1 are sinks (with no
outgoing edges) and the n vertices m, . . . ,m+ n− 1 are called aerial. The set of edges
of the graph is endowed with a total ordering.
65
66 CHAPTER 3. IMPLEMENTATION OF FORMALITY
Every Formality graph is encoded by the vertex numbers (m,n) followed by the ordered
list of directed edges (represented by the vertex pairs).
Example 1. The wedge is the main example of a Formality graph; in fact it is a building
block in the Kontsevich graphs.
[2]: wedge = FormalityGraph(2,1,[(2,0),(2,1)]); wedge.show(figsize=2)
Definition 2. A Kontsevich graph is a Formality graph built of wedges, i.e. with each
aerial vertex having exactly two outgoing edges.
Example 2. Here is another example of a Kontsevich graph:
[3]: g = FormalityGraph(2,2,[(2,0),(2,1),(3,0),(3,2)]); g.show(figsize=2)
Definition 3. An automorphism of a Formality graph is an automorphism of the directed
graph which preserves the ground vertices pointwise. A Formality graph is a zero graph
if it admits an automorphism that induces an odd permutation on the set of directed
edges.
Example 3. Zero graphs exist. Here is an example:
[4]: g_zero = FormalityGraph(2,3,[(2,0),(2,1),(3,0),(3,1),(4,2),(4,3)]); g_zero.
↪→show(figsize=3)
3.1. FORMALITY GRAPHS 67
Let us relabel the aerial vertices but preserve the ordering of edges, so that we obtain a
“different” Kontsevich graph:
[5]: g_zero_relabeled = g_zero.relabeled({0:0,1:1,2:3,3:2,4:4}); g_zero_relabeled
[5]: FormalityGraph(2, 3, [(3, 0), (3, 1), (2, 0), (2, 1), (4, 3), (4, 2)])
We repeat: the ordering of the edges is the old one (same as in g_zero); every directed
edge is encoded by an ordered pair of labeled vertices; the labeling of aerial vertices in
g_zero_relabeled is new with respect to the original labeling in g_zero; this is why the
ordering of edges in g_zero_relabeled is no longer lexicographic.
The method canonicalize_edges (re)orders the edges lexicographically –that is, it mod-
ifies the graph– and returns the sign of the permutation needed to reach that new order.
[6]: g_zero_relabeled.canonicalize_edges()
[6]: -1
That is, a parity odd permutation was needed to reorder the edges lexicographically.
Now, the graphs g_zero and g_zero_relabeled coincide identically:
[7]: g_zero_relabeled
[7]: FormalityGraph(2, 3, [(2, 0), (2, 1), (3, 0), (3, 1), (4, 2), (4, 3)])
[8]: g_zero_relabeled == g_zero
[8]: True
Originally, g_zero and g_zero_relabeled had different edge orderings, and it took a par-
ity odd permutation to bring the edge ordering of g_zero_relabeled to that of g_zero.
This shows that g_zero is a zero Formality graph.
Definition 4. A Leibniz graph is a Formality graph built of at least one tripod and
further (if at all) built from wedges, i.e. with at least one aerial vertex having exactly
three outgoing edges and the other aerial vertices (if any) having exactly two outgoing
edges.
Example 4. The tripod:
68 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[9]: FormalityGraph(3,1,[(3,0),(3,1),(3,2)]).show(figsize=4)
3.2 The bi-colored operad of Formality graphs
The Formality graphs assemble to a bi-colored operad; see [36] and [16].
Definition 5. Consider the tri-graded k-vector space with components of grading
(m,n, e) spanned by Formality graphs with m ground vertices, n aerial vertices, and
e edges, modulo the relation that a permutation of edges in a graph amounts to multi-
plication of this graph by the sign of that permutation. We now construct the bi-colored
Formality graph operad FGO; the two colors are ‘ground’ and ‘air’. The ground compo-
nent of the operad is the direct sum of all the tri-graded quotient spaces with Formality
graphs. The aerial component of the operad contains only the quotient spaces of vacuum
Formality graphs (without ground vertices). In this section we give the two definitions of
the operadic insertions: a ground-component graph into a sink of a ground-component
graph, and secondly an aerial-component graph into an aerial vertex of a Formality graph
from any component.
We begin by discussing the k-vector space structure.
[10]: FGO = FormalityGraphOperad(QQ); FGO
[10]: Operad of formality graphs over Rational Field
Zero graphs are equal to the zero element in the quotient vector space, because they equal
minus themselves:
[11]: FGO(g_zero)
[11]: 0
Example 5. The wedge is a nonzero element of FGO:
[12]: FGO(wedge).show()
3.2. THE BI-COLORED OPERAD OF FORMALITY GRAPHS 69
Convention. When a graph is converted into an element of the operad FGO, it is
automatically expressed in terms of an internally chosen basis.
The above graph with the reversed ordering of the two edges is equal to minus that graph:
[13]: FGO(FormalityGraph(2,1,[(2,1),(2,0)]))
[13]: (-1)*FormalityGraph(2, 1, [(2, 0), (2, 1)])
This does not make the wedge “equal to minus itself”, because swapping the ground
vertices does not amount to a Formality graph automorphism (in such an automorphism,
the ground vertices must always be preserved pointwise).
We can insert a graph into a vertex of another graph: into a ground vertex (first case)
and into an aerial vertex (second case).
Definition 6 (graph insertion into a ground vertex). Let Γ1 be a Formality graph with
a ground vertex ℓ, and let Γ2 be another Formality graph with λ ⩾ 0 ground vertices.
We define the right-into-left insertion Γ1 ◦` Γ2 of Γ2 into the ℓth ground vertex of Γ1. By
definition, the result is a sum of graphs in FGO. In each term, the sink ℓ of Γ1 is replaced
by the whole inserted graph Γ2 (with shifted labels), every edge originally coming into
that sink ℓ becomes aimed —consecutively, over all vertices of Γ2— at a vertex (aerial
or sink) of Γ2; appointing a target for one incoming edge is completely independent from
choosing a target for any other incoming edge.
• In every term, the labeling of sinks becomes as follows: first go the sinks 0, . . . , ℓ − 1
of Γ1, preceding the sink ℓ into which a whole new graph Γ2 is inserted, then follow the
sinks of the inserted graph Γ2 (their labels are shifted by +ℓ), and finally the remaining
sinks (if any) of the graph Γ1 follow in their original order (their labels shifted by +λ−1).
• Likewise, in every term the labeling of aerial vertices becomes as follows: all the aerial
vertices of Γ1 go first (their labels may be shifted by the count of ground vertices, as
above, if λ > 0), followed by all the aerial vertices of Γ2 (again, their labeling is possibly
shifted); in either case the consecutive ordering of aerial vertices is not interrupted.
• Finally, the edge ordering: first go all the edges of the graph Γ1, last go all the edges
of the graph Γ2; neither of the two orderings of the edges is anywhere broken.
Example 6. Let the wedge (with its own ordering of the edges) be inserted into the
leftmost ground vertex 0 of another wedge (with its own edge ordering):
[14]: FGO(wedge).insertion(0, FGO(wedge)).show()
70 CHAPTER 3. IMPLEMENTATION OF FORMALITY
By construction, the Left ≺ Right edges of the wedge whose top is vertex 3 precede the
Left ≺ Right edges of the wedge over sinks 0, 1 and top in vertex 4 in every term. The
internal choice of the basis by the software is such that the third graph acquires the
coefficient −1 with respect to that basis. Indeed, the ordering of edges in basic graphs is
lexicographic.
Definition 7 (graph insertion into an aerial vertex). Into aerial vertices, only vacuum
Formality graphs (without ground vertices) can be inserted. Let Γ1 be a Formality graph
with an aerial vertex k, and let Γ2 be a vacuum Formality graph on κ aerial vertices. Let
us define the right-into-left insertion Γ1 ◦k Γ2 of Γ2 into the kth aerial vertex of Γ1. By
definition, the result is a sum of graphs in FGO. In each term, the aerial vertex k of Γ1
is replaced by the whole inserted graph Γ2 (with shifted labels of aerial vertices). Every
edge originally incident to that aerial vertex k becomes incident —consecutively over
vertices of Γ2, which all are aerial— to an aerial vertex of Γ2; appointing a new source or
target for one such edge is completely independent from choosing a source (respectively,
target) for any other such edges.
• In every term, the labeling of sinks is preserved from the labeling of sinks in Γ1, but
the labeling of aerial vertices becomes as follows: first go the aerial vertices up to and
including (if any) k− 1 preceding the aerial vertex k into which the whole new graph Γ2
is inserted, then follow the aerial vertices of the inserted graph Γ2 (their labels are shifted
by +k), and finally go the remaining aerial vertices (if any) of the graph Γ1: they follow
in their original ordering (now, their labels are shifted by +κ− 1).
• About the edge ordering: first go all the edges of the graph Γ1, last go all the edges of
the graph Γ2; neither of the two orderings of the edges is anywhere changed.
Example 7. Insertion of a directed stick graph into the aerial vertex of a wedge:
[15]: right_stick = FGO(FormalityGraph(0,2,[(0,1)])); right_stick.show()
[16]: FGO(wedge).insertion(2, right_stick).show(ncols=2)
3.2. THE BI-COLORED OPERAD OF FORMALITY GRAPHS 71
Definitions 6 and 7 are extended to sums of graphs by linearity in the respective compo-
nent. We claim that both operadic insertions respect the quotient vector space structure
(e.g. insertion of a zero graph into any graph produces zero graphs, and the other way
around); the proof is similar to [33].
Remark. Note that aerial vertices are still distinguishable by their labels in the operad
FGO:
[17]: sum_of_graphs = FGO([(1, FormalityGraph(3,2,[(3,0),(3,1),(4,1),(4,2)])), (1,␣
↪→FormalityGraph(3,2,[(4,0),(4,1),(3,1),(3,2)]))])
sum_of_graphs.show()
The output confirms that the two graphs are linearly independent (otherwise, the sum
would have simplified): by definition, the labeling of the (aerial) vertices is part of the
data. In the above two graphs, vertices 3 and 4 stand on different pairs of sinks.
72 CHAPTER 3. IMPLEMENTATION OF FORMALITY
3.3 Formality graph complex
By taking the quotient modulo aerial vertex labeling, we obtain a graph complex. With
the graphs originally by Kontsevich, this graphical construction was studied by Willwa-
cher in [36] and by Dolgushev in [16].
Definition 8. First, from now on we restrict to the ground (non-aerial) component of
the Formality graph operad FGO. Next, we quotient every vector space in this ground
component modulo aerial vertex labeling. This collection of quotient vector spaces forms
a differential graded Lie algebra (dgLa), with the graphical Gerstenhaber bracket and
with the graphical Hochschild differential (see below). On this differential graded Lie
algebra we thus obtain the structure of Formality graph complex FGC.
[18]: FGC = FormalityGraphComplex(QQ); FGC
[18]: Formality graph complex over Rational Field with Basis consisting of representatives
of isomorphism classes of formality graphs with no automorphisms that induce an odd
permutation on edges
By having constructed FGC, we obtain an object which represents a basis of nonzero
Formality graphs:
[19]: FGC.basis()
[19]: Basis consisting of representatives of isomorphism classes of formality graphs with no
automorphisms that induce an odd permutation on edges
The software is now able to generate a basis of directed graphs for the tri-graded (m,n, e)
homogeneity component with a given number m of ground vertices, n aerial vertices, and e
edges. Those bases are automatically created by using the nauty programs geng, directg
and pickg to generate the respective isomorphism classes of directed graphs, followed by
permutations of the ground vertices and filtering out zero graphs.
The graph bases are important for us, e.g. because the cyclic weight relations for the
weights of Formality graphs will be explicitly referred to those bases, i.e. to the ordered
lists of basic graphs with ordered sets of edges.
Example 8 (wedge). In the basis at tri-grading (m,n, e) = (2, 1, 2) there is one graph:
[20]: list(FGC.basis().graphs(2,1,2))
[20]: [FormalityGraph(2, 1, [(2, 0), (2, 1)])]
So the list of graphs in the basis consists of just one line: the wedge itself.
Example 9.
[21]: list(FGC.basis().graphs(1,2,4))
[21]: [FormalityGraph(1, 2, [(1, 0), (1, 2), (2, 0), (2, 1)])]
This graph was used in Kontsevich’s breakthrough paper [28] to construct the first ex-
ample of a universal gauge transformation for Kontsevich’s star-product.
Example 10.
3.3. FORMALITY GRAPH COMPLEX 73
[22]: list(FGC.basis().graphs(2,2,4))
[22]: [FormalityGraph(2, 2, [(2, 1), (2, 3), (3, 0), (3, 1)]),
FormalityGraph(2, 2, [(2, 0), (2, 3), (3, 0), (3, 1)]),
FormalityGraph(2, 2, [(2, 1), (3, 0), (3, 1), (3, 2)]),
FormalityGraph(2, 2, [(2, 0), (3, 0), (3, 1), (3, 2)]),
FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 0), (3, 1)]),
FormalityGraph(2, 2, [(2, 3), (3, 0), (3, 1), (3, 2)]),
FormalityGraph(2, 2, [(2, 1), (2, 3), (3, 0), (3, 2)]),
FormalityGraph(2, 2, [(2, 1), (2, 3), (3, 1), (3, 2)]),
FormalityGraph(2, 2, [(2, 0), (2, 3), (3, 0), (3, 2)])]
Not all of these Formality graphs on 2 sinks, 2 aerial vertices and 4 edges are built of
wedges. Indeed, three of them contain a tripod vertex — and they are not Leibniz graphs!
Convention. Whenever a Formality graph is converted into an element of FGC, it is
automatically expressed in terms of the internally chosen basis. This also defines our
normal form for Kontsevich graphs and Leibniz graphs.
For example:
[23]: FGC(FormalityGraph(3,2,[(3,0),(3,1),(4,1),(4,2)]))
[23]: 1*FormalityGraph(3, 2, [(3, 1), (3, 2), (4, 0), (4, 1)])
In the above, aerial vertices were relabeled (3 ⇄ 4) and pairs of outgoing edges were
reordered.
[24]: FGC(FormalityGraph(3,2,[(3,1),(3,2),(4,1),(4,0)]))
[24]: (-1)*FormalityGraph(3, 2, [(3, 1), (3, 2), (4, 0), (4, 1)])
This time, the ordering of outgoing edges at aerial vertex 4 is swapped, at a price of the
coefficient −1 appearing in front of the basic graph.
Remark. In gcaops the graph bases are generated by using the nauty software, as men-
tioned above. In [9] (§1.1, Definition 2) a different convention was used to generate graph
bases (only containing graphs with wedges): it refers to minimal base-(m + n) integer
numbers with respect to all permutations of labels for n aerial vertices and reordering of
Left ≺ Right outgoing edges in the n pairs making a Kontsevich graph.
Example 11. In cell [23] just above, our input is the minimal encoding of a Kontsevich
graph, whereas its internal storage by gcaops, as seen from the output in cell [23], is not
in the minimal base-(n+m) form.
Corrigendum. Normal forms for Leibniz graphs with one Jacobiator were introduced
in [10] (Definition 5): the idea was to re-use the normal form for Kontsevich graphs.
Namely, the Jacobiator is expanded into the sum of three Kontsevich graphs (built of
wedges), all the incoming arrows (to the top of the tripod) are formally directed to the
top of the lower wedge in each Kontsevich graph, and then we find the normal forms of
the resulting three Kontsevich graphs, while also remembering where the internal edge of
the Jacobiator is located in those normal forms. The normal form of the Leibniz graph
then was: choose the minimal (w.r.t. base-(m + n) numbers) Kontsevich graph encod-
ing, supplemented with the indication of the internal Jacobiator edge. This definition,
i.e. the pair (Kontsevich graph, marked edge) is unfortunately not a true normal form of
74 CHAPTER 3. IMPLEMENTATION OF FORMALITY
the Leibniz graph. Namely, it can happen that the resulting Kontsevich graph has an
automorphism that maps the marked edge elsewhere, to a new place in the graph. Con-
sequently, two isomorphic Leibniz graphs could have different “normal forms” (differing
only by the marking where the internal Jacobiator edge is). This led to a visible pathol-
ogy, namely to redundant parameters in the systems of equations: one and the same
Leibniz graph, encoded differently, acquired two unrelated coefficients. Fortunately, the
effect disappeared when Leibniz graphs were expanded to sums of Kontsevich graphs and
similar terms were collected. (Besides, it is necessary to pay attention to whether the
internal Jacobiator edge is labeled Left or Right, in order to expand the Leibniz graph
with the correct sign ±1.) In consequence, that normal form was abandoned in favor of
inambiguous (and fast) description of Leibniz graphs by using the nauty software.
Examples of data files containing Formality graphs can be found at the following locations:
• In Appendix B there is the encoding of Kontsevich’s ⋆-product modulo ō(h̄4).
• In Appendix C there is the encoding of Kontsevich’s affine ⋆-product modulo ō(h̄7).
• In Appendix D there are the encodings of flows Qγ built of Kontsevich graphs and
operators ♢γ built of Leibniz graphs.
• In §3.6 there are links to files with bases of graphs, with weights of graphs, and
with linear cyclic weight relations referred to those graph bases.
3.4 Kontsevich star product
In this section we construct a fourth order expansion ⋆ mod ō(h̄4) of Kontsevich’s star-
product, first with undetermined coefficients and then with coefficients as determined
by Panzer’s kontsevint software from the joint paper [1] by Banks, Panzer, Pym (2018).
3.4.1 Multiplicity
In this section we will take sums over all Formality graphs of a certain type (namely,
Kontsevich graphs and Leibniz graphs). We keep in mind that every Formality graph is
a topological combinatorial structure (a graph) equipped with a global ordering of edges
(in particular, ordering of outgoing edges at every aerial vertex); every Formality graph is
taken, in the quotient vector space FGC, modulo labeling of aerial vertices. To make this
process —of taking sums of graphs— more efficient, we can instead sum over isomorphism
classes of Formality graphs, with multiplicities.
Definition. The multiplicity m(Γ) of a Formality graph Γ is the number of Formality
graphs isomorphic to Γ, under permutations of aerial vertex labels (leaving the ground
vertices invariant) and reorderings of the list of edges issued from each particular vertex.
The count of aerial vertex relabelings yields the number n! /#Aut(Γ), i.e. not merely n!
for a Formality graph Γ on n aerial vertices and with a non-trivial automorphism group
Aut(Γ).
Example 1. Recall the graph g:
[1]: from gcaops.graph.formality_graph import FormalityGraph
g = FormalityGraph(2,2,[(2,0),(2,1),(3,0),(3,2)]); g.show(figsize=2)
3.4. KONTSEVICH STAR PRODUCT 75
[2]: g.multiplicity()
[2]: 8
Indeed, for the graph g there are two ways to (re)label the internal vertices and 2 · 2 = 4
ways to (re)order the edges. The number of options is therefore equal to 2 · 4 = 8.
Example 2.
[3]: wedgewedge = FormalityGraph(2,2,[(2,0),(2,1),(3,0),(3,1)]); wedgewedge.show(figsize=2)
[4]: wedgewedge.multiplicity()
[4]: 4
There are 2 · 2 = 4 ways to (re)order the edges; there are 2 ways, {id, 2 ⇄ 3}, to (re)label
the aerial vertices, but the swap 2 ⇄ 3 does not produce any different Formality graph
because the swap is a symmetry of wedgewedge, so 2! /#Aut(Γ) = 1 (here n = 2).
3.4.2 Star product
The non-commutative associative star-product [28] of two functions f, g ∈ C∞(M) is
realized by a sum over Kontsevich graphs,
∑ h̄n ∑
f ⋆ g = f · g + w(Γ) Γ(P, . . . , P )(f, g).
n⩾ n!1 Γ∈Ĝn2
76 CHAPTER 3. IMPLEMENTATION OF FORMALITY
The Kontsevich formula for the weight of a Formality graph Γ on m ⩾ 0 ground vertices
and n ⩾ 0 aerial vertices is: ∫
1 ∧
w(Γ) := dφe.
(2π)2n+m−2 C̄+n,m e∈E(Γ)
The wedge product of angle 1-forms dφe is integrated over the compactified configuration
space C̄+n,m of n points inside the hyperbolic plane H2 and m strictly ordered sinks on
∂H2 = R, modulo the affine group action. That construction is given in [28].
Remark. The weights in Kontsevich’s original construction from [28] are defined using
a different convention, (∏n 1 )
WΓ =
(#Star w(Γ),(k))!
k=1
where # Star(k) is the number of arrows starting at the aerial vertex m+ k − 1.
The weight function in the kontsevint software by Panzer [1] computes the weights w(Γ).
The n-linear polydifferential operator Γ(P, . . . , P ) and the weight w(Γ) behave un-
der permutations of edges and of aerial vertex labels in such a way that the product
w(Γ) Γ(P, . . . , P ) is invariant under both types of permutations. Hence, the star-product
can be expressed in a more computationally efficient way as a sum over graphs with
multiplicities, i.e. ∑ h̄n ∑
f ⋆ g = f · g + m(Γ)w(Γ) Γ(P, . . . , P )(f, g),
n⩾ n!1 [Γ]
where m(Γ) is the multiplicity defined above, and the sum now runs over equivalence
classes [Γ] of Formality graphs in Ĝn2 modulo labeling of aerial vertices and ordering of
edges.
[5]: from gcaops.graph.formality_graph_complex import FormalityGraphComplex
#FGC = FormalityGraphComplex(SR, implementation='vector'); FGC
FGC = FormalityGraphComplex(SR, lazy=True); FGC
[5]: Formality graph complex over Symbolic Ring with Basis consisting of representatives of
isomorphism classes of formality graphs with no automorphisms that induce an odd
permutation on edges
Example. Let us convert the known expansion ⋆ mod ō(h̄3), given as a list of encodings
of Kontsevich graphs (in terms of ordered pairs of target vertices for edges [9]), into the
gcaops format of weighted Formality graphs.
[6]: star3 = FGC.element_from_kgs_encoding("""h^0:
2 0 1 1
h^1:
2 1 1 0 1 1
h^2:
2 2 1 0 3 1 2 -1/6
2 2 1 0 1 1 2 -1/3
2 2 1 0 1 0 2 1/3
2 2 1 0 1 0 1 1/2
h^3:
3.4. KONTSEVICH STAR PRODUCT 77
2 3 1 0 3 1 2 2 3 -1/6
2 3 1 0 1 1 2 1 2 1/6
2 3 1 0 1 0 1 1 2 -1/3
2 3 1 0 1 0 2 0 2 1/6
2 3 1 0 1 0 1 0 2 1/3
2 3 1 0 1 0 1 0 1 1/6
2 3 1 0 1 1 2 2 3 -1/6
2 3 1 0 3 1 2 1 2 1/6
2 3 1 0 1 0 2 2 3 -1/6
2 3 1 0 3 1 2 0 3 -1/6
2 3 1 0 1 0 4 1 3 -1/6
2 3 1 0 1 0 2 1 3 -1/6
2 3 1 0 1 0 4 1 2 -1/6""")
Let us see how this graph expansion looks like; another drawing of this formula for ⋆ mod
ō(h̄3) is given in Figure 1 of Chapter 11.
[7]: star3.show()
78 CHAPTER 3. IMPLEMENTATION OF FORMALITY
First we construct the associator (f ⋆g)⋆h−f ⋆ (g ⋆h) modulo ō(h̄3) for the star product
expansion ⋆ mod ō(h̄3) and for f, g, h ∈ C∞(M).
[8]: %time star3_assoc = star3.insertion(0, star3, max_num_aerial=3) - star3.insertion(1,␣
↪→star3, max_num_aerial=3)
3.4. KONTSEVICH STAR PRODUCT 79
CPU times: user 258 ms, sys: 7.76 ms, total: 266 ms
Wall time: 264 ms
Let us inspect the leading order term in the associator.
[9]: star3_assoc.homogeneous_part(3, 0, 0)
[9]: 0
So, the coefficient of h̄0 = 1 vanishes. Next, let us inspect the leading deformation term
at h̄1 in the associator.
[10]: star3_assoc.homogeneous_part(3, 1, 2)
[10]: 0
In fact, the associator nontrivially starts at the order two in its h̄-expansion.
[11]: star3_assoc.homogeneous_part(3, 2, 4).show()
Obviously, this is the Jacobiator appearing with nonzero coefficient 2/3. This tells us
at once that for the ⋆-product to be associative, the Jacobi identity (for the bi-vector P
whose copies are placed in the aerial vertices) is a necessary condition.
At the order h̄3 we have more Formality graphs in the associator:
[12]: len(star3_assoc.homogeneous_part(3, 3, 6))
[12]: 39
These 39 graphs are discussed in full detail in Chapter 10; see also Chapter 12.
[13]: #star3_assoc.homogeneous_part(3, 3, 6).show()
The associativity of the ⋆-product can be expressed as the equation 1 [⋆, ⋆]G = 0, where2
the bracket is the Gerstenhaber bracket on the space of polydifferential operators on
C∞(M)[[h̄]].
[14]: star3_assoc == (1/2)*star3.gerstenhaber_bracket(star3, max_num_aerial=3)
[14]: True
80 CHAPTER 3. IMPLEMENTATION OF FORMALITY
Equivalently, the associativity of ⋆ = µ + B, where µ is the ordinary multiplication in
C∞(M)[[h̄]], is the Maurer– Cartan equation d (B) + 1H [B,B]G = 0 for the deformation2
tail B.
[15]: mu = FGC(FormalityGraph(2,0,[])); mu.show()
We recall that by construction, the content f, g of the respective sinks 0, 1 is here not
differentiated (indeed, there are no edges), and then the usual product of the sinks’
content is taken with coefficient +1, thus giving f · g.
[16]: B = star3 - mu
[17]: star3_assoc == B.hochschild_differential() + (1/2)*B.gerstenhaber_bracket(B,␣
↪→max_num_aerial=3)
[17]: True
3.4.3 Star product from Formality morphism
Now, by using the gcaops software, let us construct a third order expansion of Kontsevich’s
star product.
We take for granted the known expansion of ⋆ mod ō(h̄2): each coefficient is obtained by
direct integration of Kontsevich’s formula (see Appendix A.1 in Chapter 11).
[18]: star2 = FGC(FormalityGraph(2,0,[])) + FGC(FormalityGraph(2,1,[(2,0),(2,1)])) + \
FGC([(1/2, FormalityGraph(2,2,[(2,0),(2,1),(3,0),(3,1)])), (1/3,␣
↪→FormalityGraph(2,2,[(2,0),(2,1),(3,0),(3,2)])), (-1/3,␣
↪→FormalityGraph(2,2,[(2,0),(2,1),(3,1),(3,2)])), (-1/6,␣
↪→FormalityGraph(2,2,[(2,0),(2,3),(3,1),(3,2)]))]); star2
[18]: 1*FormalityGraph(2, 0, []) + 1*FormalityGraph(2, 1, [(2, 0), (2, 1)]) +
(1/2)*FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 0), (3, 1)]) + (1/3)*FormalityGraph(2,
2, [(2, 0), (2, 3), (3, 0), (3, 1)]) + (-1/3)*FormalityGraph(2, 2, [(2, 1), (2, 3),
(3, 0), (3, 1)]) + (-1/6)*FormalityGraph(2, 2, [(2, 1), (2, 3), (3, 0), (3, 2)])
Example. By evaluating ⋆ mod ō(h̄2) at the Poisson structure {x, y} = xy/2 on R2 we
reproduce the polydifferential operator from Example 2 in Section 1.2.
[19]: from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
from gcaops.algebra.polydifferential_operator import PolyDifferentialOperatorAlgebra
SA.<xi1,xi2> = SuperfunctionAlgebra(SR, var('x,y'))
PA.<ddx,ddy> = PolyDifferentialOperatorAlgebra(SR, var('x,y'))
3.4. KONTSEVICH STAR PRODUCT 81
[20]: from gcaops.graph.formality_graph_operator import FormalityGraphOperator
star2_operator = FormalityGraphOperator(SA, PA, star2)
[21]: P = (x*y/2)*xi1*xi2; P
[21]: (1/2*x*y)*xi1*xi2
[22]: star2_op = star2_operator.value_at_copies_of(var('hbar')*P); star2_op
[22]: (1/8*hbar^2*x^2*y^2)*(ddx^2 ⊗ ddy^2) + (id ⊗ id) + (-1/4*hbar^2*x^2*y^2)*(ddx*ddy ⊗
ddx*ddy) + (1/8*hbar^2*x^2*y^2)*(ddy^2 ⊗ ddx^2) + (1/12*hbar^2*x^2*y)*(ddx^2 ⊗ ddy) +
(-1/12*hbar^2*x^2*y)*(ddx*ddy ⊗ ddx) + (-1/12*hbar^2*x*y^2)*(ddx*ddy ⊗ ddy) +
(1/12*hbar^2*x*y^2)*(ddy^2 ⊗ ddx) + (-1/12*hbar^2*x^2*y)*(ddx ⊗ ddx*ddy) +
(1/12*hbar^2*x^2*y)*(ddy ⊗ ddx^2) + (1/12*hbar^2*x*y^2)*(ddx ⊗ ddy^2) +
(-1/12*hbar^2*x*y^2)*(ddy ⊗ ddx*ddy) + (-1/24*hbar^2*x^2)*(ddx ⊗ ddx) +
(1/24*hbar^2*x*y - 1/2*hbar*x*y)*(ddy ⊗ ddx) + (1/24*hbar^2*x*y + 1/2*hbar*x*y)*(ddx ⊗
ddy) + (-1/24*hbar^2*y^2)*(ddy ⊗ ddy)
This is literally the formula in Cell [36] in Section 1.3.
Next, we add Kontsevich graphs with undetermined coefficients at h̄3 to the already
known ⋆ mod ō(h̄2) — again, with a generic Poisson structure P .
[23]: from gcaops.graph.formality_graph_basis import KontsevichGraphBasis
KGB = KontsevichGraphBasis(positive_differential_order=True)
[24]: star3c = star2 + 1/6*FGC([(g.multiplicity()*var('c{}'.format(k)), g) for (k,g) in␣
↪→enumerate(KGB.graphs(2,3))]); star3c
[24]: 1*FormalityGraph(2, 0, []) + 1*FormalityGraph(2, 1, [(2, 0), (2, 1)]) +
(1/2)*FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 0), (3, 1)]) + (1/3)*FormalityGraph(2,
2, [(2, 0), (2, 3), (3, 0), (3, 1)]) + (-1/3)*FormalityGraph(2, 2, [(2, 1), (2, 3),
(3, 0), (3, 1)]) + (-1/6)*FormalityGraph(2, 2, [(2, 1), (2, 3), (3, 0), (3, 2)]) +
(4*c0)*FormalityGraph(2, 3, [(2, 1), (2, 4), (3, 1), (3, 4), (4, 0), (4, 1)]) +
(4*c1)*FormalityGraph(2, 3, [(2, 0), (2, 4), (3, 0), (3, 4), (4, 0), (4, 1)]) +
(8*c2)*FormalityGraph(2, 3, [(2, 3), (2, 4), (3, 0), (3, 4), (4, 0), (4, 1)]) +
(8*c3)*FormalityGraph(2, 3, [(2, 3), (2, 4), (3, 1), (3, 4), (4, 0), (4, 1)]) +
(4/3*c4)*FormalityGraph(2, 3, [(2, 0), (2, 1), (3, 0), (3, 1), (4, 0), (4, 1)]) +
(8*c5)*FormalityGraph(2, 3, [(2, 1), (2, 4), (3, 0), (3, 4), (4, 0), (4, 1)]) +
(8*c6)*FormalityGraph(2, 3, [(2, 1), (2, 3), (3, 0), (3, 4), (4, 0), (4, 1)]) +
(8*c7)*FormalityGraph(2, 3, [(2, 1), (2, 4), (3, 0), (3, 2), (4, 0), (4, 1)]) +
(8*c8)*FormalityGraph(2, 3, [(2, 1), (2, 4), (3, 0), (3, 1), (4, 0), (4, 1)]) +
(8*c9)*FormalityGraph(2, 3, [(2, 0), (2, 4), (3, 0), (3, 1), (4, 0), (4, 1)]) +
(8*c10)*FormalityGraph(2, 3, [(2, 1), (2, 3), (3, 1), (3, 4), (4, 0), (4, 2)]) +
(8*c11)*FormalityGraph(2, 3, [(2, 1), (2, 3), (3, 0), (3, 4), (4, 0), (4, 2)]) +
(8*c12)*FormalityGraph(2, 3, [(2, 1), (2, 3), (3, 1), (3, 4), (4, 0), (4, 1)]) +
(8*c13)*FormalityGraph(2, 3, [(2, 0), (2, 3), (3, 0), (3, 4), (4, 0), (4, 1)]) +
(4*c14)*FormalityGraph(2, 3, [(2, 3), (2, 4), (3, 2), (3, 4), (4, 0), (4, 1)]) +
(8*c15)*FormalityGraph(2, 3, [(2, 1), (2, 4), (3, 1), (3, 4), (4, 0), (4, 3)]) +
(8*c16)*FormalityGraph(2, 3, [(2, 1), (2, 4), (3, 0), (3, 2), (4, 0), (4, 2)]) +
(8*c17)*FormalityGraph(2, 3, [(2, 3), (2, 4), (3, 0), (3, 2), (4, 0), (4, 1)]) +
(8*c18)*FormalityGraph(2, 3, [(2, 3), (2, 4), (3, 1), (3, 2), (4, 0), (4, 1)]) +
(8*c19)*FormalityGraph(2, 3, [(2, 3), (2, 4), (3, 1), (3, 4), (4, 0), (4, 3)]) +
(8*c20)*FormalityGraph(2, 3, [(2, 3), (2, 4), (3, 1), (3, 4), (4, 0), (4, 2)]) +
(8*c21)*FormalityGraph(2, 3, [(2, 3), (2, 4), (3, 1), (3, 2), (4, 0), (4, 3)]) +
(8*c22)*FormalityGraph(2, 3, [(2, 1), (2, 4), (3, 0), (3, 4), (4, 0), (4, 2)]) +
82 CHAPTER 3. IMPLEMENTATION OF FORMALITY
(8*c23)*FormalityGraph(2, 3, [(2, 1), (2, 3), (3, 1), (3, 4), (4, 0), (4, 3)]) +
(8*c24)*FormalityGraph(2, 3, [(2, 1), (2, 4), (3, 0), (3, 4), (4, 0), (4, 3)]) +
(8*c25)*FormalityGraph(2, 3, [(2, 1), (2, 3), (3, 1), (3, 2), (4, 0), (4, 3)]) +
(4*c26)*FormalityGraph(2, 3, [(2, 1), (2, 3), (3, 1), (3, 2), (4, 0), (4, 1)]) +
(4*c27)*FormalityGraph(2, 3, [(2, 0), (2, 3), (3, 0), (3, 2), (4, 0), (4, 1)]) +
(8*c28)*FormalityGraph(2, 3, [(2, 1), (2, 3), (3, 0), (3, 2), (4, 0), (4, 1)]) +
(8*c29)*FormalityGraph(2, 3, [(2, 3), (2, 4), (3, 1), (3, 2), (4, 0), (4, 2)])
This time, we take the associator modulo ō(h̄3) for the ansatz ⋆ mod ō(h̄3) and insert the
3D Nambu–Poisson structure into the associator. The term at h̄2 must vanish because
the Jacobi identity holds for the Nambu–Poisson structure. There remains only the term
near h̄3: it is differential polynomial in a and ρ (inside P ) and it is linear with respect to
the constants ci in the top-degree ansatz. That term must vanish for ⋆ to be associative.
We solve the arising system of linear algebraic equations with respect to the top-degree
coefficients (of Kontsevich graphs at h̄3).
[25]: %time assoc3c = star3c.insertion(0, star3c, max_num_aerial=3) - star3c.insertion(1,␣
↪→star3c, max_num_aerial=3)
CPU times: user 405 ms, sys: 0 ns, total: 405 ms
Wall time: 404 ms
[26]: from gcaops.algebra.differential_polynomial_ring import DifferentialPolynomialRing
D3 = DifferentialPolynomialRing(SR, ('rho','a'), ('x','y','z'),␣
↪→max_differential_orders=[5,6])
rho, a = D3.fibre_variables()
SA3 = SuperfunctionAlgebra(D3, D3.base_variables())
xi1,xi2,xi3 = SA3.odd_coordinates()
x,y,z = SA3.even_coordinates()
PA3 = PolyDifferentialOperatorAlgebra(D3, D3.base_variables())
P3 = rho*(xi1*xi2*diff(a,z) + xi2*xi3*diff(a,x) + xi3*xi1*diff(a,y)); P3
[26]: (rho*a_z)*xi0*xi1 + (rho*a_x)*xi1*xi2 + (-rho*a_y)*xi0*xi2
[27]: assoc3c3 = assoc3c.homogeneous_part(3, 3, 6)
[28]: %%time
linear_system = []
for diff_order in assoc3c3.differential_orders():
print(diff_order)
assoc3c3_operator = FormalityGraphOperator(SA3, PA3, assoc3c3.
↪→part_of_differential_order(diff_order))
assoc3c3_op = assoc3c3_operator.value_at_copies_of(P3)
assoc3c3_op_coeffs_diffpoly = [assoc3c3_op[m] for m in assoc3c3_op.
↪→multi_indices()]
assoc3c3_op_coeffs_consts = sum([diffpoly.coefficients() for diffpoly in␣
↪→assoc3c3_op_coeffs_diffpoly], [])
linear_system.extend(list(set(assoc3c3_op_coeffs_consts)))
(2, 1, 1)
(1, 2, 1)
(1, 1, 1)
(2, 1, 3)
3.4. KONTSEVICH STAR PRODUCT 83
(1, 2, 3)
(1, 1, 2)
(1, 1, 3)
(2, 1, 2)
(1, 2, 2)
(3, 1, 2)
(3, 1, 1)
(2, 2, 1)
(3, 2, 1)
CPU times: user 1min 37s, sys: 14.6 ms, total: 1min 37s
Wall time: 1min 37s
[29]: c = [var('c{}'.format(k)) for k in range(len(KGB.graphs(2,3)))]
[30]: solve(linear_system, c)
[30]: [[c0 == (1/24), c1 == (1/24), c2 == r5, c3 == r3, c4 == (1/8), c5 == 0, c6 == -r2 -
1/24, c7 == r2, c8 == (-1/24), c9 == (1/24), c10 == r4, c11 == r4, c12 == 0, c13 == 0,
c14 == r8, c15 == -r3 + r5 + r6 + 1/48, c16 == 2*r5 + r6 - 1/16, c17 == r5 - 1/48, c18
== r3 - 1/48, c19 == r7, c20 == r10, c21 == r9, c22 == r5 + r6 - 1/48, c23 == r5 + r6
- 1/48, c24 == r6, c25 == r3 + r5 + r6 - 1/24, c26 == 0, c27 == 0, c28 == (-1/48), c29
== r1]]
So, there remain only 10 parameters not yet constrained by the postulate of associa-
tivity for the 3D Nambu–Poisson structure (some of them will never be constrained by
associativity alone, due to the gauge freedom).
We can obtain the missing values either by direct integration (see Appendix A.1 in Chap-
ter 11), or by importing these values from [1], or by using all the methods in Chapter 11
(specifically, in Example 26) to find the few missing values.
[31]: c_values = [1/24, 1/24, 1/48, 1/48, 1/8, 0, -1/48, -1/48, -1/24, 1/24, 0, 0, 0, 0, 0,␣
↪→1/48, -1/48, 0, 0, 1/48, 0, 0, 0, 0, 0, 0, 0, 0, -1/48, 0]
c_subs = dict(zip(c,c_values))
This results in the cubic expansion ⋆ mod ō(h̄3) as seen from Cell [6], [7] in Section 3.4.2.
[32]: star3c_substituted = star3c.map_coefficients(lambda d: d.subs(c_subs))
[33]: star3c_substituted == star3
[33]: True
Remark. The associativity of ⋆ mod ō(h̄3) has been established (with these values of
coefficients in the top-degree ansatz) only for a restriction of ⋆ to a particular (however,
large) class of Poisson brackets. We claim nevertheless that the built star-product is
associative modulo ō(h̄3) for generic Poisson structures in any finite dimension. Let us
demonstrate this in the next section.
3.4.4 Star product associativity via Leibniz graphs
Now we consider at once Kontsevich’s star product ⋆ mod ō(h̄4): we have it from Chap-
ter 11.
84 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[34]: star4_txt = open('data/star4.txt').read().rstrip()
star4 = FGC.element_from_kgs_encoding(star4_txt) #; star4
%time star4_assoc = star4.insertion(0, star4, max_num_aerial=4) - star4.insertion(1,␣
↪→star4, max_num_aerial=4)
CPU times: user 3.03 s, sys: 4 ms, total: 3.03 s
Wall time: 3.03 s
In what follows, the associator of ⋆ is expressed as the sum of Leibniz graphs,
∑ n+1 ∑
Assoc h̄(⋆(P ))(f, g, h) = coeff(n) m(L)w(L)L(P, . . . , P, [[P, P ]])(f, g, h).
⩾ (n+ 1)!n 1 [L]∈Ln3
The weights w(L) are calculated by using the kontsevint software by Panzer [1].
To show the factorization Assoc(⋆(P )) = ♢(P, [[P, P ]]) mod ō(h̄4) for generic Poisson
brackets P we must expand Leibniz graphs in ♢ in the right-hand side to Kontsevich
graphs. Leibniz graphs contain tripod vertices (and possibly wedges), and Kontsevich
graphs are built of wedges: expanding the Jacobiator 1 [[P, P ]] amounts to inserting the
2
stick (edge) graph into the trident vertex. In effect, we can insert the stick graph into
every aerial vertex of L and then select only those graphs which are built of wedges.
[35]: stick = FGC(FormalityGraph(0,2,[(0,1)])); stick
[35]: 1*FormalityGraph(0, 2, [(0, 1)])
[36]: from gcaops.graph.formality_graph_basis import LeibnizGraphBasis
LGB = LeibnizGraphBasis(positive_differential_order=True); LGB
[36]: Basis consisting of representatives of isomorphism classes of Leibniz graphs (of
positive differential order) with no automorphisms that induce an odd permutation on
edges
We start with order h̄2 in the associator. The list L13 of Leibniz graphs on 3 sinks and 1
aerial vertex amounts to the tripod itself:
[37]: l31 = list(LGB.graphs(3,1)); l31
[37]: [FormalityGraph(3, 1, [(3, 0), (3, 1), (3, 2)])]
We will use a helper function to import the graph weights:
[38]: def vector_from_file(filename):
with open(filename) as f:
return vector(sage_eval('[{}]'.format(','.join(f.readlines()))))
The list of weights w(L) (here, consisting of just one number) is this:
[39]: wl31 = vector_from_file('data/weights_leibniz_3_1.txt'); wl31
[39]: (1/6)
We are able to present the sum of Leibniz graphs to balance the associator for ⋆ at h̄2 for
generic Poisson structure P :
3.4. KONTSEVICH STAR PRODUCT 85
[40]: L31 = -2/3 * FGC([(c*g.multiplicity(),g) for (c,g) in zip(wl31,l31)]); L31 #.show()
[40]: (-2/3)*FormalityGraph(3, 1, [(3, 0), (3, 1), (3, 2)])
As said, the Leibniz graphs must now be expanded into Kontsevich graphs:
[41]: L31_expanded = L31.insertion(3,stick,max_out_degree=2)
And they balance the order h̄2 in the associator:
[42]: L31_expanded == star4_assoc.homogeneous_part(3, 2, 4)
[42]: True
• Next order: h̄3 in the associator and Leibniz graphs on 3 sinks and 2 aerial vertices.
[43]: l32 = list(LGB.graphs(3,2))
wl32 = vector_from_file('data/weights_leibniz_3_2.txt') #; wl32
L32 = -1/3 * FGC([(c*g.multiplicity(),g) for (c,g) in zip(wl32,l32)]); L32 #.show()
[43]: (1/3)*FormalityGraph(3, 2, [(3, 2), (3, 4), (4, 0), (4, 1), (4, 2)]) +
(-1/3)*FormalityGraph(3, 2, [(3, 0), (3, 4), (4, 0), (4, 1), (4, 2)]) +
(-2/3)*FormalityGraph(3, 2, [(3, 0), (3, 1), (4, 0), (4, 1), (4, 2)]) +
(-2/3)*FormalityGraph(3, 2, [(3, 0), (3, 2), (4, 0), (4, 1), (4, 2)]) +
(-2/3)*FormalityGraph(3, 2, [(3, 1), (3, 2), (4, 0), (4, 1), (4, 2)]) +
(-1/6)*FormalityGraph(3, 2, [(3, 0), (3, 2), (4, 0), (4, 1), (4, 3)]) +
(1/6)*FormalityGraph(3, 2, [(3, 0), (3, 2), (3, 4), (4, 0), (4, 1)]) +
(-1/3)*FormalityGraph(3, 2, [(3, 1), (3, 2), (4, 0), (4, 1), (4, 3)]) +
(-1/3)*FormalityGraph(3, 2, [(3, 1), (3, 2), (3, 4), (4, 0), (4, 1)]) +
(1/6)*FormalityGraph(3, 2, [(3, 1), (3, 2), (4, 0), (4, 2), (4, 3)]) +
(-1/6)*FormalityGraph(3, 2, [(3, 1), (3, 2), (3, 4), (4, 0), (4, 2)]) +
(1/6)*FormalityGraph(3, 2, [(3, 2), (3, 4), (4, 0), (4, 1), (4, 3)]) +
(-1/6)*FormalityGraph(3, 2, [(3, 1), (3, 2), (3, 4), (4, 0), (4, 3)])
The number of Leibniz graphs with nonzero coefficients is 13:
[44]: len(L32)
[44]: 13
[45]: L32_expanded = sum(L32.insertion(k,stick,max_out_degree=2) for k in [3,4])
These are precisely the 39 Kontsevich graphs at order h̄3 in the associator of ⋆ for generic
Poisson structure.
[46]: len(L32_expanded)
[46]: 39
[47]: L32_expanded == star4_assoc.homogeneous_part(3, 3, 6)
[47]: True
The claim of associator’s factorization modulo ō(h̄3) is thus established.
Let us proceed to the next, fourth, order!
86 CHAPTER 3. IMPLEMENTATION OF FORMALITY
As usual, we generate the list of Leibniz graphs, import their weights w(L), and take the
(weighted) sum with multiplicities and with an overall leading coefficient coeff(3) = −8/3.
[48]: l33 = list(LGB.graphs(3,3))
[49]: wl33 = vector_from_file('data/weights_leibniz_3_3.txt') #; wl33
[50]: len(wl33) - list(wl33).count(0)
[50]: 241
[51]: L33 = -1/9 * FGC([(c*g.multiplicity(),g) for (c,g) in zip(wl33,l33)])
The number of Leibniz graphs with nonzero coefficients is 241:
[52]: len(L33)
[52]: 241
Leibniz graphs are now expanded into Kontsevich’s graphs built of wedges:
[53]: L33_expanded = sum(L33.insertion(k,stick,max_out_degree=2) for k in [3,4,5])
[54]: len(L33_expanded), len(star4_assoc.homogeneous_part(3, 4, 8))
[54]: (740, 740)
The two lengths of lists match, and moreover: these are the same sums of Kontsevich
graphs with the same coefficients!
[55]: L33_expanded == star4_assoc.homogeneous_part(3, 4, 8)
[55]: True
This proves that the star product expansion ⋆ mod ō(h̄4) is associative modulo ō(h̄4) for
generic Poisson structure.
Example. Let us inspect the order (3, 3, 1) in the ⋆-product associator in particular:
[56]: star4_assoc.homogeneous_part(3, 4, 8).part_of_differential_order((3,3,1))
[56]: (-1/3)*FormalityGraph(3, 4, [(3, 1), (3, 2), (4, 0), (4, 3), (5, 0), (5, 1), (6, 0),
(6, 1)]) + (1/3)*FormalityGraph(3, 4, [(3, 1), (3, 4), (4, 0), (4, 2), (5, 0), (5, 1),
(6, 0), (6, 1)]) + (-1/3)*FormalityGraph(3, 4, [(3, 2), (3, 6), (4, 0), (4, 1), (5,
0), (5, 1), (6, 0), (6, 1)])
It is realized by the expansion of a single Leibniz graph:
[57]: L_331 = l33[114]; L_331
[57]: FormalityGraph(3, 3, [(3, 0), (3, 1), (4, 0), (4, 1), (5, 0), (5, 1), (5, 2)])
[58]: -1/9 * wl33[114] * L_331.multiplicity() * sum(FGC(L_331).
↪→insertion(k,stick,max_out_degree=2) for k in [3,4,5])
3.5. LEIBNIZ GRAPH EXPANSION AND FACTORIZATION(S) 87
[58]: (-1/3)*FormalityGraph(3, 4, [(3, 1), (3, 2), (4, 0), (4, 3), (5, 0), (5, 1), (6, 0),
(6, 1)]) + (1/3)*FormalityGraph(3, 4, [(3, 1), (3, 4), (4, 0), (4, 2), (5, 0), (5, 1),
(6, 0), (6, 1)]) + (-1/3)*FormalityGraph(3, 4, [(3, 2), (3, 6), (4, 0), (4, 1), (5,
0), (5, 1), (6, 0), (6, 1)])
In the next section we will express the associator for Kontsevich’s star-product
⋆ mod ō(h̄6) as a sum of Leibniz graphs with some coefficients. The coefficients we
will obtain there are not necessarily (and probably not) those guaranteed by the Formal-
ity theorem (which we had in this section), yet any such coefficients do suffice to prove
the associativity up to ō(h̄6).
3.5 Leibniz graph expansion and factorization(s)
In this section we discuss two further aspects of Leibniz graphs: the iterative production
of Leibniz graphs from Kontsevich graphs, and the (non)uniqueness of Leibniz graph
factorizations.
3.5.1 Iterative production of Leibniz graphs
Some sums of Kontsevich graphs amount to the zero polydifferential operator, when
evaluated at copies of a Poisson structure. This is the case e.g. for the associator (f ⋆ g)⋆
h− f ⋆ (g ⋆ h) of Kontsevich’s ⋆-product. To prove that a sum of Kontsevich graphs built
of wedges amounts to zero when evaluated at copies of a Poisson structure, it suffices to
realize it as (the expansion of) a sum of Leibniz graphs. Here we consider the problem
of finding the (potentially) necessary Leibniz graphs by using an iterative process; see
Chapter 14.
Example. First we import Kontsevich’s ⋆ mod ō(h̄4):
[1]: from gcaops.graph.formality_graph_complex import FormalityGraphComplex
FGC = FormalityGraphComplex(SR, lazy=True)
[2]: star4 = FGC.element_from_kgs_encoding(open('data/star4.txt').read().rstrip())
We calculate the associator:
[3]: star4_assoc = star4.insertion(0, star4, max_num_aerial=4) - star4.insertion(1, star4,␣
↪→max_num_aerial=4)
[4]: star4_assoc.homogeneous_part(3, 2, 4)
[4]: (-2/3)*FormalityGraph(3, 2, [(3, 1), (3, 2), (4, 0), (4, 3)]) +
(2/3)*FormalityGraph(3, 2, [(3, 1), (3, 4), (4, 0), (4, 2)]) +
(-2/3)*FormalityGraph(3, 2, [(3, 2), (3, 4), (4, 0), (4, 1)])
So at h̄2 in the associator there are three Kontsevich graphs which themselves assimilate
to the Jacobiator on the associator’s three sinks. This was known from the seminal works
of [2] and [28]; we discuss the third order expansion of Assoc(⋆) mod ō(h̄3) in Chapter 10.
88 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[5]: star4_assoc4 = star4_assoc.homogeneous_part(3, 4, 8)
We generate the necessary Leibniz graphs from the Kontsevich graphs at h̄4, i.e. built
of 4 wedges over 3 sinks, by contraction of one internal edge in every such Kontsevich
graph:
[6]: from gcaops.graph.leibniz_graph_expansion import␣
↪→_kontsevich_graph_sum_to_leibniz_graphs
leibniz_graphs = []
_kontsevich_graph_sum_to_leibniz_graphs(star4_assoc4, leibniz_graphs)
len(leibniz_graphs)
[6]: 274
Referring to Chapter 12 below, we claim that these 274 Leibniz graphs, but not their
neighbors (obtained by contracting one edge between aerial vertices in each Kontsevich
graph in the expansion of those Leibniz graphs), are already enough to factorize the
respective part of the associator.
For optimization purposes, we can split the problem into many (here 23) smaller parts of
fixed differential orders (in-degrees of ground vertices):
[7]: len(list(star4_assoc4.differential_orders()))
[7]: 23
In general we can expand the found Leibniz graphs to Kontsevich graphs and contract
one edge (in all possible ways) in the resulting Kontsevich graphs. At each iterative step,
we form the linear system with the left-hand side containing (with one undetermined
coefficient for each Leibniz graph) the Kontsevich graph expansion of the Leibniz graphs,
and the right-hand side containing the Kontsevich graphs with their coefficients from the
input (here, the associator). If there is a solution to the linear system, we are happy and
we return it. If there is no solution to the linear system, then we either go to the next
step (if still new Leibniz graphs can be found by contracting edges in Kontsevich graphs
in the left-hand side) or report that there is no solution.
The following lines of output contain the grading of a tri-differential component of the
associator at h̄4 for ⋆ mod ō(h̄4), followed by the number of Kontsevich graphs in that
component, followed by the number of new Leibniz graphs and new Kontsevich graphs
obtained in each step. As soon as there appears a solution, the program reports writes
True about the existence of some factorization in that tri-differential order.
[8]: from gcaops.graph.leibniz_graph_expansion import␣
↪→kontsevich_graph_sum_to_leibniz_graph_sum
from gcaops.graph.leibniz_graph_expansion import␣
↪→leibniz_graph_sum_to_kontsevich_graph_sum
for diff_order in star4_assoc4.differential_orders():
print(diff_order, end=': ')
part = star4_assoc4.part_of_differential_order(diff_order)
part_Leibniz = kontsevich_graph_sum_to_leibniz_graph_sum(part, verbose=True)
print(leibniz_graph_sum_to_kontsevich_graph_sum(part_Leibniz) == part)
(3, 1, 3): 3K -> +1L -> +0K
True
3.5. LEIBNIZ GRAPH EXPANSION AND FACTORIZATION(S) 89
(2, 2, 3): 3K -> +1L -> +0K
True
(1, 3, 3): 3K -> +1L -> +0K
True
(3, 1, 2): 12K -> +4L -> +0K
True
(2, 2, 2): 26K -> +12L -> +7K
True
(1, 3, 2): 7K -> +4L -> +5K
True
(2, 1, 3): 12K -> +4L -> +0K
True
(1, 2, 3): 12K -> +4L -> +0K
True
(3, 1, 1): 26K -> +9L -> +1K
True
(2, 2, 1): 53K -> +24L -> +6K
True
(1, 3, 1): 26K -> +9L -> +1K
True
(1, 1, 3): 26K -> +9L -> +1K
True
(2, 1, 2): 55K -> +24L -> +4K
True
(1, 2, 2): 53K -> +24L -> +6K
True
(2, 1, 1): 94K -> +35L -> +7K
True
(1, 2, 1): 90K -> +35L -> +11K
True
(1, 1, 1): 117K -> +28L -> +6K
True
(1, 1, 2): 94K -> +35L -> +7K
True
(3, 2, 1): 12K -> +4L -> +0K
True
(3, 2, 2): 3K -> +1L -> +0K
True
(2, 3, 2): 3K -> +1L -> +0K
True
(2, 3, 1): 7K -> +4L -> +5K
True
(3, 3, 1): 3K -> +1L -> +0K
True
The number of actually used layers of Leibniz graphs can be read from the number of
times the count of Leibniz graphs is printed: single time appearance means that the 0th
layer is enough. Indeed we observe that the 0th layer of Leibniz graphs is enough to find
a factorization in this case of Assoc(⋆) mod ō(h̄4).
Example. We repeat the above example for Kontsevich’s ⋆ mod ō(h̄5), known from [1].
[9]: star5 = FGC.element_from_kgs_encoding(open('data/star5.txt').read().rstrip())
[10]: star5_assoc = star5.insertion(0, star5, max_num_aerial=5) - star5.insertion(1, star5,␣
↪→max_num_aerial=5)
90 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[11]: star5_assoc5 = star5_assoc.homogeneous_part(3, 5, 10)
The Leibniz graph factorization problem splits at h̄5 into 54 tri-differential parts within
the associator.
[12]: len(list(star5_assoc5.differential_orders()))
[12]: 54
For every tri-differential order, we solve each Leibniz graph factorization problem itera-
tively over the layers of Leibniz graphs:
[13]: for diff_order in star5_assoc5.differential_orders():
print(diff_order, end=': ')
part = star5_assoc5.part_of_differential_order(diff_order)
part_Leibniz = kontsevich_graph_sum_to_leibniz_graph_sum(part, verbose=True)
print(leibniz_graph_sum_to_kontsevich_graph_sum(part_Leibniz) == part)
(3, 1, 4): 12K -> +4L -> +0K
True
(2, 2, 4): 15K -> +5L -> +0K
True
(2, 1, 4): 38K -> +13L -> +1K
True
(1, 3, 4): 12K -> +4L -> +0K
True
(1, 2, 4): 38K -> +13L -> +1K
True
(1, 1, 4): 74K -> +27L -> +7K
True
(4, 2, 3): 3K -> +1L -> +0K
True
(4, 1, 3): 12K -> +4L -> +0K
True
(3, 3, 3): 3K -> +1L -> +0K
True
(3, 2, 3): 32K -> +14L -> +7K
True
(3, 1, 3): 79K -> +35L -> +4K
True
(2, 4, 3): 3K -> +1L -> +0K
True
(2, 3, 3): 32K -> +14L -> +7K
True
(2, 2, 3): 165K -> +79L -> +15K
True
(2, 1, 3): 316K -> +151L -> +23K
True
(1, 4, 3): 7K -> +4L -> +5K
True
(1, 3, 3): 72K -> +35L -> +11K
True
(1, 2, 3): 317K -> +151L -> +22K
True
(1, 1, 3): 486K -> +223L -> +64K
True
(4, 1, 4): 3K -> +1L -> +0K
3.5. LEIBNIZ GRAPH EXPANSION AND FACTORIZATION(S) 91
True
(3, 2, 4): 3K -> +1L -> +0K
True
(2, 3, 4): 3K -> +1L -> +0K
True
(1, 4, 4): 3K -> +1L -> +0K
True
(4, 2, 2): 15K -> +5L -> +0K
True
(4, 1, 2): 38K -> +13L -> +1K
True
(3, 3, 2): 32K -> +14L -> +7K
True
(3, 2, 2): 165K -> +79L -> +15K
True
(3, 1, 2): 316K -> +151L -> +23K
True
(2, 3, 2): 166K -> +78L -> +14K
True
(2, 2, 2): 638K -> +340L -> +76K
True
(2, 1, 2): 958K -> +483L -> +116K
True
(1, 4, 2): 33K -> +13L -> +6K
True
(1, 3, 2): 296K -> +151L -> +43K
True
(1, 2, 2): 964K -> +481L -> +108K
True
(1, 1, 2): 1213K -> +530L -> +201K
True
(2, 4, 2): 10K -> +5L -> +5K
True
(4, 2, 1): 38K -> +13L -> +1K
True
(4, 1, 1): 74K -> +27L -> +7K
True
(3, 3, 1): 72K -> +35L -> +11K
True
(3, 2, 1): 317K -> +151L -> +22K
True
(3, 1, 1): 486K -> +223L -> +64K
True
(2, 4, 1): 33K -> +13L -> +6K
True
(2, 3, 1): 296K -> +151L -> +43K
True
(2, 2, 1): 964K -> +481L -> +108K
True
(2, 1, 1): 1213K -> +530L -> +201K
True
(1, 4, 1): 54K -> +27L -> +27K
True
(1, 3, 1): 355K -> +200L -> +170K
True
(1, 2, 1): 934K -> +462L -> +415K
True
(1, 1, 1): 1028K -> +400L -> +449K
92 CHAPTER 3. IMPLEMENTATION OF FORMALITY
True
(4, 3, 1): 12K -> +4L -> +0K
True
(4, 3, 2): 3K -> +1L -> +0K
True
(3, 4, 2): 3K -> +1L -> +0K
True
(3, 4, 1): 7K -> +4L -> +5K
True
(4, 4, 1): 3K -> +1L -> +0K
True
Again we observe that the 0th layer of Leibniz graphs is enough to find a factorization in
this case of Assoc(⋆) mod ō(h̄5)
Example. We import Kontsevich’s tetrahedral flow from the future Section 5.1.
[14]: from gcaops.graph.formality_graph import FormalityGraph
Q_tetra = FGC([(-1, FormalityGraph(2, 4, [(2, 4), (2, 5), (3, 2), (3, 5), (4, 3), (4,␣
↪→5), (5, 0), (5, 1)])),
(-3, FormalityGraph(2, 4, [(2, 3), (2, 5), (3, 4), (3, 5), (4, 1), (4,␣
↪→2), (5, 0), (5, 4)])),
(-3, FormalityGraph(2, 4, [(2, 3), (2, 4), (3, 4), (3, 5), (4, 1), (4,␣
↪→5), (5, 0), (5, 2)]))])
We calculate its Schouten bracket with the wedge graph:
[15]: wedge = FGC(FormalityGraph(2,1,[(2,0),(2,1)]))
[16]: P_Q_tetra = wedge.schouten_bracket(Q_tetra)
[17]: len(P_Q_tetra)
[17]: 39
This time, let us repeat the iterations until saturation when no new Leibniz graphs are
produced any longer:
[18]: kontsevich_graph_sum_to_leibniz_graph_sum(P_Q_tetra, force_saturation=True,␣
↪→verbose=True);
39K -> +46L -> +306K -> +209L -> +570K -> +138L -> +459K -> +67L -> +189K -> +3L ->
+12K
This iterative count means the following: starting with a given sum of 39 Kontsevich’s
graphs in [[P,Q]] and contracting one internal edge in each graph, we obtain 46 Leibniz
graphs in the (initial) 0th layer. Expanding them into Kontsevich’s graphs we do ob-
tain 306 new such graphs. Repeating the contraction and expansion procedure we then
generate 209, 138, etc. new Leibniz graphs in every next layer of neighbors — until the
saturation, when no new Leibniz graphs are produced. By construction, the factorization
of [[P,Q]] via Leibniz graphs cannot refer to any Leibniz graphs other than the ones which
have been produced by the iterative algorithm. For more details, we refer to Chapter 14.
3.5. LEIBNIZ GRAPH EXPANSION AND FACTORIZATION(S) 93
Remark. In Section 5.1 below, we illustrate another method (by Kontsevich, 1996) to
find all the Leibniz graphs which are sufficient for a factorization of the Poisson cocycle
condition, [[P,Q]] = ♢(P, [[P, P ]]). This is achieved by orienting the tetrahedron over the
three sinks such that there are three wedges and one trident; the extra edges (not in the
tetrahedron) go to the three ground vertices.
3.5.2 Leibniz graph factorization (non)uniqueness
Some sums of Leibniz graphs amount to zero when expanded to Kontsevich graphs; this
can lead to non-uniqueness of solutions to the factorization problems. Let us illustrate
this effect by examples.
We begin with a count. Leibniz graphs are expanded to sums of Kontsevich graphs
by inserting the stick graph into the trivalent vertex. This expansion map may have a
nontrivial kernel. We now compute the dimension of the kernel of that map restricted to
Leibniz graphs of a given bi-grading.
[13]: def leibniz_graph_expansion_nullity(num_ground, num_aerial, skew=False):
from gcaops.graph.formality_graph_basis import LeibnizGraphBasis
LGB = LeibnizGraphBasis(positive_differential_order=True,␣
↪→mod_ground_permutations=skew)
from gcaops.graph.formality_graph_basis import KontsevichGraphBasis
KGB = KontsevichGraphBasis(positive_differential_order=True)
from gcaops.graph.formality_graph_complex import FormalityGraphComplex
FGC = FormalityGraphComplex(QQ, lazy=True)
from gcaops.graph.formality_graph import FormalityGraph
stick = FGC(FormalityGraph(0,2,[(0,1)]))
K = KGB.graphs(num_ground, num_aerial+1)
#print(list(K))
LL = LGB.graphs(num_ground, num_aerial)
if skew:
LL = [L for L in LL if FGC(L).ground_skew_symmetrization() != 0]
#print(len(K), len(LL))
M = matrix(QQ, len(K), len(LL), sparse=True)
for i, L in enumerate(LL):
L = FGC(L)
if skew:
L = L.ground_skew_symmetrization()
L_expanded = sum(L.insertion(k,stick,max_out_degree=2) for k in␣
↪→range(num_ground,num_ground+num_aerial))
for (c,g) in L_expanded:
#print(g)
M[K.index(g), i] = c # NOTE: uses that normal form of graphs in KGB is␣
↪→same as in FGC
return M.right_nullity()
#for v in M.right_kernel().basis():
# if list(v).count(0) != len(v) - 1:
# print(v)
# break
[14]: for n in range(1,5):
print((3, n), '--> nullity', leibniz_graph_expansion_nullity(3, n))
(3, 1) --> nullity 0
(3, 2) --> nullity 0
94 CHAPTER 3. IMPLEMENTATION OF FORMALITY
(3, 3) --> nullity 12
(3, 4) --> nullity 538
So, e.g. for the Leibniz graphs on 3 sinks and 3 aerial vertices (of which one is a trident
and two are wedges), there are twelve linearly independent linear combinations of Leibniz
graphs that expand to zero sums of Kontsevich graphs built of wedges.
The bi-grading (3, 4) is relevant for (the Poisson cocycle condition for) Kontsevich’s tetra-
hedral flow. Yet, let us point out, referring to the second example in the preceding section
3.5.1, that many of the identities for the Leibniz graphs are specific about the graphs
which did not show up in the iterative process of building the layers of Leibniz graphs
that started with [[P,Q]] for the tetrahedral flow. That is, the number of solutions ♢ for
the cocycle condition factorization problem which have the property that they are in the
span of the graphs produced by the iterative process, would be less than 538; we know 2.
Example. Here is a sum of nine nonzero Leibniz graphs on 3 sinks and 3 aerial vertices
that expands to zero.
[15]: from gcaops.graph.formality_graph_basis import LeibnizGraphBasis
LGB = LeibnizGraphBasis(positive_differential_order=True)
[16]: from gcaops.graph.formality_graph import FormalityGraph
stick = FGC(FormalityGraph(0,2,[(0,1)]))
Here is the promised nontrivial linear combination of nine nonzero Leibniz graphs:
[17]: mystery = FGC([(s, LGB.graphs(3,3)[k]) for k,s in␣
↪→zip([6,28,29,84,113,159,161,165,167], [1,1,-1,-1,1,-1,1,1,-1])]); mystery
[17]: 1*FormalityGraph(3, 3, [(3, 4), (3, 5), (4, 0), (4, 1), (5, 0), (5, 1), (5, 2)]) +
1*FormalityGraph(3, 3, [(3, 1), (3, 4), (4, 0), (4, 2), (4, 5), (5, 0), (5, 1)]) +
(-1)*FormalityGraph(3, 3, [(3, 1), (3, 2), (3, 5), (4, 0), (4, 3), (5, 0), (5, 1)]) +
(-1)*FormalityGraph(3, 3, [(3, 2), (3, 4), (4, 0), (4, 1), (4, 5), (5, 0), (5, 1)]) +
1*FormalityGraph(3, 3, [(3, 2), (3, 5), (4, 0), (4, 1), (4, 3), (5, 0), (5, 1)]) +
(-1)*FormalityGraph(3, 3, [(3, 1), (3, 5), (4, 0), (4, 2), (4, 3), (5, 0), (5, 1)]) +
1*FormalityGraph(3, 3, [(3, 1), (3, 2), (3, 4), (4, 0), (4, 5), (5, 0), (5, 1)]) +
1*FormalityGraph(3, 3, [(3, 1), (3, 4), (3, 5), (4, 0), (4, 2), (5, 0), (5, 1)]) +
(-1)*FormalityGraph(3, 3, [(3, 1), (3, 2), (4, 0), (4, 3), (4, 5), (5, 0), (5, 1)])
[18]: mystery.show()
3.5. LEIBNIZ GRAPH EXPANSION AND FACTORIZATION(S) 95
[19]: sum(mystery.insertion(k,stick,max_out_degree=2) for k in [3,4,5])
[19]: 0
Equivalently, we can expand sums of Leibniz graphs into Kontsevich graphs by using
another method:
[ ]: #leibniz_graph_sum_to_kontsevich_graph_sum(mystery)
Example. Here is a sum of 14 skew Leibniz graphs on 2 sinks and 4 aerial vertices, such
that its expansion into Kontsevich graphs vanishes.
[20]: LGB_skew = LeibnizGraphBasis(positive_differential_order=True,␣
↪→mod_ground_permutations=True)
[21]: skew_mystery = FGC([(s, LGB_skew.graphs(2,4)[k]) for k,s in␣
↪→zip([2,52,93,128,129,193,199,328,338,349,422,438,479,484],␣
↪→[1,-2,-1,-1,1,1,1,-1,-1,-1,-1,-1,1,-1])]); skew_mystery
[21]: 1*FormalityGraph(2, 4, [(2, 4), (2, 5), (3, 1), (3, 4), (3, 5), (4, 0), (4, 5), (5,
0), (5, 1)]) + (-2)*FormalityGraph(2, 4, [(2, 1), (2, 4), (2, 5), (3, 1), (3, 5), (4,
0), (4, 3), (5, 0), (5, 4)]) + (-1)*FormalityGraph(2, 4, [(2, 4), (2, 5), (3, 1), (3,
2), (3, 4), (4, 0), (4, 5), (5, 0), (5, 1)]) + (-1)*FormalityGraph(2, 4, [(2, 1), (2,
3), (3, 1), (3, 5), (4, 0), (4, 2), (4, 5), (5, 0), (5, 2)]) + 1*FormalityGraph(2, 4,
[(2, 1), (2, 3), (3, 1), (3, 5), (4, 0), (4, 3), (4, 5), (5, 0), (5, 2)]) +
1*FormalityGraph(2, 4, [(2, 4), (2, 5), (3, 1), (3, 4), (3, 5), (4, 0), (4, 2), (5,
0), (5, 1)]) + 1*FormalityGraph(2, 4, [(2, 4), (2, 5), (3, 1), (3, 5), (4, 0), (4, 3),
(5, 0), (5, 1), (5, 3)]) + (-1)*FormalityGraph(2, 4, [(2, 1), (2, 5), (3, 1), (3, 4),
96 CHAPTER 3. IMPLEMENTATION OF FORMALITY
(3, 5), (4, 0), (4, 5), (5, 0), (5, 4)]) + (-1)*FormalityGraph(2, 4, [(2, 4), (2, 5),
(3, 1), (3, 2), (4, 0), (4, 5), (5, 0), (5, 1), (5, 4)]) + (-1)*FormalityGraph(2, 4,
[(2, 1), (2, 5), (3, 1), (3, 5), (4, 0), (4, 3), (4, 5), (5, 0), (5, 3)]) +
(-1)*FormalityGraph(2, 4, [(2, 3), (2, 4), (3, 1), (3, 5), (4, 0), (4, 5), (5, 0), (5,
1), (5, 4)]) + (-1)*FormalityGraph(2, 4, [(2, 1), (2, 4), (2, 5), (3, 1), (3, 5), (4,
0), (4, 3), (5, 0), (5, 3)]) + 1*FormalityGraph(2, 4, [(2, 1), (2, 3), (2, 5), (3, 1),
(3, 4), (4, 0), (4, 5), (5, 0), (5, 4)]) + (-1)*FormalityGraph(2, 4, [(2, 3), (2, 5),
(3, 1), (3, 2), (4, 0), (4, 2), (4, 3), (5, 0), (5, 1)])
[22]: skew_mystery.show()
When we skew-symmetrize the above sum of Leibniz graphs over the content of two sinks,
the result remains nonzero:
[23]: skew_mystery.ground_skew_symmetrization() == 0
[23]: False
3.6. CYCLIC WEIGHT RELATIONS 97
Finally, let us expand the skew-symmetrized Leibniz graphs into Kontsevich’s graphs
built of wedges:
[24]: leibniz_graph_sum_to_kontsevich_graph_sum(skew_mystery.ground_skew_symmetrization())
[24]: 0
This results in zero, as promised.
On 3 sinks and 4 or 6 aerial vertices (i.e. 3 or 5 wedges and one trident) the same
mechanism is responsible for the coexistence of solutions ♢1 and ♢2 in the factorization
problem for the Poisson cocycle condition .[[P,Q ⊗nγ(P )]] = 0 on Jac(P ) = 0 (here γ = γ3
or γ5, respectively), see Chapters 5, 15, 16 and 17.
3.6 Cyclic weight relations
On the basis of previous work by Shoikhet on cyclic formality [20], Willwacher and
Felder [21] showed that the weights of graphs in Kontsevich’s deformation quantization
[28] satisfy a class of linear relations, namely the cyclic weight relations. In this section
we obtain the linear algebraic system of cyclic weight relations between the weights of
Kontsevich graphs in Kontsevich’s star-product f ⋆ g mod ō(h̄5) and between the weights
of Leibniz graphs in the associator (f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) mod ō(h̄5).
3.6.1 From a basis to relations
We define a basis for the vector space spanned by Kontsevich graphs built of wedges,
that is by the Formality graphs which show up in Kontsevich’s star-product f ⋆ g and its
associator (f ⋆ g) ⋆ h− f ⋆ (g ⋆ h):
[2]: from gcaops.graph.formality_graph_basis import FormalityGraphComplexBasis,␣
↪→KontsevichGraphBasis
KGB = KontsevichGraphBasis(positive_differential_order=True); KGB
[2]: Basis consisting of representatives of isomorphism classes of Kontsevich graphs (of
positive differential order) with no automorphisms that induce an odd permutation on
edges
Recall that we can then generate a basis of directed graphs for the bi-graded homogeneity
component of degree (m,n) with a given number m of ground vertices, n aerial vertices,
and e = 2n edges.
Here is an example:
[3]: list(KGB.graphs(2,2))
[3]: [FormalityGraph(2, 2, [(2, 1), (2, 3), (3, 0), (3, 1)]),
FormalityGraph(2, 2, [(2, 0), (2, 3), (3, 0), (3, 1)]),
FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 0), (3, 1)]),
FormalityGraph(2, 2, [(2, 1), (2, 3), (3, 0), (3, 2)])]
Using this basis and treating the graphs as placeholders for their Kon∑tsevich weights, we
can generate the linear algebraic system of cyclic weight relations j cijw(Γj) = 0 by
98 CHAPTER 3. IMPLEMENTATION OF FORMALITY
calling the cyclic_weight_relations method on the graph basis, passing the bi-grading
in the input:
[4]: KGB.cyclic_weight_relations(2,2)
[4]: [-1 2 -1 0]
[ 1 1 0 0]
[ 0 0 0 0]
[ 0 -1 0 -2]
By construction, there is one cyclic weight relation per graph (although not all of the
relations in the resulting system are necessarily linearly independent). The above method
returns the square matrix C = (cij) in which the cyclic weight relations are encoded in
the following way: every row (ci1, . . . , cik) corresponds to the graph Γi that marks the
relation; the entries of that row are the coefficients cij of weights w(∑Γj) of the graphs Γj
in the basis at the given bi-grading. Every such linear combination j cijw(Γj) vanishes
(under the default assumption that the list of graphs in the basis is enough to make the
relation well-defined). Let us illustrate this construction.
3.6.2 Kontsevich graphs in f ⋆ g
First we consider the Kontsevich graphs appearing at each order in Kontsevich’s star-
product, i.e. graphs built of wedges, and having two sinks. We now start producing
(and listing) and counting the cyclic weight relations for Kontsevich’s graphs on n aerial
vertices, i.e. at h̄n in the ⋆-product.
[5]: len(list(KGB.graphs(2,1)))
[5]: 1
[6]: ck21 = KGB.cyclic_weight_relations(2,1); ck21
[6]: [0]
So, there are no cyclic weight relations other than 0 = 0 for Kontsevich’s wedge graph at
h̄1 in the star product.
[7]: len(list(KGB.graphs(2,2)))
[7]: 4
We recognize all these four graphs on two sinks and two aerial vertices at h̄2 in the star-
product expansion, see Figure 1 in Chapter 11. We now establish that the Kontsevich
weights of these four graphs are constrained by four cyclic weight relations, of which three
are linearly independent, and the remaining one is a tautology: the linear combination
of weights with zero coefficients equals zero.
[8]: ck22 = KGB.cyclic_weight_relations(2,2); ck22
[8]: [-1 2 -1 0]
[ 1 1 0 0]
[ 0 0 0 0]
[ 0 -1 0 -2]
3.6. CYCLIC WEIGHT RELATIONS 99
The tautology in the third line is produced by the Moyal graph at h̄2 in ⋆.
[9]: ck22.right_nullity()
[9]: 1
That is, the corank of the linear algebraic system equals one.
Next, let us count nonzero Kontsevich’s graphs built of three wedges over two sinks. This
time, not all of them show up – in the authentic ⋆-product – with nonzero weights.
[10]: len(list(KGB.graphs(2,3)))
[10]: 30
[11]: ck23 = KGB.cyclic_weight_relations(2,3)
ck23.right_nullity()
[11]: 11
So the rank of the system is 30− 11 = 19.
Remark. Also nonzero graphs with zero Kontsevich weights can produce cyclic weight
relations that increase the rank of the system! This is very important for constraining the
weights of Kontsevich graphs which actually show up in the star-product. For example,
only 13 graphs are seen at h̄3 in ⋆ (cf. Figure 1 in Chapter 11); still the rank of the linear
system at hand is 19 > 13. The missing six nontrivial relations were produced by the
invisible nonzero graphs.
[12]: len(list(KGB.graphs(2,4)))
[12]: 331
[13]: ck24 = KGB.cyclic_weight_relations(2,4)
ck24.right_nullity()
[13]: 103
Hence the rank of the system is 228. From the main result in Chapter 11, i.e. the explicit
formula ⋆ mod ō(h̄4) we remember that there are 247 Kontsevich’s graphs showing up
with nonzero coefficients at h̄4 in the star-product. Hence the system of cyclic weight
relations, taken alone, does not yet fully constrain those weights: firstly because there are
more nonzero weights than the rank of the system. Secondly the cyclic weight relations
constrain the weights of all the relevant graphs, including those weights which are a
posteriori found to be zero numbers. The overall number of unknowns (331) is thus much
greater than 247.
[14]: len(list(KGB.graphs(2,5)))
[14]: 4907
[15]: ck25 = KGB.cyclic_weight_relations(2,5)
ck25.right_nullity()
100 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[15]: 1561
Likewise, here the rank equals 3346.
[16]: len(list(KGB.graphs(2,6)))
[16]: 91694
[17]: #%time ck26 = KGB.cyclic_weight_relations(2,6)
[18]: #%time ck26.right_nullity()
[19]: %time len(list(KGB.graphs(2,7)))
CPU times: user 3min 21s, sys: 1.68 s, total: 3min 22s
Wall time: 3min 22s
[19]: 2053511
All the cyclic weight relations corresponding to all the Kontsevich graphs (of pos-
itive differential order) in the star-product modulo ō(h̄5) are available from https:
//rburing.nl/gcaops. The files come in pairs: for each order n ⩽ 6 there is a file
kontsevich_2_n.txt containing an ordered basis of Kontsevich graphs built of n wedges
over two sinks, and a file cyclic_kontsevich_2_n.txt with all cyclic weight relations
between those graphs (as a sparse matrix C with respect to the ordered basis: each line
is of the form i j C[i,j]). The full list of pairs of files is as follows:
• kontsevich_2_1.txt and cyclic_kontsevich_2_1.txt
• kontsevich_2_2.txt and cyclic_kontsevich_2_2.txt
• kontsevich_2_3.txt and cyclic_kontsevich_2_3.txt
• kontsevich_2_4.txt and cyclic_kontsevich_2_4.txt
• kontsevich_2_5.txt and cyclic_kontsevich_2_5.txt
• kontsevich_2_6.txt and cyclic_kontsevich_2_6.txt
3.6.3 Leibniz graphs in (f ⋆ g) ⋆ h− f ⋆ (g ⋆ h)
We consider the Leibniz graphs appearing in the associator of Kontsevich’s star-product:
[20]: from gcaops.graph.formality_graph_basis import LeibnizGraphBasis
[21]: LGB = LeibnizGraphBasis(positive_differential_order=True); LGB
[21]: Basis consisting of representatives of isomorphism classes of Leibniz graphs (of
positive differential order) with no automorphisms that induce an odd permutation on
edges
Note that there is a shift of degrees: Leibniz graphs with n− 1 aerial vertices appear at
h̄n in the associator for ⋆.
First of all we can count such graphs; recall that all aerial vertices but one are the tops
of wedges and one aerial vertex is the top of a tripod.
3.6. CYCLIC WEIGHT RELATIONS 101
[22]: len(list(LGB.graphs(3,1)))
[22]: 1
Indeed, this is the tripod itself, standing on three sinks.
[23]: cl31 = LGB.cyclic_weight_relations(3,1); cl31
[23]: [0]
[24]: len(list(LGB.graphs(3,2)))
[24]: 15
The ordered list of encodings of these 15 nonzero Leibniz graphs on three sinks and two
aerial vertices is available as file leibniz_graphs_3_1.txt from https://rburing.nl/
gcaops. We notice that with two aerial vertices, no Leibniz graph can be a zero graph
(because the aerial vertices have different nature: a wedge versus a tripod, hence there is
no automorphism preserving the sinks).
[25]: cl32 = LGB.cyclic_weight_relations(3,2); cl32
[25]: [-1 1 1 0 -1 0 0 0 0 0 0 0 0 0 0]
[ 0 -1 2 -1 0 0 0 0 0 0 0 0 0 0 0]
[ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0]
[ 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0]
[ 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0]
[ 0 0 0 1 0 0 -1 1 0 -1 0 0 0 0 0]
[ 0 0 0 0 0 0 0 -1 0 -1 -1 0 0 0 0]
[ 0 0 0 0 1 0 -1 0 -1 0 0 -1 0 0 0]
[ 0 0 0 0 0 0 -1 0 0 0 0 0 -1 -1 0]
[ 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 -1]
[ 0 0 1 0 0 0 0 0 0 0 0 0 1 0 -1]
Remark. By construction, such matrices always have integer entries; the entries are
bounded by the number of graphs (that is, by the size of the square matrix). For more
information about these relations, interpretations and their origin in the cyclic coho-
mology theory we refer to Willwacher–Felder [21] and the original paper by Shoikhet
[20].
[26]: cl32.right_nullity()
[26]: 3
In consequence, the rank of the system of 15 linear algebraic equations at h̄3 is 12.
[27]: len(list(LGB.graphs(3,3)))
[27]: 301
102 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[28]: cl33 = LGB.cyclic_weight_relations(3,3)
cl33.right_nullity()
[28]: 66
The rank of the system of cyclic weight relations for the Leibniz graphs at h̄4 in the
factorization of the associator is equal to 235.
[29]: len(list(LGB.graphs(3,4)))
[29]: 6741
[30]: cl34 = LGB.cyclic_weight_relations(3,4)
cl34.right_nullity()
[30]: 1469
Thus by having reached 4 aerial vertices in Leibniz graphs, we have reached the order h̄5
in the associator; the rank of the system for the weights of Leibniz graphs is 5272.
[31]: len(list(LGB.graphs(3,5)))
[31]: 171528
[32]: #cl35 = LGB.cyclic_weight_relations(3,5)
#cl35.right_nullity()
We are approaching terra incognita; at h̄6 in the associator’s factorization we know the
full list of suitable Leibniz graphs (with 5 aerial vertices), yet we have not computed the
rank of the linear system of cyclic weight relations.
[33]: %time len(list(LGB.graphs(3,6)))
CPU times: user 7min 59s, sys: 3.62 s, total: 8min 2s
Wall time: 8min 2s
[33]: 4902838
The normal form encodings (with respect to the gcaops internal format, based on nauty) of
bases of Leibniz graphs (of positive differential order) with n ⩽ 5 aerial vertices at h̄n+1
are stored in the files leibniz_3_n.txt at https://rburing.nl/gcaops. The sparse
coefficient matrices C of cyclic weight relations, referred to the ordering of graphs in the
bases, are contained in the plain text files cyclic_leibniz_3_n.txt (each line is of the
form i j C[i,j]). The full list of pairs of files is as follows:
• leibniz_3_1.txt and cyclic_leibniz_3_1.txt
• leibniz_3_2.txt and cyclic_leibniz_3_2.txt
• leibniz_3_3.txt and cyclic_leibniz_3_3.txt
• leibniz_3_4.txt and cyclic_leibniz_3_4.txt
• leibniz_3_5.txt and cyclic_leibniz_3_5.txt
3.6. CYCLIC WEIGHT RELATIONS 103
3.6.4 Known weights satisfy the cyclic weight relations
Proposition. We confirm that all these cyclic weight relations up to n = 5 vertices
are satisfied by the weights of Kontsevich graphs in the authentic ⋆-product and by the
weights of Leibniz graphs in its associator; we calculated all these weights by using the
kontsevint program by Panzer [1].
Notation. In Panzer’s kontsevint the m ground vertices of a Formality graph are labeled
by p1, . . . , pm and the n internal vertices by integers 1, . . . , n; the encoding then consists
of the list of lists of targets of the aerial vertices.
[34]: def kontsevint_weights_maple_program(graph_list):
return '[{}];'.format(','.join(['weight({})'.format(g.kontsevint_encoding()) for␣
↪→g in graph_list]))
[35]: def vector_from_file(filename):
with open(filename) as f:
return vector(sage_eval('[{}]'.format(','.join(f.readlines()))))
Example.
[36]: print(kontsevint_weights_maple_program(KGB.graphs(2,1)))
[weight([[p1,p2]])];
[37]: wk21 = vector_from_file('data/weights_kontsevich_2_1.txt'); wk21
[37]: (1/2)
[38]: len(wk21) - list(wk21).count(0)
[38]: 1
[39]: ck21*wk21
[39]: (0)
Example.
[40]: print(kontsevint_weights_maple_program(KGB.graphs(2,2)))
[weight([[p2,2],[p1,p2]]),weight([[p1,2],[p1,p2]]),weight([[p1,p2],[p1,p2]]),weight([[
p2,2],[p1,1]])];
[41]: wk22 = vector_from_file('data/weights_kontsevich_2_2.txt'); wk22
[41]: (-1/12, 1/12, 1/4, -1/24)
[42]: len(wk22) - list(wk22).count(0)
[42]: 4
104 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[43]: ck22*wk22
[43]: (0, 0, 0, 0)
[44]: print(kontsevint_weights_maple_program(KGB.graphs(2,3)))
[weight([[p2,3],[p2,3],[p1,p2]]),weight([[p1,3],[p1,3],[p1,p2]]),weight([[2,3],[p1,3],
[p1,p2]]),weight([[2,3],[p2,3],[p1,p2]]),weight([[p1,p2],[p1,p2],[p1,p2]]),weight([[p2
,3],[p1,3],[p1,p2]]),weight([[p2,2],[p1,3],[p1,p2]]),weight([[p2,3],[p1,1],[p1,p2]]),w
eight([[p2,3],[p1,p2],[p1,p2]]),weight([[p1,3],[p1,p2],[p1,p2]]),weight([[p2,2],[p2,3]
,[p1,1]]),weight([[p2,2],[p1,3],[p1,1]]),weight([[p2,2],[p2,3],[p1,p2]]),weight([[p1,2
],[p1,3],[p1,p2]]),weight([[2,3],[1,3],[p1,p2]]),weight([[p2,3],[p2,3],[p1,2]]),weight
([[p2,3],[p1,1],[p1,1]]),weight([[2,3],[p1,1],[p1,p2]]),weight([[2,3],[p2,1],[p1,p2]])
,weight([[2,3],[p2,3],[p1,2]]),weight([[2,3],[p2,3],[p1,1]]),weight([[2,3],[p2,1],[p1,
2]]),weight([[p2,3],[p1,3],[p1,1]]),weight([[p2,2],[p2,3],[p1,2]]),weight([[p2,3],[p1,
3],[p1,2]]),weight([[p2,2],[p2,1],[p1,2]]),weight([[p2,2],[p2,1],[p1,p2]]),weight([[p1
,2],[p1,1],[p1,p2]]),weight([[p2,2],[p1,1],[p1,p2]]),weight([[2,3],[p2,1],[p1,1]])];
[45]: wk23 = vector_from_file('data/weights_kontsevich_2_3.txt'); wk23
[45]: (1/24, 1/24, 1/48, 1/48, 1/8, 0, -1/48, -1/48, -1/24, 1/24, 0, 0, 0, 0, 0, 1/48,
-1/48, 0, 0, 1/48, 0, 0, 0, 0, 0, 0, 0, 0, -1/48, 0)
[46]: len(wk23) - list(wk23).count(0)
[46]: 13
[47]: ck23*wk23
[47]: (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0)
[48]: #print(kontsevint_weights_maple_program(Kgraphs(2,4)))
[49]: wk24 = vector_from_file('data/weights_kontsevich_2_4.txt') #; wk24
[50]: len(wk24) - list(wk24).count(0)
[50]: 247
[51]: ck24*wk24
[51]: (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
3.6. CYCLIC WEIGHT RELATIONS 105
[52]: #print(kontsevint_weights_maple_program(Kgraphs(2,5)))
[53]: wk25 = vector_from_file('data/weights_kontsevich_2_5.txt') #; wk25
[54]: len(wk25) - list(wk25).count(0)
[54]: 2356
[55]: (ck25*wk25).norm()
[55]: 0
The files kontsevich_2_n.txt with the normal form encodings of all these Kontsevich
graphs (of positive differential order) and the files weights_kontsevich_2_n.txt with
their weights (up to n = 6) are stored at https://rburing.nl/gcaops. The full list of
pairs of files is as follows:
• kontsevich_2_1.txt and weights_kontsevich_2_1.txt
• kontsevich_2_2.txt and weights_kontsevich_2_2.txt
• kontsevich_2_3.txt and weights_kontsevich_2_3.txt
• kontsevich_2_4.txt and weights_kontsevich_2_4.txt
• kontsevich_2_5.txt and weights_kontsevich_2_5.txt
• kontsevich_2_6.txt and weights_kontsevich_2_6.txt
Now we deal with the Leibniz graphs.
[56]: print(kontsevint_weights_maple_program(LGB.graphs(3,1)))
[weight([[p1,p2,p3]])];
This is the tripod.
[57]: wl31 = vector_from_file('data/weights_leibniz_3_1.txt'); wl31
[57]: (1/6)
[58]: len(wl31) - list(wl31).count(0)
[58]: 1
[59]: cl31*wl31
[59]: (0)
[60]: print(kontsevint_weights_maple_program(LGB.graphs(3,2)))
[weight([[p3,2],[p1,p2,p3]]),weight([[p2,2],[p1,p2,p3]]),weight([[p1,2],[p1,p2,p3]]),w
eight([[p1,p2],[p1,p2,p3]]),weight([[p1,p3],[p1,p2,p3]]),weight([[p2,p3],[p1,p2,p3]]),
weight([[p1,p3],[p1,p2,1]]),weight([[p1,p3,2],[p1,p2]]),weight([[p2,p3],[p1,p2,1]]),we
ight([[p2,p3,2],[p1,p2]]),weight([[p2,p3],[p1,p3,1]]),weight([[p2,p3,2],[p1,p3]]),weig
ht([[p3,2],[p1,p2,1]]),weight([[p2,2],[p1,p3,1]]),weight([[p2,p3,2],[p1,1]])];
106 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[61]: wl32 = vector_from_file('data/weights_leibniz_3_2.txt'); wl32
[61]: (-1/24, 0, 1/24, 1/12, 1/12, 1/12, 1/48, -1/48, 1/24, 1/24, -1/48, 1/48, -1/48, 0,
1/48)
This list of weights refers to the ordering of the list LGB.graphs(3,2).
[62]: len(wl32) - list(wl32).count(0)
[62]: 13
[63]: cl32*wl32
[63]: (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
[64]: #print(kontsevint_weights_maple_program(LGB.graphs(3,3)))
[65]: wl33 = vector_from_file('data/weights_leibniz_3_3.txt'); wl33
[65]: (1/60, 1/120, 1/60, 11/1440, 1/180, 11/1440, 1/144, 0, -1/144, -41/5760, -1/384,
-1/90, -1/90, -1/384, -41/5760, 1/240, -1/120, 1/240, -7/360, -13/1440, 11/1440,
-11/1440, 13/1440, 7/360, -1/80, -1/480, -1/160, 1/160, 1/480, 1/80, -1/90, -1/720,
-1/120, -1/120, -1/720, -1/90, 1/48, 1/48, 0, 0, -1/48, -1/48, -17/1440, -13/720,
17/1440, -17/1440, 13/720, 17/1440, 5/576, -11/2880, 1/120, 1/120, -11/2880, 5/576,
1/480, -1/240, -1/480, 1/480, 1/240, -1/480, -11/11520, 13/11520, 1/480, 1/480,
13/11520, -11/11520, 17/2880, 1/576, -1/240, 1/240, 1/576, -17/2880, 11/11520, 1/3840,
-1/1440, -1/1440, 1/3840, 11/11520, -1/48, 0, 1/48, 0, 0, 0, 0, 0, 0, -1/1440, -1/480,
-1/720, 1/720, 1/480, 1/1440, 1/160, -1/480, -1/120, 1/120, 1/480, -1/160, 0, 0, 0,
1/1440, 1/720, 1/1440, 1/720, -1/360, 1/720, -1/36, 0, 1/36, 1/96, 0, -1/96, 1/24,
1/24, 1/24, 1/24, 1/24, 1/24, 1/96, 1/48, -1/96, -1/96, 1/48, 1/96, -1/96, 1/48, 1/96,
1/96, 1/48, -1/96, 1/48, -1/96, 1/96, 1/96, -1/96, 1/48, -1/240, 1/120, -1/240,
1/1440, -1/720, -1/1440, 1/1440, 1/720, -1/1440, -1/72, -1/72, -1/72, 1/72, 1/72,
1/72, 1/1440, 1/720, 1/1440, -1/1440, -1/720, -1/1440, 1/160, -7/1440, 7/720, -7/720,
7/1440, -1/160, 1/240, 1/480, -1/160, -1/160, 1/480, 1/240, 0, 0, 0, 0, 0, 0, 0,
1/120, -1/240, 1/120, 1/1440, -1/720, 1/1440, -47/5760, -13/2880, 47/5760, 1/11520,
-1/5760, -1/11520, -1/5760, 1/2880, 1/5760, -1/1280, 1/640, 1/1280, 0, 1/1440,
-1/1440, -1/1440, 1/1440, 0, 1/480, -1/480, -1/240, -1/240, -1/480, 1/480, 0, 0, 0, 0,
0, 0, -1/96, -13/1440, 1/720, 1/720, -13/1440, -1/96, 1/960, -1/960, -1/480, -1/480,
-1/960, 1/960, -17/2880, 17/2880, -13/1440, -13/1440, 17/2880, -17/2880, -1/96, 0,
-1/96, 1/96, 0, 1/96, -1/144, -1/144, -1/144, 11/11520, -1/1152, -7/3840, -7/3840,
-1/1152, 11/11520, -7/1440, 23/2880, 23/2880, -1/96, 0, 1/96, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1/2880, -1/960, -1/1440, -1/1440, -1/960, -1/2880,
1/320, -1/960, -1/240, 1/240, -1/960, -1/320, 1/1440, 1/720, 1/1440, 0, 0, 0, 0, 0, 0,
0, 0, 0, -1/2880, 1/1440, 1/2880, -1/960, -1/480, -1/960)
[66]: len(wl33) - list(wl33).count(0)
[66]: 241
[67]: cl33*wl33
[67]: (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
3.7. KONTSEVICH’S ⋆ PRODUCT FOR AFFINE POISSON STRUCTURES 107
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
[68]: #kontsevint_weights_maple_program(LGB.graphs(3,4))
[70]: wl34 = vector_from_file('data/weights_leibniz_3_4.txt') #; wl34
[71]: len(wl34) - list(wl34).count(0)
[71]: 4609
[73]: (cl34*wl34).norm()
[73]: 0
The files leibniz_3_n.txt with the normal form encodings of all these Leibniz graphs
(of positive differential order) and the files weights_leibniz_3_n.txt with their weights
(up to n+1 = 6) are stored at https://rburing.nl/gcaops. The full list of pairs of files
is as follows:
• leibniz_3_1.txt and weights_leibniz_3_1.txt
• leibniz_3_2.txt and weights_leibniz_3_2.txt
• leibniz_3_3.txt and weights_leibniz_3_3.txt
• leibniz_3_4.txt and weights_leibniz_3_4.txt
• leibniz_3_5.txt and weights_leibniz_3_5.txt
3.7 Kontsevich’s ⋆ product for affine Poisson
structures
In this section we calculate the Kontsevich ⋆-product up to ō(h̄7) in the case where
the Poisson structure coefficients P ij are affine functions on Rd. We start with all the
potentially needed graphs with undetermined coefficients at h̄7, we reduce the number of
unknowns by using all the relations between Kontsevich graph weights which are known
to us (and which are relevant here), and then we calculate the weights of the remaining
unknowns using Panzer’s program kontsevint. Finally we prove the associativity of the
star product expansion up to ō(h̄7) explicitly, by producing a suitable sum of Leibniz
graphs enough to represent the associator up to ō(h̄7).
3.7.1 Affine Kontsevich star product mod ō(h̄6)
The Kontsevich ⋆-product, known from Banks–Panzer–Pym [1] up to ō(h̄6), can be re-
stricted to Poisson structures P with (generic) affine structure coefficients P ij by omitting
those terms containing second or higher derivatives of P ij. In terms of graphs, this means
we can restrict to those graphs with aerial vertices of in-degrees ⩽ 1, which actually con-
tribute to the affine ⋆-product.
108 CHAPTER 3. IMPLEMENTATION OF FORMALITY
Definition. A Kontsevich graph (built of wedges) is called affine if the in-degree of each
of its aerial vertices is ⩽ 1.
We define the basis of relevant affine graphs:
[1]: KGB_affine = KontsevichGraphBasis(positive_differential_order=True,␣
↪→max_aerial_in_degree=1); KGB_affine
[1]: Basis consisting of representatives of isomorphism classes of Kontsevich graphs (of
positive differential order, with aerial vertices of in-degree <= 1) with no
automorphisms that induce an odd permutation on edges
The affine ⋆-product expansion and the expansion of its associator will be defined as
elements of FGC:
[2]: from gcaops.graph.formality_graph_complex import FormalityGraphComplex
FGC = FormalityGraphComplex(SR, lazy=True); FGC
[2]: Formality graph complex over Symbolic Ring with Basis consisting of representatives of
isomorphism classes of formality graphs with no automorphisms that induce an odd
permutation on edges
We can restrict the full Kontsevich ⋆-product mod ō(h̄6) to affine graphs, and store the
result in a file:
[3]: #star6_txt = open('data/star6.txt').read().rstrip()
#star6 = FGC.element_from_kgs_encoding(star6_txt) #; star6
#affine_star6 = star6.filter(max_aerial_in_degree=1)
#with open('data/affine_star6.txt', 'w') as f:
# f.write(affine_star6.kgs_encoding())
After this is done once and the result is stored (or the file data/affine_star6.txt is
imported from elsewhere), we can read back the result:
[4]: affine_star6_txt = open('data/affine_star6.txt').read().rstrip()
affine_star6 = FGC.element_from_kgs_encoding(affine_star6_txt) #; affine_star6
We obtain 465 graphs in total in the affine star product up to ō(h̄6)  — only about 0.678%
of the original 68663 graphs showing up in Kontsevich’s ⋆ mod ō(h̄6) with the harmonic
weights:
[5]: len(affine_star6)
[5]: 465
3.7.2 Relations between the weights at h̄7
In the basis of affine Kontsevich graphs suitable for ⋆-products there are 1731 graphs
with 7 aerial vertices on 2 ground vertices:
[6]: len(KGB_affine.graphs(2,7))
[6]: 1731
Let us find the relations between the Kontsevich weights, to reduce the number of un-
knowns.
3.7. KONTSEVICH’S ⋆ PRODUCT FOR AFFINE POISSON STRUCTURES 109
Lemma. Upon flipping a graph on n vertices, i.e. interchanging the two sinks, its weight
is multiplied by (−1)n.
[7]: %time flipping_matrix = KGB_affine.flipping_weight_relations(2,7)
CPU times: user 1.11 s, sys: 0 ns, total: 1.11 s
Wall time: 1.11 s
Lemma. The cyclic weight relations restrict to the subset of affine graphs.
Proof. Re-directing edges to ground vertices does not affect the in-degree of aerial ver-
tices.
[8]: %time cyclic_matrix = KGB_affine.cyclic_weight_relations(2,7)
CPU times: user 1min 9s, sys: 1.46 s, total: 1min 10s
Wall time: 1min 10s
Lemma. The weight of a graph with an “eye on ground”, i.e. containing a 2-cycle
between aerial vertices such that both vertices in the 2-cycle are connected to the same
ground vertex, vanishes.
Proof. By a dimension count.
[9]: %time eye_on_ground_matrix = KGB_affine.eye_on_ground_weight_relations(2,7)
CPU times: user 266 ms, sys: 4.05 ms, total: 270 ms
Wall time: 267 ms
Lemma. The weight of a disconnected graph vanishes.
(Note that such graphs can consist of two components, each standing on one sink, but
also e.g., of one component standing on two sinks and a purely aerial component.)
[10]: %time disconnected_indices = [k for k,g in enumerate(KGB_affine.graphs(2,7)) if not␣
↪→DiGraph(g.edges()).is_connected()]
%time disconnected_matrix = Matrix(ZZ, len(disconnected_indices), len(KGB_affine.
↪→graphs(2,7)), {(i,k) : 1 for (i,k) in enumerate(disconnected_indices)})
CPU times: user 236 ms, sys: 91 µs, total: 236 ms
Wall time: 232 ms
CPU times: user 1.4 ms, sys: 0 ns, total: 1.4 ms
Wall time: 1.34 ms
Lemma. The weight of a composite Kontsevich graph Γ = Γ1 ×̄ Γ2 on n = n1+n2 aerial
vertices is (by the graph weights’ multiplicativity) equal to the product of the weights:
w(Γ1 ×̄ Γ2) = w(Γ1) · w(Γ2).
The multiplicativity of weights yields (inhomogenenous) linear relations for the weights
of composite graphs at order n as soon as the weights of graphs at lower orders k < n
are known. The weights of graphs at lower orders can be read off from the ⋆-product at
h̄k by multiplying the coefficient of a graph Γ by k! , where m(Γ) is the multiplicity.
m(Γ)
110 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[11]: affine_weight_vectors = {k : vector(SR, len(KGB_affine.graphs(2, k)), {KGB_affine.
↪→graphs(2,k).index(g) : c*factorial(k)/g.multiplicity() for c, g in affine_star6.
↪→homogeneous_part(2, k, 2*k)}, sparse=True) for k in range(1,7)}
We can store these weight vectors in plain text files for posterity:
[12]: #for k in range(1,7):
# with open('data/affine_weights_kontsevich_2_{}.txt'.format(k), 'w') as f:
# for c in affine_weight_vectors[k]:
# f.write(str(c).replace(' ','') + '\n')
For k = 1, 2, 3, 4, 5 the harmonic weights of graphs in the (affine) Kontsevich ⋆-product
at h̄k are rational numbers. At h̄6 the harmonic weights of graphs in the ⋆-product are Q-
linear combinations of 1 and ζ(3)2/π6. We will express each of the composite weights at
h̄7 as such a Q-linear combination, so that we obtain relations with rational coefficients.
(See also the Remark at the end of this section.)
[13]: %%time
w0 = SR.wild() # Wildcard, used for substitution.
composite_vectors = {}
for (p1,p2) in Partitions(7,length=2):
for (g_idx, h_idx, plusminus_gh_idx, plusminus) in KGB_affine.
↪→multiplication_table(2,p1,p2):
prod_weight =␣
↪→plusminus*affine_weight_vectors[p1][g_idx]*affine_weight_vectors[p2][h_idx]
# Replace `zeta(3)^2/pi^6` by `z`.
prod_weight = prod_weight.subs({zeta(3)^2/pi^6 : var('z'), w0*zeta(3)^2/pi^6 :
↪→ w0*var('z')})
prod_weight_a = prod_weight.subs({var('z'):0})
prod_weight_b = prod_weight.coefficient(var('z'))
v = vector(QQ, len(KGB_affine.graphs(2,7)) + 2, {
plusminus_gh_idx : 1,
len(KGB_affine.graphs(2,7)) : -prod_weight_a,
len(KGB_affine.graphs(2,7)) + 1 : -prod_weight_b
}, sparse=True)
if plusminus_gh_idx in composite_vectors:
# We already expressed this weight as a constant. let's make sure it's␣
↪→the same constant:
assert v == composite_vectors[plusminus_gh_idx]
else:
composite_vectors[plusminus_gh_idx] = v
composites_matrix = matrix(QQ, composite_vectors.values())
CPU times: user 4.26 s, sys: 72.1 ms, total: 4.34 s
Wall time: 4.34 s
Now, by construction, every linear relation between the Kontesvich graph weights is a row
with, firstly, rational coefficients of the unknown weights (for graphs which are ordered in
a basis), followed by the last two columns (for the future right-hand side) with the rational
coefficients of 1 and of ζ(3)2/π6 respectively. All these rows are combined into a matrix.
We thus obtain a matrix with rational entries, which will be beneficial for performance
when solving the linear system in the next section (e.g., a one-hour computation over
Q[z] becomes an equivalent one-second calculation over Q).
3.7. KONTSEVICH’S ⋆ PRODUCT FOR AFFINE POISSON STRUCTURES 111
( )
Remark. At orders ⩾ 12, the number 2ζ(3)2/π6 will show up (with a nonzero rational
coefficient) in some of the weights of composite graphs. In general we need generators
over Q of multiple zeta values to express the multiplicativity of the weight as a relation
over Q.
3.7.3 From weight relations to master parameters
We put all the matrices together:
[14]: %%time
big_matrix = block_matrix([[flipping_matrix, zero_matrix(flipping_matrix.nrows(), 2)],
[cyclic_matrix, zero_matrix(cyclic_matrix.nrows(), 2)],
[eye_on_ground_matrix, zero_matrix(eye_on_ground_matrix.nrows(), 2)],
[disconnected_matrix, zero_matrix(disconnected_matrix.nrows(), 2)]]).
↪→stack(composites_matrix)
big_matrix
CPU times: user 5.22 s, sys: 60 ms, total: 5.28 s
Wall time: 5.28 s
[14]: 3936 x 1733 sparse matrix over Rational Field (use the '.str()' method to see the
entries)
In the above, 3936 is the total number of relations constraining the 1731 unknown weights
of affine Kontsevich graphs on n = 7 aerial vertices in ⋆aff mod ō(h̄7), and 1733 is the
width of each row of known coefficients (extended by the rational coefficients of 1 and of
ζ(3)2/π6 in the right-hand side).
We compute the reduced row echelon form of this non-square matrix:
[15]: %time big_matrix_rref = big_matrix.rref() #algorithm='scaled_partial_pivoting')
CPU times: user 982 ms, sys: 12 ms, total: 994 ms
Wall time: 996 ms
The general solution of the big Q-linear system contains 78 parameters:
[16]: big_matrix_rref.right_nullity()
[16]: 78
By construction two of the parameters can be chosen to be 1 and ζ(3)2/π6, so all the
weights can be expressed as Q-linear combinations of 1, ζ(3)2/π6, and 76 parameters.
[17]: %time weight_directions = big_matrix_rref.right_kernel().basis() # fast because we␣
↪→have the rref
CPU times: user 15.5 s, sys: 160 ms, total: 15.6 s
Wall time: 15.6 s
We first obtain the parts of the solution expressed directly as a Q-linear combinations of
1 and ζ(3)2/π6:
112 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[18]: inhomogeneouspart_directions = [v for v in weight_directions if v[-1] != 0 or v[-2] !
↪→= 0]
[19]: len(inhomogeneouspart_directions)
[19]: 2
Extract the coefficients of 1 and ζ(3)2/π6:
[20]: inhomogeneouspart_submatrix = matrix(QQ, [v[-2:] for v in␣
↪→inhomogeneouspart_directions]); inhomogeneouspart_submatrix
[20]: [ 128 1216/6075]
[ 0 512/15]
Make a Q-linear transformation to obtain one part proportional to 1 and another part
proportional to ζ(3)2/π6:
[21]: inhomogeneouspart_new = [sum(scale*v for scale,v in zip(scales,␣
↪→inhomogeneouspart_directions)) for scales in inhomogeneouspart_submatrix.inverse().
↪→rows()]
[22]: matrix(QQ, [v[-2:] for v in inhomogeneouspart_new])
[22]: [1 0]
[0 1]
We now choose parameters for the homogeneous part of the solution:
[23]: homogeneouspart_directions = [v for v in weight_directions if v[-1] == 0 and v[-2] ==␣
↪→0]
The linear solver is such that the first nonzero entry of each vector is a 1:
[24]: all([next(v_i for v_i in v if v_i != 0) == 1 for v in homogeneouspart_directions])
[24]: True
Hence the index of that first 1 is the index of a graph (in the basis) whose weight can be
chosen as a master parameter.
[25]: master_indices = [list(v).index(1) for v in homogeneouspart_directions]
[26]: len(master_indices)
[26]: 76
3.7.4 Substitute master parameters into ⋆ 7aff mod ō(h̄ ) and its
associator
Introduce undetermined weights for all affine graphs at h̄7:
[27]: w = [var('w{}'.format(k)) for k in range(len(KGB_affine.graphs(2,7)))]
3.7. KONTSEVICH’S ⋆ PRODUCT FOR AFFINE POISSON STRUCTURES 113
[28]: master_weights = [w[i] for i in master_indices]
[29]: affine_star7w = FGC(affine_star6) + 1/factorial(7)*FGC([(g.multiplicity()*w[k], g)␣
↪→for (k,g) in enumerate(KGB_affine.graphs(2,7))]) #; affine_star7w
[30]: len(affine_star7w)
[30]: 2196
This is how many graphs we have with their weights known at orders ⩽ 6 and none of
the weights known yet at order 7.
Create a substitution in terms of the master parameters:
[31]: inhomogeneouspart_symbolic = vector(SR, inhomogeneouspart_new[0][:-2]) +␣
↪→var('z')*vector(SR, inhomogeneouspart_new[1][:-2])
[32]: homogeneouspart_symbolic = sum(w[idx]*vector(SR, v[:-2]) for idx, v in␣
↪→zip(master_indices, homogeneouspart_directions))
[33]: my_subs = dict(zip(w, inhomogeneouspart_symbolic + homogeneouspart_symbolic))
Apply the subsitution:
[34]: affine_star7_master = affine_star7w.map_coefficients(lambda c: c.subs(my_subs))
[35]: len(affine_star7_master)
[35]: 1458
So, by taking into account all the so far known relations we already reduce the number
of nonzero coefficients of the topmost graphs. The coefficients of the remaining topmost
graphs will be fixed in the near future, and many of them will vanish as well.
Compute the associator mod ō(h̄7):
[36]: %time affine_assoc7_master = affine_star7_master.insertion(0, affine_star7_master,␣
↪→max_num_aerial=7, max_aerial_in_degree=1) - affine_star7_master.insertion(1,␣
↪→affine_star7_master, max_num_aerial=7, max_aerial_in_degree=1)
CPU times: user 2min, sys: 681 ms, total: 2min
Wall time: 2min
[37]: affine_assoc7_master7 = affine_assoc7_master.homogeneous_part(3,7,14)
[38]: len(affine_assoc7_master7)
[38]: 50429
This is how many Kontsevich graphs, with their coefficients expressed in terms of the 76
master parameters, currently survive into the associator at h̄7 (if all the topmost graph
coefficients are undetermined, the associator contains 69843 terms at h̄7).
[39]: len(list(affine_assoc7_master7.differential_orders()))
114 CHAPTER 3. IMPLEMENTATION OF FORMALITY
[39]: 161
Likewise, this is how many tri-differential orders survive into the associator at h̄7 so far
— out of 203.
3.7.5 Restrict onto generic affine Poisson bivector on R2
We consider a generic affine Poisson bivector P = (Ax+By + C) ∂ 2x ∧ ∂y on R :
[40]: R.<x,y,A,B,C> = SR[]
[41]: SA.<xi1,xi2> = SuperfunctionAlgebra(R, [x,y])
[42]: PA.<ddx,ddy> = PolyDifferentialOperatorAlgebra(R, [x,y])
[43]: P = (A*x+B*y+C)*xi1*xi2; P
[43]: (x*A + y*B + C)*xi1*xi2
The part at order h̄7 of the associator (f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) for Kontsevich’s ⋆ will be
evaluated at this P . This results in a tri-differential operator AI1I2I3∂I1 ⊗ ∂I2 ⊗ ∂I3 acting
on f ⊗ g ⊗ h ∈ C∞(R2)⊗3 with coefficients AI1I2I3 which are polynomials in x, y with
coefficients in SR[A,B,C]. The vanishing of this tri-differential operator and hence of
all these polynomials yields relations between the master parameters.
[44]: from gcaops.graph.formality_graph_operator import FormalityGraphOperator
[45]: #%%time
#assoc2_eqns = []
#for diff_order in affine_assoc7_master7.differential_orders():
# part = affine_assoc7_master7.part_of_differential_order(diff_order)
# print(diff_order, ':', len(part))
# operat = FormalityGraphOperator(SA, PA, part)
# op = operat.value_at_copies_of(P)
# assoc2_eqns.extend(sum([op[m].coefficients() for m in op.multi_indices()], []))
[46]: #solve(assoc2_eqns, master_weights)
Thus we would reduce the number of master parameters from 76 down to 74. Since this
is only a small improvement, we ignore it.
3.7.6 Restrict onto affine rescaled Nambu–Poisson bivector on
R3
O∣∣n R3 t∣∣here are two families of affine (rescaled) Nambu–Poisson brackets {f, g} =ρ ∂(ϕ,f,g) , with (ρ, φ) polynomial of degrees (deg(ρ), deg(φ)) equal to (1, 1) or (0, 2).
∂(x,y,z)
Since φ is always differentiated once we can omit its constant term in either case.
First let ρ = ax+ by + cz + d and φ = Ax+By + Cz:
[47]: D.<ddx,ddy,ddz> = PolyDifferentialOperatorAlgebra(SR, var('x,y,z'))
S.<xi1,xi2,xi3> = SuperfunctionAlgebra(SR, var('x,y,z'), simplify='expand',␣
↪→is_zero='is_trivial_zero')
3.7. KONTSEVICH’S ⋆ PRODUCT FOR AFFINE POISSON STRUCTURES 115
phi = var('A')*x + var('B')*y + var('C')*z
P_1_1 = (var('a')*x+var('b')*y+var('c')*z+var('d'))*(diff(phi,x)*xi2*xi3 +␣
↪→diff(phi,y)*xi3*xi1 + diff(phi,z)*xi1*xi2)
[48]: #%%time
#assoc3_1_1_eqns = []
#for diff_order in star7.differential_orders():
# print(diff_order)
# part = star7.part_of_differential_order(diff_order)
# operat = FormalityGraphOperator(S, D, part)
# op = operat.value_at_copies_of(P_1_1)
# assoc3_1_1_eqns.extend(sum([op[m].coefficients() for m in op.multi_indices()],␣
↪→[]))
[49]: #solve(assoc3_1_1_eqns, master_weights)
Now let ρ = 1 and φ = Ax2 +By2 + Cz2 +Dxy + Exz + Fyz:
[50]: phi = var('A')*x^2 + var('B')*y^2 + var('C')*z^2 + var('D')*x*y + var('E')*x*z +␣
↪→var('F')*y*z
P_0_2 = (diff(phi,x)*xi2*xi3 + diff(phi,y)*xi3*xi1 + diff(phi,z)*xi1*xi2)
[51]: #%%time
#assoc3_0_2_eqns = []
#for diff_order in star7.differential_orders():
# print(diff_order)
# part = star7.part_of_differential_order(diff_order)
# operat = FormalityGraphOperator(S, D, part)
# op = operat.value_at_copies_of(P_0_2)
# assoc3_0_2_eqns.extend(sum([op[m].coefficients() for m in op.multi_indices()],␣
↪→[]))
[52]: #solve(assoc3_0_2_eqns, master_weights)
It would be interesting to see how far the number of parameters drops in these cases.
3.7.7 Direct calculation of master parameter values
We calculate the values of the 76 master parameters directly using Panzer’s kontsevint
program:
[53]: #master_graphs = [KGB_affine.graphs(2,7)[i] for i in master_indices]
#print('[' + ','.join('weight({})'.format(g.kontsevint_encoding().replace('p1','L').
↪→replace('p2','R')) for g in master_graphs) + '];')
[54]:
116 CHAPTER 3. IMPLEMENTATION OF FORMALITY
master_values = [0, 3/2048*zeta(3)^2/pi^6 + 943/46448640, 1/2048*zeta(3)^2/pi^6 + 257/
↪→46448640, 65/2048*zeta(3)^2/pi^6 - 437/9289728, 149/2048*zeta(3)^2/pi^6 - 5239/
↪→46448640, -99/2048*zeta(3)^2/pi^6 + 53/737280, 27/2048*zeta(3)^2/pi^6 - 289/
↪→15482880, -31/2048*zeta(3)^2/pi^6 + 131/6635520, -35/1024*zeta(3)^2/pi^6 + 121/
↪→1935360, -5/512*zeta(3)^2/pi^6 + 131/9289728, -27/2048*zeta(3)^2/pi^6 + 43/
↪→15482880, 1/256*zeta(3)^2/pi^6 + 1/573440, -17/2048*zeta(3)^2/pi^6 + 9/573440, 1/
↪→512*zeta(3)^2/pi^6 + 1/1658880, -49/2048*zeta(3)^2/pi^6 + 1849/46448640, -31/
↪→1024*zeta(3)^2/pi^6 + 893/11612160, 143/2048*zeta(3)^2/pi^6 - 53/516096, 71/
↪→2048*zeta(3)^2/pi^6 - 1/20480, 53/1024*zeta(3)^2/pi^6 - 163/2322432, 29/
↪→1024*zeta(3)^2/pi^6 - 41/1161216, 5/512*zeta(3)^2/pi^6 - 1019/46448640, -1/
↪→128*zeta(3)^2/pi^6 + 293/23224320, 95/1024*zeta(3)^2/pi^6 - 1301/9289728, -5/
↪→128*zeta(3)^2/pi^6 + 71/1290240, 179/2048*zeta(3)^2/pi^6 - 2029/15482880, -3/
↪→1024*zeta(3)^2/pi^6 - 109/7741440, -57/2048*zeta(3)^2/pi^6 + 85/2322432, 17/
↪→512*zeta(3)^2/pi^6 - 121/3317760, -7/512*zeta(3)^2/pi^6 + 17/860160, -1/241920, 0,␣
↪→1/256*zeta(3)^2/pi^6 - 89/3870720, 13/1024*zeta(3)^2/pi^6 - 1/5806080, 0, -55/
↪→1024*zeta(3)^2/pi^6 + 1021/13271040, 57/2048*zeta(3)^2/pi^6 - 3659/92897280, 5/
↪→1024*zeta(3)^2/pi^6 + 199/11612160, 1/2048*zeta(3)^2/pi^6 + 41/23224320, -7/
↪→2048*zeta(3)^2/pi^6 + 263/46448640, 1/64*zeta(3)^2/pi^6 - 37/2903040, -25/
↪→1024*zeta(3)^2/pi^6 + 25/663552, 53/2048*zeta(3)^2/pi^6 - 853/15482880, 3/
↪→128*zeta(3)^2/pi^6 - 3131/92897280, -47/2048*zeta(3)^2/pi^6 + 3229/92897280, 67/
↪→5806080, 1/256*zeta(3)^2/pi^6 - 157/11612160, -19/1024*zeta(3)^2/pi^6 + 199/
↪→5806080, 3/128*zeta(3)^2/pi^6 - 7/221184, -7/2048*zeta(3)^2/pi^6 + 31/5806080, -27/
↪→1024*zeta(3)^2/pi^6 + 3961/92897280, -65/2048*zeta(3)^2/pi^6 + 2123/46448640, -37/
↪→1024*zeta(3)^2/pi^6 + 5011/92897280, -35/2048*zeta(3)^2/pi^6 + 37/2211840, -31/
↪→2048*zeta(3)^2/pi^6 + 461/18579456, -29/1024*zeta(3)^2/pi^6 + 733/18579456, 3/
↪→512*zeta(3)^2/pi^6 + 1/241920, -7/512*zeta(3)^2/pi^6 + 13/573440, 5/512*zeta(3)^2/
↪→pi^6 - 17/1161216, 13/2048*zeta(3)^2/pi^6 - 149/5806080, -5/1024*zeta(3)^2/pi^6 + 7/
↪→1105920, 3/2048*zeta(3)^2/pi^6 - 71/46448640, -87/4096*zeta(3)^2/pi^6 + 751/
↪→26542080, -39/512*zeta(3)^2/pi^6 + 1807/15482880, 1/241920, 1/512*zeta(3)^2/pi^6 -␣
↪→89/7741440, -1/512*zeta(3)^2/pi^6 + 1/11612160, -83/4096*zeta(3)^2/pi^6 + 1427/
↪→46448640, 0, 1/241920, 0, -1/241920, 0, 1/241920, 5/2048*zeta(3)^2/pi^6 - 7/
↪→2211840, 15/2048*zeta(3)^2/pi^6 - 31/3870720, 11/2048*zeta(3)^2/pi^6 - 179/23224320]
[55]: #master_values_subs = dict({w[idx] : master_values[k] for k, idx in␣
↪→enumerate(master_indices)})
#affine_star7 = affine_star7_master.map_coefficients(lambda c: c.
↪→subs(master_values_subs))
We have finally obtained the affine star-product ⋆ 7aff mod ō(h̄ ) with the harmonic weights!
It is also contained in Appendix C.1 of the dissertation.
We write the result to the file data/affine_star7.txt:
[56]: #with open('data/affine_star7.txt', 'w') as f:
# f.write(affine_star7.kgs_encoding())
After the file is saved (or imported from elsewhere) the result can be read back:
[57]: affine_star7 = FGC.element_from_kgs_encoding(open('data/affine_star7.txt').read().
↪→rstrip())
Remark. In Section 3.7.9 below we compare the formula ⋆aff mod ō(h̄7) with an earlier
result of Ben Amar (2003) about the rationality of coefficients in it.
3.7.8 Certificate of associativity
We calculate the associator for Kontsevich’s affine ⋆-product up to ō(h̄7):
3.7. KONTSEVICH’S ⋆ PRODUCT FOR AFFINE POISSON STRUCTURES 117
[58]: %time affine_assoc7 = affine_star7.insertion(0, affine_star7, max_num_aerial=7,␣
↪→max_aerial_in_degree=1) - affine_star7.insertion(1, affine_star7, max_num_aerial=7,␣
↪→max_aerial_in_degree=1)
CPU times: user 1min 54s, sys: 733 ms, total: 1min 55s
Wall time: 1min 54s
[59]: affine_assoc7_7 = affine_assoc7.homogeneous_part(3,7,14)
[60]: len(affine_assoc7_7)
[60]: 49621
This is how many Kontsevich graphs truly survive into the associator at order h̄7 for the
genuine affine ⋆-product modulo ō(h̄7) with harmonic propagators in the graph weights.
We now obtain some Leibniz graph factorization of (each tri-differential component of)
the associator at h̄7 for ⋆aff mod ō(h̄7), as seen before in Section 3.5. Namely, we contract
an edge between aerial vertices (in all possible ways) in the Kontsevich graphs to obtain
Leibniz graphs, and we expand the resulting Leibniz graphs to Kontsevich graphs again
(as explained in [10]). (Each time we expand Leibniz graphs we necessarily reproduce the
Kontsevich graphs which were seen previously, but we possibly also meet new Kontsevich
graphs.) At each stage where Leibniz graphs are obtained, we check if they are enough
to factor the respective tri-differential component Ad1d2d3 of the associator A at h̄7, by
equating Ad1d2d3 to the Kontsevich graph expansion of the obtained Leibniz graphs with
undetermined coefficients, and trying to solve the linear system. If the so far available
set of Leibniz graphs is not enough, the program builds the next layer of (neighbor)
Leibniz graphs by contracting edges in all the Kontsevich graphs from the expansions
of previously available Leibniz graphs. This is repeated until a solution appears for the
factorization problem in that specific tri-differential order.
To obtain and solve the respective linear system over Q, we use the following helper
functions.
[63]: from gcaops.graph.leibniz_graph_expansion import␣
↪→kontsevich_graph_sum_to_leibniz_graph_sum
def coefficient_to_vector(c):
c_poly = QQ['zzz'](str(SR(c).expand()).replace('zeta(3)^2/pi^6', 'zzz'))
return vector(QQ, [c_poly.constant_coefficient(), c_poly.
↪→monomial_coefficient(QQ['zzz'].gen())])
def vector_to_coefficient(v):
return v[0] + v[1]*zeta(3)^2/pi^6
The following lines of output contain the grading of a tri-differential component at of
the associator at h̄7 for ⋆aff mod ō(h̄7), followed by the number of Kontsevich graphs in
that component, followed by the number of new Leibniz graphs and Kontsevich graphs
obtained in each step (usually one step, exceptionally two steps in four cases).
[64]: f_leibniz = open('data/affine_assoc7_leibniz.txt', 'w')
f_leibniz_coeffs = open('data/affine_assoc7_leibniz_coeffs.txt', 'w')
for diff_order in affine_assoc7_7.differential_orders():
print(diff_order, end=': ', flush=True)
118 CHAPTER 3. IMPLEMENTATION OF FORMALITY
part = affine_assoc7_7.part_of_differential_order(diff_order)
part_Leibniz = kontsevich_graph_sum_to_leibniz_graph_sum(part,␣
↪→coefficient_to_vector=coefficient_to_vector,␣
↪→vector_to_coefficient=vector_to_coefficient, max_aerial_in_degree=1, verbose=True)
for c,L in part_Leibniz:
f_leibniz.write(str(L.edges()).replace(' ','') + '\n')
f_leibniz_coeffs.write(str(c).replace(' ','') + '\n')
f_leibniz.close()
f_leibniz_coeffs.close()
(4, 3, 4): 236K -> +164L -> +21K
(3, 4, 4): 232K -> +164L -> +25K
(3, 3, 4): 835K -> +835L -> +116K
(2, 5, 4): 111K -> +80L -> +16K
(2, 4, 4): 539K -> +535L -> +79K
(5, 2, 4): 117K -> +81L -> +10K
(4, 2, 4): 552K -> +541L -> +66K
(5, 3, 4): 32K -> +16L -> +4K
(4, 4, 4): 38K -> +18L -> +4K
(3, 5, 4): 32K -> +16L -> +4K
(5, 4, 3): 32K -> +16L -> +4K
(5, 3, 3): 143K -> +98L -> +14K
(4, 5, 3): 32K -> +16L -> +4K
(4, 4, 3): 232K -> +164L -> +25K
(4, 3, 3): 835K -> +835L -> +116K
(3, 6, 3): 10K -> +5L -> +2K
(3, 5, 3): 135K -> +97L -> +22K
(3, 4, 3): 811K -> +827L -> +139K
(2, 6, 3): 23K -> +14L -> +5K
(2, 5, 3): 286K -> +274L -> +60K
(6, 3, 3): 12K -> +5L -> +0K
(6, 2, 3): 27K -> +14L -> +1K
(5, 2, 3): 320K -> +280L -> +27K
(6, 4, 3): 3K -> +1L -> +0K
(5, 5, 3): 3K -> +1L -> +0K
(4, 6, 3): 3K -> +1L -> +0K
(2, 6, 4): 10K -> +5L -> +2K
(6, 2, 4): 12K -> +5L -> +0K
(6, 3, 4): 3K -> +1L -> +0K
(5, 4, 4): 3K -> +1L -> +0K
(4, 5, 4): 3K -> +1L -> +0K
(3, 6, 4): 3K -> +1L -> +0K
(2, 3, 4): 1387K -> +1721L -> +148K
(3, 2, 4): 1411K -> +1730L -> +125K
(5, 4, 2): 115K -> +81L -> +12K
(5, 3, 2): 319K -> +280L -> +28K
(4, 5, 2): 111K -> +80L -> +16K
(4, 4, 2): 539K -> +535L -> +79K
(4, 3, 2): 1387K -> +1721L -> +148K
(3, 6, 2): 23K -> +14L -> +5K
(3, 5, 2): 286K -> +274L -> +60K
(3, 4, 2): 1300K -> +1691L -> +229K
(2, 6, 2): 36K -> +27L -> +11K
(2, 5, 2): 414K -> +508L -> +112K
(6, 3, 2): 27K -> +14L -> +1K
(6, 2, 2): 40K -> +27L -> +7K
(5, 2, 2): 479K -> +522L -> +51K
3.7. KONTSEVICH’S ⋆ PRODUCT FOR AFFINE POISSON STRUCTURES 119
(6, 4, 2): 12K -> +5L -> +0K
(5, 5, 2): 26K -> +14L -> +4K
(4, 6, 2): 10K -> +5L -> +2K
(2, 2, 4): 1431K -> +2111L -> +98K
(3, 3, 3): 2216K -> +2897L -> +296K
(2, 4, 3): 1300K -> +1691L -> +229K
(4, 2, 3): 1411K -> +1730L -> +125K
(5, 4, 1): 135K -> +118L -> +9K
(5, 3, 1): 268K -> +291L -> +28K
(4, 5, 1): 122K -> +116L -> +22K
(4, 4, 1): 473K -> +558L -> +47K
(4, 3, 1): 790K -> +1140L -> +57K
(3, 6, 1): 24K -> +20L -> +10K
(3, 5, 1): 230K -> +280L -> +63K
(3, 4, 1): 758K -> +1124L -> +86K
(2, 6, 1): 30K -> +28L -> +12K
(2, 5, 1): 207K -> +329L -> +84K
(6, 3, 1): 28K -> +20L -> +6K
(6, 2, 1): 34K -> +28L -> +8K
(5, 2, 1): 274K -> +346L -> +20K
(6, 4, 1): 18K -> +10L -> +1K
(5, 5, 1): 41K -> +30L -> +6K
(4, 6, 1): 16K -> +10L -> +3K
(2, 3, 3): 2294K -> +3584L -> +221K -> +123L -> +35K
(3, 2, 3): 2331K -> +3603L -> +191K -> +106L -> +30K
(3, 3, 2): 2294K -> +3584L -> +221K -> +123L -> +35K
(2, 4, 2): 1246K -> +2041L -> +273K -> +111L -> +23K
(4, 2, 2): 1431K -> +2111L -> +98K
(2, 2, 3): 1095K -> +1967L -> +47K
(3, 3, 1): 616K -> +1091L -> +32K
(2, 4, 1): 353K -> +636L -> +49K
(4, 2, 1): 389K -> +652L -> +17K
(2, 3, 2): 1056K -> +1950L -> +81K
(3, 2, 2): 1095K -> +1967L -> +47K
(1, 5, 3): 230K -> +280L -> +63K
(5, 1, 3): 273K -> +291L -> +23K
(6, 5, 2): 3K -> +1L -> +0K
(5, 6, 2): 3K -> +1L -> +0K
(1, 6, 2): 30K -> +28L -> +12K
(6, 1, 2): 34K -> +28L -> +8K
(1, 6, 3): 24K -> +20L -> +10K
(6, 1, 3): 28K -> +20L -> +6K
(1, 5, 2): 207K -> +329L -> +84K
(5, 1, 2): 276K -> +346L -> +18K
(6, 1, 6): 3K -> +1L -> +0K
(5, 2, 6): 3K -> +1L -> +0K
(4, 3, 6): 3K -> +1L -> +0K
(3, 4, 6): 3K -> +1L -> +0K
(2, 5, 6): 3K -> +1L -> +0K
(1, 6, 6): 3K -> +1L -> +0K
(5, 1, 6): 9K -> +4L -> +0K
(4, 2, 6): 12K -> +5L -> +0K
(3, 3, 6): 12K -> +5L -> +0K
(2, 4, 6): 12K -> +5L -> +0K
(1, 5, 6): 9K -> +4L -> +0K
(6, 1, 5): 9K -> +4L -> +0K
(5, 2, 5): 26K -> +14L -> +4K
(4, 3, 5): 32K -> +16L -> +4K
120 CHAPTER 3. IMPLEMENTATION OF FORMALITY
(3, 4, 5): 32K -> +16L -> +4K
(2, 5, 5): 26K -> +14L -> +4K
(1, 6, 5): 7K -> +4L -> +2K
(6, 1, 4): 18K -> +10L -> +1K
(1, 6, 4): 16K -> +10L -> +3K
(4, 1, 6): 18K -> +10L -> +1K
(3, 2, 6): 27K -> +14L -> +1K
(2, 3, 6): 27K -> +14L -> +1K
(1, 4, 6): 18K -> +10L -> +1K
(5, 1, 5): 44K -> +30L -> +3K
(4, 2, 5): 117K -> +81L -> +10K
(3, 3, 5): 143K -> +98L -> +14K
(2, 4, 5): 115K -> +81L -> +12K
(1, 5, 5): 41K -> +30L -> +6K
(3, 1, 6): 28K -> +20L -> +6K
(2, 2, 6): 40K -> +27L -> +7K
(1, 3, 6): 28K -> +20L -> +6K
(5, 1, 4): 139K -> +118L -> +5K
(1, 5, 4): 122K -> +116L -> +22K
(4, 1, 5): 139K -> +118L -> +5K
(3, 2, 5): 320K -> +280L -> +27K
(2, 3, 5): 319K -> +280L -> +28K
(1, 4, 5): 135K -> +118L -> +9K
(3, 1, 5): 273K -> +291L -> +23K
(2, 2, 5): 479K -> +522L -> +51K
(1, 3, 5): 268K -> +291L -> +28K
(4, 1, 4): 505K -> +564L -> +15K
(1, 4, 4): 473K -> +558L -> +47K
(3, 1, 4): 825K -> +1148L -> +22K
(1, 3, 4): 790K -> +1140L -> +57K
(2, 1, 5): 276K -> +346L -> +18K
(1, 2, 5): 274K -> +346L -> +20K
(4, 1, 3): 825K -> +1148L -> +22K
(1, 4, 3): 758K -> +1124L -> +86K
(2, 1, 6): 34K -> +28L -> +8K
(1, 2, 6): 34K -> +28L -> +8K
(1, 1, 6): 14K -> +18L -> +13K
(1, 1, 5): 78K -> +119L -> +13K
(4, 1, 2): 397K -> +657L -> +9K
(1, 4, 2): 353K -> +636L -> +49K
(2, 1, 4): 397K -> +657L -> +9K
(1, 2, 4): 389K -> +652L -> +17K
(3, 1, 3): 626K -> +1095L -> +22K
(1, 3, 3): 616K -> +1091L -> +32K
(6, 2, 5): 3K -> +1L -> +0K
(5, 3, 5): 3K -> +1L -> +0K
(4, 4, 5): 3K -> +1L -> +0K
(3, 5, 5): 3K -> +1L -> +0K
(2, 6, 5): 3K -> +1L -> +0K
(6, 1, 1): 14K -> +18L -> +13K
(1, 6, 1): 12K -> +18L -> +15K
(5, 1, 1): 78K -> +119L -> +13K
(1, 5, 1): 40K -> +91L -> +48K
(6, 6, 1): 3K -> +1L -> +0K
(6, 5, 1): 9K -> +4L -> +0K
(5, 6, 1): 7K -> +4L -> +2K
3.7. KONTSEVICH’S ⋆ PRODUCT FOR AFFINE POISSON STRUCTURES 121
We observe that four tri-differential orders, namely (2, 4, 2), (3, 3, 2), (2, 3, 3), (3, 2, 3), re-
quire the use of Leibniz graphs from the 1st layer of neighbors for a solution to the
factorization problem to appear.
The Leibniz graphs are now stored in data/affine_assoc7_leibniz.txt and their found
coefficients are in data/affine_assoc7_leibniz_coeffs.txt; together, these form a
certificate of the affine ⋆-product associativity up to ō(h̄7).
Given the data files with the coefficients and graphs in the Leibniz graph factorization,
the associativity up to ō(h̄7) can be verified instantly and directly:
[65]: affine_assoc7_leibniz_coeffs = vector(SR, open('data/affine_assoc7_leibniz_coeffs.
↪→txt').readlines())
[68]: affine_assoc7_leibniz_graphs = [FormalityGraph(3,6,sage_eval(line)) for line in␣
↪→open('data/affine_assoc7_leibniz.txt').readlines()]
[69]: affine_assoc7_leibniz = FGC([(c,L) for c, L in zip(affine_assoc7_leibniz_coeffs,␣
↪→affine_assoc7_leibniz_graphs)])
[70]: from gcaops.graph.leibniz_graph_expansion import␣
↪→leibniz_graph_sum_to_kontsevich_graph_sum
affine_assoc7_leibniz_expanded =␣
↪→leibniz_graph_sum_to_kontsevich_graph_sum(affine_assoc7_leibniz,␣
↪→max_aerial_in_degree=1)
[71]: affine_assoc7_leibniz_expanded == affine_assoc7_7
[71]: True
3.7.9 Rationality of ⋆aff mod ō(h̄7)
It is known from Ben Amar [3] that the Kontsevich ⋆-product formula for the duals of
Lie algebras g∗ contains only rational coefficients which are known explicitly, expressed
in terms of Bernoulli numbers and factorials. Specifically, all coefficients are determined
by the weights of the following graphs [4]:
• The Bernoulli graphs:
[1]: def bernoulli_graph(n):
return FormalityGraph(2, n, [(2,0),(2,1)] + sum([[(k+3, k+2),(k+3, 1)] for k in␣
↪→range(n-1)], []))
The weights of the Bernoulli graphs are Bk/(k!)2:
[4]: for n in range(1,8):
B = bernoulli_graph(n)
B_sign, B_normal = list(FGC(B))[0]
B_coeff = sum([B_sign*c for c,g in affine_star7 if g == B_normal], 0)
W = B_coeff/B.multiplicity()
print(W, bernoulli(n)/factorial(n)^2)
122 CHAPTER 3. IMPLEMENTATION OF FORMALITY
1/2 -1/2
1/24 1/24
0 0
-1/17280 -1/17280
0 0
1/21772800 1/21772800
0 0
• The wheel graphs:
[5]: def wheel_graph(n):
return FormalityGraph(2, n, [(2,0),(2,1+n)] + sum([[(k+3,k+2),(k+3,1)] for k in␣
↪→range(n-1)], []))
The weights of the wheel graphs are 1Bk/(k!)2:2
[6]: for n in range(2,8):
W = wheel_graph(n)
W_sign, W_normal = list(FGC(W))[0]
W_coeff = sum([W_sign*c for c,g in affine_star7 if g == W_normal], 0)
Wt = W_coeff/W.multiplicity()
print(Wt, 1/2*bernoulli(n)/factorial(n)^2)
1/48 1/48
0 0
-1/34560 -1/34560
0 0
1/43545600 1/43545600
0 0
We now observe that the Kontsevich weights of some graphs in ⋆aff mod ō(h̄7) contain
the conjecturally irrational number ζ(3)2/π6 with a nonzero rational coefficient.
While this may seem at odds with the fact of rationality of coefficients appearing in the
expansion, we prove here that these claims are in fact consistent. We express the part
of ⋆aff mod ō(h̄7) proportional to ζ(3)2/π6 as a sum of Leibniz graphs, proving that this
part makes a zero contribution when evaluated at a generic affine Poisson structure.
[72]: def zetapart(c):
return QQ['zzz'](str(SR(c).expand()).replace('zeta(3)^2/pi^6', 'zzz')).
↪→monomial_coefficient(QQ['zzz'].gen())
[73]: affine_star7_zetapart = affine_star7.map_coefficients(zetapart)
[74]: for diff_order in affine_star7_zetapart.differential_orders():
print(diff_order, end=': ', flush=True)
part = affine_star7_zetapart.part_of_differential_order(diff_order)
part_Leibniz = kontsevich_graph_sum_to_leibniz_graph_sum(part,␣
↪→max_aerial_in_degree=1, verbose=True)
print(leibniz_graph_sum_to_kontsevich_graph_sum(part_Leibniz,␣
↪→max_aerial_in_degree=1) == part)
3.7. KONTSEVICH’S ⋆ PRODUCT FOR AFFINE POISSON STRUCTURES 123
(3, 4): 119K -> +179L -> +3K
True
(4, 3): 119K -> +179L -> +3K
True
(2, 4): 19K -> +24L -> +4K
True
(4, 2): 19K -> +24L -> +4K
True
(3, 3): 40K -> +54L -> +1K
True
(4, 5): 22K -> +24L -> +2K
True
(5, 4): 22K -> +24L -> +2K
True
(4, 4): 106K -> +141L -> +4K
True
(3, 5): 62K -> +76L -> +7K
True
(5, 3): 62K -> +76L -> +7K
True
(2, 5): 28K -> +41L -> +6K
True
(5, 2): 28K -> +41L -> +6K
True
In conclusion, the number ζ(3)2/π6 showed up at h̄6 and h̄7 in the coefficients of Kontse-
vich’s graphs at 12 bi-differential orders; in all the cases those Kontsevich graphs with
their rational coefficients standing near ζ(3)2/π6 themselves—i.e. not needing higher lay-
ers of Leibniz graph neighbors—assimilate into Leibniz graphs, so that whenever the
graph formula ⋆aff mod ō(h̄7) is restricted to any affine Poisson bracket, the entire ana-
lytic expression near ζ(3)2/π6 vanishes identically, by force of the Jacobi identity and its
differential consequences.
In fact there are more terms in the affine ⋆-product that can be eliminated, by assimilating
them into Leibniz graphs. This process of elimination results in a reduced graph expansion
⋆redaff mod ō(h̄
7) with only rational coefficients, which is reported in Appendix C.2.
[75]: reduced_affine_star7 = FGC.zero()
for diff_order in affine_star7.differential_orders():
#print(diff_order, end=': ', flush=True)
part = affine_star7.part_of_differential_order(diff_order)
part_Leibniz = kontsevich_graph_sum_to_leibniz_graph_sum(part,␣
↪→max_aerial_in_degree=1, force_saturation=True, exact=False, verbose=True)
if part_Leibniz is not None:
# NOTE: by having passed exact=False, we get a (least squares?) "solution"
part_Leibniz_expanded =␣
↪→leibniz_graph_sum_to_kontsevich_graph_sum(part_Leibniz, max_aerial_in_degree=1)
new_part = part - part_Leibniz_expanded
reduced_affine_star7 += new_part
#print(len(part), 'down to', len(new_part))
else:
#print('no reduction')
124 CHAPTER 3. IMPLEMENTATION OF FORMALITY
reduced_affine_star7 += part
We write the reduced star product to a file:
[ ]: #with open('data/affine_star7_reduced.txt', 'w') as f:
# f.write(reduced_affine_star7.kgs_encoding())
We see at once that the reduced affine star product formula (in particular, having its
ζ(3)2/π6-slice equal to zero) is much shorter: at all orders up to h̄7, there remain only
326 terms:
[7]: affine_star7_reduced = FGC.element_from_kgs_encoding(open('data/affine_star7_reduced.
↪→txt').read().rstrip())
[8]: len(affine_star7_reduced)
[8]: 326
We calculate the associator:
[9]: %time affine_assoc7_reduced = affine_star7_reduced.insertion(0, affine_star7_reduced,␣
↪→max_num_aerial=7, max_aerial_in_degree=1) - affine_star7_reduced.insertion(1,␣
↪→affine_star7_reduced, max_num_aerial=7, max_aerial_in_degree=1)
CPU times: user 49.7 s, sys: 0 ns, total: 49.7 s
Wall time: 49.3 s
[10]: len(affine_assoc7_reduced)
[10]: 29371
Again, the associator becomes twice smaller: there were 59905 Kontsevich graphs showing
up at all orders in the associator for ⋆aff mod ō(h̄7).
[11]: len(list(affine_assoc7_reduced.differential_orders()))
[11]: 181
This is an overall count of tri-differential orders; specifically at h̄7, there are 161 (that is,
discarding the ζ(3)2/π6-slice does not decrease the number).
In Appendix C.3 we inspect that the reduced star product ⋆redaff mod ō(h̄
7) remains asso-
ciative.
We conclude that the Kontsevich graph expansion of the reduced affine ⋆-product itself
is associative up to ō(h̄7); the analytic formula which one writes for f ⋆redaff g mod ō(h̄
7)
with arbitrary arguments f, g ∈ C∞(M) and any affine Poisson structure P in ⋆aff mod
ō(h̄7) is, we establish in this section, identically equal to the formula f ⋆aff g mod ō(h̄7):
all the coefficients of differential polynomials in f, g and P ij are rational numbers, and
ζ(3)2/π6 is not met at all.
Chapter 4
Implementation of the graph
complex
This chapter is an introduction to the graph operad and graph complex. By using exam-
ples we illustrate how to manipulate all the objects and structures (such as graphs, sums
of graphs, and operadic insertions of graphs) with the new software. We now learn that
graphs with a wedge ordering of edges form a vector space. Some graphs are zero graphs,
whenever they admit an automorphism which induces an odd permutation of edges. In
the quotient space modulo zero graphs, we (can) choose a basis so that (non)zero graphs
are conveniently represented as vectors. We refer to [26, 27] or [17], [32] and [33] for
general theory and more details.
4.1 Graphs
We import the implementation of undirected graphs:
[1]: from gcaops.graph.undirected_graph import UndirectedGraph
We consider undirected graphs with n vertices labeled 0, . . . , n − 1, and an ordered list
of edges between vertices. An often convenient/preferred ordering of the edges is the
lexicographic ordering.
The stick on n = 2 vertices labeled 0 and 1, and one edge (0, 1):
[2]: stick_graph = UndirectedGraph(2, [(0,1)]); stick_graph
[2]: UndirectedGraph(2, [(0, 1)])
[3]: stick_graph.show(figsize=2)
125
126 CHAPTER 4. IMPLEMENTATION OF THE GRAPH COMPLEX
[4]: tetrahedron_graph = UndirectedGraph(4, [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)]);␣
↪→tetrahedron_graph
[4]: UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
[5]: tetrahedron_graph.show(figsize=2)
[6]: g = UndirectedGraph(4, [(0,1),(1,2),(1,3),(2,3)]); g.show(figsize=2)
Relabeling of the graph’s vertices by a graph automorphism induces a permutation on
the set of edges.
For example, let us relabel two vertices in the above graph g.
[7]: g.relabeled({0:0, 1:1, 2:3, 3:2}).edges()
[7]: [(0, 1), (1, 3), (1, 2), (2, 3)]
This permutation on the set of edges has a parity:
[8]: g.relabeled({0:0, 1:1, 2:3, 3:2}).canonicalize_edges()
[8]: -1
Indeed, the edge permutation swaps the two edges (1, 2) and (1, 3); hence it is parity odd.
4.2. GRAPH OPERAD 127
4.2 Graph operad
We import the implementation of the operad of undirected graphs:
[9]: from gcaops.graph.undirected_graph_operad import UndirectedGraphOperad
The operad is a graded vector space spanned by undirected graphs, quotiented by graded
edge permutations.
[10]: Gra = UndirectedGraphOperad(QQ); Gra
[10]: Operad of undirected graphs over Rational Field
Convert a graph into an element of Gra:
[11]: stick = Gra(stick_graph); stick
[11]: 1*UndirectedGraph(2, [(0, 1)])
Create a sum of graphs from a list of (coefficient, graph) pairs:
[12]: Gra([(1, stick_graph), (4, tetrahedron_graph)])
[12]: 1*UndirectedGraph(2, [(0, 1)]) + (4)*UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1,
2), (1, 3), (2, 3)])
We can insert a given graph into a vertex of another graph; the result is a sum of terms
— to get all of them, we reattach the edges which were incident to that vertex, now
attaching them (independently one from another) to the vertices of the inserted graph.
We do this in all possible ways, and take the sum. In every resulting term, the edges
of the inserted graph go last; first go the edges of the graph into which the former was
inserted.
Here is an example.
[13]: stick.insertion(0, stick)
[13]: 0
Indeed, all terms in the result are proportional to the following graph:
[14]: stickstick = UndirectedGraph(3,[(0,1),(1,2)]); stickstick
[14]: UndirectedGraph(3, [(0, 1), (1, 2)])
But this graph is zero:
[15]: Gra(stickstick)
[15]: 0
Indeed, this is because the graph stickstick has an automorphism that induces an odd
permutation on the set of edges.
[16]: stickstick.relabeled({0:2, 1:1, 2:0}).canonicalize_edges()
128 CHAPTER 4. IMPLEMENTATION OF THE GRAPH COMPLEX
[16]: -1
Equal to minus itself, the graph is zero.
4.3 Full undirected graph complex
We import the implementation of the undirected graph complex:
[17]: from gcaops.graph.undirected_graph_complex import UndirectedGraphComplex
Define the full undirected graph complex over Q:
[18]: fGC = UndirectedGraphComplex(QQ); fGC
[18]: Undirected graph complex over Rational Field with Basis consisting of representatives
of isomorphism classes of undirected graphs with no automorphisms that induce an odd
permutation on edges
For the time being, we do not impose any restrictions on the graphs (such as “connected”,
“biconnected”, etc.) or on the vertex degrees (e.g. “at least 3”).
Convert the stick graph to an element of the graph complex:
[19]: stick = fGC(stick_graph); stick
[19]: 1*UndirectedGraph(2, [(0, 1)])
[20]: stick.show()
The graded Lie bracket of graphs is the graded commutator of insertions [a, b] = a←−◦ b−
(−)#E(a)·E(b)a−→◦ b. We recall that the reverse order of terms (a−→◦ b preceding a←−◦ b) is also
used in the literature.
[21]: stick.bracket(stick)
[21]: 0
We now define the vertex-expanding differential: this operator is the graded Lie bracket
with the stick graph as the first argument, d = [•−•, ·].
[22]: stick.differential()
[22]: 0
4.3. FULL UNDIRECTED GRAPH COMPLEX 129
We now construct some graph cocycles manually:
• The tetrahedron (3-wheel) cocycle γ3 (on 4 vertices and 6 edges):
[23]: tetrahedron_cocycle = fGC(tetrahedron_graph); tetrahedron_cocycle
[23]: 1*UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
[24]: tetrahedron_cocycle.show()
[25]: tetrahedron_cocycle.differential()
[25]: 0
• The 5-wheel cocycle γ5 (on 6 vertices and 10 edges in each term):
[26]: fivewheel_graph = UndirectedGraph(6,␣
↪→[(0,1),(1,2),(2,3),(3,4),(0,4),(0,5),(1,5),(2,5),(3,5),(4,5)])
fivewheel = fGC(fivewheel_graph)
fivewheel.differential()
[26]: (-10)*UndirectedGraph(7, [(0, 1), (0, 4), (0, 5), (1, 3), (1, 5), (2, 3), (2, 4), (2,
6), (3, 6), (4, 6), (5, 6)])
[27]: roof_graph = UndirectedGraph(6,␣
↪→[(0,1),(1,2),(2,3),(0,3),(3,4),(0,4),(4,5),(2,5),(1,5),(0,2)])
roof = fGC(roof_graph)
roof.differential()
[27]: (4)*UndirectedGraph(7, [(0, 1), (0, 4), (0, 5), (1, 3), (1, 5), (2, 3), (2, 4), (2,
6), (3, 6), (4, 6), (5, 6)])
[28]: fivewheel_cocycle = fivewheel + (5/2)*roof
fivewheel_cocycle
[28]: 1*UndirectedGraph(6, [(0, 3), (0, 4), (0, 5), (1, 2), (1, 4), (1, 5), (2, 3), (2, 5),
(3, 5), (4, 5)]) + (5/2)*UndirectedGraph(6, [(0, 1), (0, 3), (0, 5), (1, 2), (1, 4),
(2, 4), (2, 5), (3, 4), (3, 5), (4, 5)])
[29]: fivewheel_cocycle.show()
130 CHAPTER 4. IMPLEMENTATION OF THE GRAPH COMPLEX
[30]: fivewheel_cocycle.differential()
[30]: 0
We can also create a graph cochain from a list of (coefficient, graph) tuples:
[31]: fGC([(1, fivewheel_graph), (5/2, roof_graph)])
[31]: 1*UndirectedGraph(6, [(0, 3), (0, 4), (0, 5), (1, 2), (1, 4), (1, 5), (2, 3), (2, 5),
(3, 5), (4, 5)]) + (5/2)*UndirectedGraph(6, [(0, 1), (0, 3), (0, 5), (1, 2), (1, 4),
(2, 4), (2, 5), (3, 4), (3, 5), (4, 5)])
Whenever we have a graph cochain (no matter how it was constructed) we can iterate
over its terms:
[32]: for (c,g) in fivewheel_cocycle:
print(c, g)
1 UndirectedGraph(6, [(0, 3), (0, 4), (0, 5), (1, 2), (1, 4), (1, 5), (2, 3), (2, 5),
(3, 5), (4, 5)])
5/2 UndirectedGraph(6, [(0, 1), (0, 3), (0, 5), (1, 2), (1, 4), (2, 4), (2, 5), (3,
4), (3, 5), (4, 5)])
[33]: set([c for (c,g) in fivewheel_cocycle])
[33]: {1, 5/2}
4.4 Graph bases: storing them in cache
The graph complex implementation stores a choice of basis internally. The basis can be
accessed as follows:
[34]: list(fGC.basis().graphs(4,6))
[34]: [UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])]
[35]: list(fGC.basis().graphs(6,10))
[35]: [UndirectedGraph(6, [(0, 4), (0, 5), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4),
(3, 5), (4, 5)]),
UndirectedGraph(6, [(0, 4), (0, 5), (1, 3), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4),
(3, 5), (4, 5)]),
4.5. UNDIRECTED GRAPH COMPLEX 131
UndirectedGraph(6, [(0, 3), (0, 5), (1, 2), (1, 4), (1, 5), (2, 4), (2, 5), (3, 4),
(3, 5), (4, 5)]),
UndirectedGraph(6, [(0, 3), (0, 4), (0, 5), (1, 2), (1, 4), (1, 5), (2, 3), (2, 5),
(3, 5), (4, 5)]),
UndirectedGraph(6, [(0, 1), (0, 3), (0, 5), (1, 2), (1, 4), (2, 4), (2, 5), (3, 4),
(3, 5), (4, 5)]),
UndirectedGraph(6, [(0, 4), (0, 5), (1, 2), (1, 3), (1, 5), (2, 3), (2, 4), (3, 4),
(3, 5), (4, 5)])]
Instead of listing too many graphs, let us now just count the dimensions of vector spaces.
It can be noticed that we deal primarily with the spaces of undirected graphs on n vertices
and 2n− 2 edges; they will be used to construct Poisson 2-cocycles.
[36]: len(fGC.basis().graphs(7,12))
[36]: 48
[37]: len(fGC.basis().graphs(8,14))
[37]: 1006
By default, these bases are generated anew (whenever needed) in each SageMath or
Python session. To avoid generating these bases over and over, it is possible to use a
cache on disk. Making use of this functionality is achieved by specifying a directory:
[38]: #from gcaops.graph import graph_cache
#graph_cache.GCAOPS_DATA_DIR = '/home/rburing/src/gcaops_data/'
Whenever a graph basis is accessed (e.g. as above) while this directory is specified, the
program inspects first whether the needed basis is already stored in the directory. If so,
it simply returns a reference to it. If not, the program generates the basis, stores it there
for future (re)use, and returns a reference to it.
4.5 Undirected graph complex
To find interesting cohomology classes it suffices to restrict to a subcomplex spanned
by graphs which are connected, biconnected, and in which each vertex has degree ⩾ 3.
Graphs are stored as collections of vectors (one for each bi-graded component) and the
differentials (restricted to each bi-graded component) are stored as matrices. This will
allow to find a basis of cohomology automatically.
[39]: GC = UndirectedGraphComplex(QQ, connected=True, biconnected=True, min_degree=3,␣
↪→implementation='vector'); GC
[39]: Undirected graph complex over Rational Field with Basis consisting of representatives
of isomorphism classes of undirected graphs (connected, biconnected, of degree at
least 3) with no automorphisms that induce an odd permutation on edges
We can convert graph cocycles from the full graph complex fGC to this restricted graph
complex GC.
[40]: GC(tetrahedron_cocycle)
132 CHAPTER 4. IMPLEMENTATION OF THE GRAPH COMPLEX
[40]: 1*UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
Now, with this implementation, we can test whether a cocycle is a coboundary:
[41]: GC(tetrahedron_cocycle).is_coboundary()
[41]: False
[42]: GC(fivewheel_cocycle).is_coboundary()
[42]: False
4.6 Directed graph complex
We also have an implementation of directed graphs:
[43]: from gcaops.graph.directed_graph import DirectedGraph
[44]: directed_tetra = DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]);␣
↪→directed_tetra
[44]: DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
We import the implementation of the directed graph complex:
[45]: from gcaops.graph.directed_graph_complex import DirectedGraphComplex
Define the directed graph complex over Q; the graphs are conveniently restricted to get
rid of graphs with 2-cycles, etc.
[46]: dGC = DirectedGraphComplex(QQ, connected=True, biconnected=True, min_degree=3,␣
↪→loops=False, implementation='vector', sparse=True); dGC
[46]: Directed graph complex over Rational Field with Basis consisting of representatives of
isomorphism classes of directed graphs (connected, biconnected, of degree at least 3,
without loops) with no automorphisms that induce an odd permutation on edges
A cochain of dGC can be defined by a list of (coefficient, graph) pairs:
[47]: dGC([(1, directed_tetra)])
[47]: 1*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
The canonical map from the undirected graph complex to the directed graph complex
works as follows: every edge, independently from all others (if any), is directed consecu-
tively in two opposite ways (this creates a sum of directed graphs). The edge ordering is
inherited from the undirected graph complex; this is important when we collect similar
terms.
This map is implemented as conversion (e.g. here from fGC to dGC):
[48]: tetrahedron_cocycle
[48]: 1*UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
4.6. DIRECTED GRAPH COMPLEX 133
[49]: tetrahedron_cocycle_directed = dGC(tetrahedron_cocycle); tetrahedron_cocycle_directed
[49]: (24)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]) +
(-8)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 3), (2, 1), (2, 3)]) +
(-24)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 2), (2, 3), (3, 1)]) +
(-8)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 3), (2, 1), (3, 2)])
[50]: tetrahedron_cocycle_directed.show(ncols=4)
The differential in the directed graph complex is the graded Lie bracket with the sum of
two ways to direct the stick: d = [•←•+ •→•, ·]:
[51]: tetrahedron_cocycle_directed.differential()
[51]: 0
For the purpose of future use, we can filter directed graphs by prescribing a bound upon
the outgoing degree of vertices.
[52]: tetrahedron_cocycle_directed_filtered = tetrahedron_cocycle_directed.
↪→filter(max_out_degree=2)
tetrahedron_cocycle_directed_filtered.show()
As usual, let us count the dimensions; because the suitable bases are already stored on
the disk, it takes milliseconds, microseconds, and even nanoseconds to produce them!
[53]: %time len(dGC.basis().graphs(6,10))
CPU times: user 0 ns, sys: 1.8 ms, total: 1.8 ms
Wall time: 4.83 ms
[53]: 732
134 CHAPTER 4. IMPLEMENTATION OF THE GRAPH COMPLEX
[54]: %time len(dGC.basis().graphs(7,12))
CPU times: user 1.55 ms, sys: 156 µs, total: 1.7 ms
Wall time: 10.2 ms
[54]: 25638
[55]: %time len(dGC.basis().graphs(8,14))
CPU times: user 6.23 ms, sys: 20.1 ms, total: 26.3 ms
Wall time: 57.5 ms
[55]: 1126232
[56]: %time len(dGC.basis().graphs(9,16))
CPU times: user 243 ms, sys: 1.1 s, total: 1.34 s
Wall time: 2.5 s
[56]: 59381077
[57]: %time len(dGC.basis().graphs(9,15))
CPU times: user 18.3 ms, sys: 110 ms, total: 128 ms
Wall time: 273 ms
[57]: 5247208
4.7 Undirected graph operations
From Chapter 2 we recall the construction of the superfunction algebra.
[58]: from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
SA.<xi1,xi2,xi3> = SuperfunctionAlgebra(SR, var('x1,x2,x3'), simplify='expand',␣
↪→is_zero='is_trivial_zero'); SA
[58]: Superfunction algebra over Symbolic Ring with even coordinates (x1, x2, x3) and odd
coordinates (xi1, xi2, xi3)
Take some bivector field:
[59]: P = (x1^2-x2*x3)*(x1*xi2*xi3 + x2*xi3*xi1 + x3*xi1*xi2); P
[59]: (x1^3 - x1*x2*x3)*xi2*xi3 + (-x1^2*x2 + x2^2*x3)*xi1*xi3 + (x1^2*x3 - x2*x3^2)*xi1*xi2
And let there be a vector field:
[60]: X = x1*xi1
Let us calculate their Schouten bracket:
4.7. UNDIRECTED GRAPH OPERATIONS 135
[61]: P.bracket(X)
[61]: (x1^2*x2 + x2^2*x3)*xi1*xi3 + (-x1^2*x3 - x2*x3^2)*xi1*xi2 + (-3*x1^3 +
x1*x2*x3)*xi2*xi3
Kontsevich’s graph calculus tells us that the Schouten bracket π (A,B) = (−)|A|−1S [[A,B]]
comes from the stick graph •−•.
Let us illustrate this (see the seminal paper [27] for key facts and statements):
[62]: stick
[62]: 1*UndirectedGraph(2, [(0, 1)])
[63]: schouten = SA.graph_operation(stick); schouten
[63]: Symmetric operation of arity 2 and degree -1 on Superfunction algebra over Symbolic
Ring with even coordinates (x1, x2, x3) and odd coordinates (xi1, xi2, xi3)
[64]: schouten(P, X) == -P.bracket(X)
[64]: True
[65]: schouten(X, P) == schouten(P,X)
[65]: True
The tetrahedral flow on the space of Poisson bi-vectors comes from the tetrahedron graph
cocycle. Namely, when directed (and filtered for the outgoing vertex degrees ⩽ 2) the
tetrahedral graph becomes a polylinear operation on the space of multivectors (in partic-
ular, of bivectors). This is how the Kontsevich graph flow is built: Ṗ = Or(γ )(P⊗n3 ); see
Chapter 17. This is precisely the formula of Kontsevich’s tetrahedral graph flow which
we use in the entire dissertation: for instance we have seen it in Chapter 2, and we shall
use it again in Chapters 5, 6, 7, 8, 9.
[66]: tetrahedron_operation = SA.graph_operation(tetrahedron_cocycle); tetrahedron_operation
[66]: Operation of arity 4 and degree -6 on Superfunction algebra over Symbolic Ring with
even coordinates (x1, x2, x3) and odd coordinates (xi1, xi2, xi3)
[67]: %time Q_tetra = tetrahedron_operation(P,P,P,P); Q_tetra
CPU times: user 2.01 s, sys: 21.5 ms, total: 2.03 s
Wall time: 2.03 s
[67]: (288*x1^4*x2^2 - 288*x1^2*x2^3*x3 - 288*x1^4*x3^2 + 288*x1^2*x2*x3^3)*xi2*xi3 +
(-288*x1^3*x2^3 + 288*x1*x2^4*x3 + 288*x1^3*x2*x3^2 - 288*x1*x2^2*x3^3)*xi1*xi3 +
(288*x1^3*x2^2*x3 - 288*x1*x2^3*x3^2 - 288*x1^3*x3^3 + 288*x1*x2*x3^4)*xi1*xi2
Operations on the space of multivectors can also be defined using directed graph cocycles:
[68]: tetrahedron_operation_directed = SA.graph_operation(tetrahedron_cocycle_directed)
tetrahedron_operation_directed(P,P,P,P)
136 CHAPTER 4. IMPLEMENTATION OF THE GRAPH COMPLEX
[68]: (-288*x1^3*x2^3 + 288*x1*x2^4*x3 + 288*x1^3*x2*x3^2 - 288*x1*x2^2*x3^3)*xi1*xi3 +
(288*x1^3*x2^2*x3 - 288*x1*x2^3*x3^2 - 288*x1^3*x3^3 + 288*x1*x2*x3^4)*xi1*xi2 +
(288*x1^4*x2^2 - 288*x1^2*x2^3*x3 - 288*x1^4*x3^2 + 288*x1^2*x2*x3^3)*xi2*xi3
The result is the same as in the previous cell; this is the tetrahedral graph flow on the
space of bivectors.
Chapter 5
Examples of graph cocycles
In this chapter we find explicit representatives of some graph cocycles in low degrees,
which will be necessary to compute the action on Poisson structures in the following
chapters. A theorem of Willwacher [35] shows that the degree 0 cohomology of the
graph complex is isomorphic to the the Grothendieck–Teichmüller Lie algebra grt, and
in particular for each odd n there exists a nontrivial graph cohomology class containing
the wheel graph with n spokes with a nonzero coefficient. Moreover, the isomorphism to
grt combined with a theorem of F. Brown [7] shows that these graph cohomology classes
generate a free Lie subalgebra in the cohomology of the graph complex. It is an open
problem whether these cohomology classes are all of the graph cohomology classes (the
Deligne–Drinfeld conjecture states that they are). The dimensions of (finite-dimensional)
graded parts of graph cohomology can be found using rank computations (Willwacher–
Živković [38], partly numerical). Using linear algebra, we find bases of the graded parts
of graph cohomology. We note that the coefficients of these graph cocycles can also be
expressed using integral formulas, as shown by Willwacher–Rossi [37]. A closed form
or combinatorial interpretation of these coefficients is an open problem, which provides
another reason for listing them explicitly. In addition to finding explicit formulas for
graph cocycles, we prove the factorization of the Poisson cocycle condition via the Jacobi
identity in each case, by providing the necessary Leibniz graphs.
Remark. Of course a cocycle plus a coboundary is again a cocycle, so the bases we find
are generally not canonical. But in low degrees there are also not that many coboundaries,
so we list some of them as well. This will also be instructive when calculating the action
on Poisson structures in the next chapters.
We import the relevant functionality from the gcaops package:
[1]: from gcaops.graph.undirected_graph import UndirectedGraph
from gcaops.graph.undirected_graph_complex import UndirectedGraphComplex
from gcaops.graph.directed_graph import DirectedGraph
from gcaops.graph.directed_graph_complex import DirectedGraphComplex
from gcaops.graph.formality_graph import FormalityGraph
from gcaops.graph.formality_graph_complex import FormalityGraphComplex
To find interesting cohomology classes it suffices to restrict to a subcomplex spanned
by graphs which are connected, biconnected (i.e. which remain connected even when
any single vertex is deleted), and in which each vertex has degree ⩾ 3. We also switch
to a different implementation: first we fix a basis in the bi-graded space of graphs, so
that graphs are stored as collections of vectors (one vector for each bi-graded component);
137
138 CHAPTER 5. EXAMPLES OF GRAPH COCYCLES
likewise, the differentials (restricted to each bi-graded component) are stored as matrices.
This will allow us to find a basis of cohomology classes automatically.
[2]: GC = UndirectedGraphComplex(QQ, connected=True, biconnected=True, min_degree=3,␣
↪→implementation='vector', sparse=True); GC
[2]: Undirected graph complex over Rational Field with Basis consisting of representatives
of isomorphism classes of undirected graphs (connected, biconnected, of degree at
least 3) with no automorphisms that induce an odd permutation on edges
[3]: dGC = DirectedGraphComplex(QQ, connected=True, biconnected=True, min_degree=3,␣
↪→loops=False, implementation='vector', sparse=True); dGC
[3]: Directed graph complex over Rational Field with Basis consisting of representatives of
isomorphism classes of directed graphs (connected, biconnected, of degree at least 3,
without loops) with no automorphisms that induce an odd permutation on edges
[4]: FGC = FormalityGraphComplex(QQ, lazy=True)
We recall that there is a Lie algebra generated by the wheel graph cocycles; every such
generator is nontrivial, it contains with nonzero coefficient a wheel graph with an odd
number of spokes.
5.1 The tetrahedron cocycle γ3
The graph cohomology in the vertex-edge bi-grading (4, 6) is one-dimensional:
[5]: len(GC.cohomology_basis(4,6))
[5]: 1
The generator is the tetrahedron (3-wheel) cocycle γ3, the full graph on 4 vertices and 6
edges:
[6]: tetrahedron_cocycle = GC.cohomology_basis(4,6)[0]; tetrahedron_cocycle.show()
This was the first nontrivial graph cocycle ever found; it was introduced by Kontsevich
in his paper [27].
The graph differential which we study here is the vertex-expanding differential; another
differential (edge contracting) is also studied by others.
5.1. THE TETRAHEDRON COCYCLE γ3 139
[7]: tetrahedron_cocycle.differential()
[7]: 0
[8]: tetrahedron_cocycle.is_coboundary()
[8]: False
Now we study the directed graph complex with the differential which is the Lie bracket
of a graph with the sum of two directed sticks •→• + •←•. The edges of tetrahedron
oriented in all possible ways, it becomes a directed graph cocycle. The overall ordering
of edges is inherited from the undirected graphs.
[9]: tetrahedron_cocycle_directed = dGC(tetrahedron_cocycle); tetrahedron_cocycle_directed
[9]: (24)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]) +
(-8)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 3), (2, 1), (3, 2)]) +
(-24)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 2), (2, 3), (3, 1)]) +
(-8)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 3), (2, 1), (2, 3)])
In the following picture, the ordering of edges is lexicographic.
[10]: tetrahedron_cocycle_directed.show(ncols=4)
[11]: tetrahedron_cocycle_directed.differential()
[11]: 0
Next, we prove the factorization of the Poisson cocycle condition via the Jacobi identity
for the tetrahedral flow.
[12]: wedge = FGC(FormalityGraph(2,1,[(2,0),(2,1)])); wedge
[12]: 1*FormalityGraph(2, 1, [(2, 0), (2, 1)])
[13]: formality_stick = FGC(DirectedGraph(2,[(0,1)])); formality_stick
[13]: 1*FormalityGraph(0, 2, [(0, 1)])
This is the tetrahedral flow:
[14]: Q_tetrahedron = FGC(tetrahedron_cocycle_directed.filter(max_out_degree=2)).
↪→attach_to_ground((2,2,2,2))
140 CHAPTER 5. EXAMPLES OF GRAPH COCYCLES
[15]: len(Q_tetrahedron)
[15]: 3
The necessary Leibniz graphs are these:
[16]: Q_tetrahedron_Leibniz = FGC(tetrahedron_cocycle_directed).attach_to_ground((2,2,2,3))
[17]: len(Q_tetrahedron_Leibniz)
[17]: 27
Now, the Poisson cocycle condition is verified:
[18]: P_Q_tetrahedron = wedge.schouten_bracket(Q_tetrahedron)
[19]: len(P_Q_tetrahedron)
[19]: 39
[20]: Q_tetrahedron_Leibniz_expanded = sum(Q_tetrahedron_Leibniz.insertion(k,␣
↪→formality_stick, max_out_degree=2) for k in [3,4,5,6])
[21]: len(Q_tetrahedron_Leibniz_expanded)
[21]: 39
[22]: P_Q_tetrahedron == -16 * Q_tetrahedron_Leibniz_expanded
[22]: True
5.2 The five-wheel cocycle γ5
It was found independently by Kontsevich and by Willwacher in the years 1996–2010.
The 5-wheel cocycle γ5 is a sum of two graphs; each of them on 6 vertices and 10 edges:
[23]: fivewheel_cocycle = GC.cohomology_basis(6,10)[0]; fivewheel_cocycle
[23]: 1*UndirectedGraph(6, [(0, 3), (0, 4), (0, 5), (1, 2), (1, 4), (1, 5), (2, 3), (2, 5),
(3, 5), (4, 5)]) + (5/2)*UndirectedGraph(6, [(0, 1), (0, 3), (0, 5), (1, 2), (1, 4),
(2, 4), (2, 5), (3, 4), (3, 5), (4, 5)])
[24]: fivewheel_cocycle.show()
5.3. THE COBOUNDARY δ6 = D(β6) 141
[25]: set([c for (c,g) in fivewheel_cocycle])
[25]: {1, 5/2}
[26]: fivewheel_cocycle.differential()
[26]: 0
[27]: fivewheel_cocycle.is_coboundary()
[27]: False
Let us orient the 5-wheel cocycle:
[28]: fivewheel_cocycle_directed = dGC(fivewheel_cocycle) #; fivewheel_cocycle_directed
[29]: len(fivewheel_cocycle_directed)
[29]: 616
Next, we prove the factorization of the Poisson cocycle condition via the Jacobi identity
for the 5-wheel flow.
[30]: Q_fivewheel = FGC(fivewheel_cocycle_directed.filter(max_out_degree=2)).
↪→attach_to_ground((2,2,2,2,2,2)); len(Q_fivewheel)
[30]: 167
[31]: Q_fivewheel_Leibniz = FGC(fivewheel_cocycle_directed.filter(max_out_degree=3)).
↪→attach_to_ground((2,2,2,2,2,3))
[32]: len(Q_fivewheel_Leibniz)
[32]: 3876
[33]: P_Q_fivewheel = wedge.schouten_bracket(Q_fivewheel)
[34]: len(P_Q_fivewheel)
[34]: 3495
[35]: Q_fivewheel_Leibniz_expanded = sum(Q_fivewheel_Leibniz.insertion(k, formality_stick,␣
↪→max_out_degree=2) for k in [3,4,5,6,7,8])
[36]: P_Q_fivewheel == -24 * Q_fivewheel_Leibniz_expanded
[36]: True
5.3 The coboundary δ6 = d(β6)
The coboundary δ6 is a sum of four graphs on 7 vertices and 12 edges:
142 CHAPTER 5. EXAMPLES OF GRAPH COCYCLES
[37]: beta6_graph = UndirectedGraph(6,␣
↪→[(0,2),(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)])
beta6_graph
[37]: UndirectedGraph(6, [(0, 2), (0, 3), (0, 4), (0, 5), (1, 3), (1, 4), (1, 5), (2, 4),
(2, 5), (3, 5), (4, 5)])
[38]: beta6_graph.show(figsize=2)
[39]: beta6 = GC(beta6_graph)
delta6_cocycle = beta6.differential()
delta6_cocycle
[39]: (-4)*UndirectedGraph(7, [(0, 1), (0, 4), (0, 6), (1, 2), (1, 6), (2, 3), (2, 5), (3,
5), (3, 6), (4, 5), (4, 6), (5, 6)]) + (4)*UndirectedGraph(7, [(0, 1), (0, 2), (0, 5),
(1, 4), (1, 5), (2, 4), (2, 6), (3, 4), (3, 5), (3, 6), (4, 6), (5, 6)]) +
(4)*UndirectedGraph(7, [(0, 3), (0, 4), (0, 5), (1, 2), (1, 4), (1, 5), (2, 4), (2,
6), (3, 5), (3, 6), (4, 6), (5, 6)]) + (2)*UndirectedGraph(7, [(0, 1), (0, 2), (0, 6),
(1, 2), (1, 5), (2, 4), (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6)])
[40]: delta6_cocycle.show(ncols=4)
Let us orient the δ6 coboundary:
[41]: delta6_cocycle_directed = dGC(delta6_cocycle) #; delta6_cocycle_directed
[42]: len(delta6_cocycle_directed)
[42]: 10960
Next, we prove the factorization of the Poisson cocycle condition via the Jacobi identity
for the δ6 flow.
5.3. THE COBOUNDARY δ6 = D(β6) 143
[43]: Q_delta6 = FGC(delta6_cocycle_directed.filter(max_out_degree=2)).
↪→attach_to_ground((2,2,2,2,2,2,2)); len(Q_delta6)
[43]: 1500
[44]: Q_delta6_Leibniz = FGC(delta6_cocycle_directed.filter(max_out_degree=3)).
↪→attach_to_ground((2,2,2,2,2,2,3)); len(Q_delta6_Leibniz)
[44]: 45965
[45]: P_Q_delta6 = wedge.schouten_bracket(Q_delta6)
[46]: len(P_Q_delta6)
[46]: 35949
[47]: Q_delta6_Leibniz_expanded = sum(Q_delta6_Leibniz.insertion(k, formality_stick,␣
↪→max_out_degree=2) for k in [3,4,5,6,7,8,9])
[48]: P_Q_delta6 == -28 * Q_delta6_Leibniz_expanded
[48]: True
In fact, the coboundary δ6 in the graph complex yields Poisson trivial flows on the spaces
of Poisson brackets:
[49]: beta6_directed = dGC(beta6); len(beta6_directed)
[49]: 1024
[50]: X_beta6 = FGC(beta6_directed.filter(max_out_degree=2)).
↪→attach_to_ground((2,2,2,2,2,2)); len(X_beta6)
[50]: 46
[51]: P_X_beta6 = wedge.schouten_bracket(X_beta6); len(P_X_beta6)
[51]: 598
[52]: X_beta6_Leibniz = FGC(beta6_directed.filter(max_out_degree=3)).
↪→attach_to_ground((2,2,2,2,2,3)); len(X_beta6_Leibniz)
[52]: 1068
[53]: X_beta6_Leibniz_expanded = sum(X_beta6_Leibniz.insertion(k, formality_stick,␣
↪→max_out_degree=2) for k in [2,3,4,5,6,7])
[54]: len(X_beta6_Leibniz_expanded)
[54]: 2098
[55]: Q_delta6 == -7*P_X_beta6 + 84*X_beta6_Leibniz_expanded
144 CHAPTER 5. EXAMPLES OF GRAPH COCYCLES
[55]: True
5.4 The heptagon-wheel cocycle γ7
The heptagon-wheel cocycle is represented by a sum of 46 graphs, each on 8 vertices and
14 edges. It is tedious to enter manually, so we generate it instead.
Check the dimension of the vector space where the cohomology class of γ7 lives:
[56]: len(GC.cohomology_basis(8,14))
[56]: 1
Get a representative of the cohomology class:
[57]: heptagon_cocycle_maybe = GC.cohomology_basis(8,14)[0]
[58]: heptagon_cocycle_maybe.show(ncols=6)
[59]: len(heptagon_cocycle_maybe)
[59]: 46
5.4. THE HEPTAGON-WHEEL COCYCLE γ7 145
[60]: %time heptagon_cocycle_maybe_directed = dGC(heptagon_cocycle_maybe)
CPU times: user 3.03 s, sys: 32.1 ms, total: 3.06 s
Wall time: 3.06 s
[61]: len(heptagon_cocycle_maybe_directed)
[61]: 595476
Next, we prove the factorization of the Poisson cocycle condition via the Jacobi identity
for the γ7 flow.
[62]: %time Q_heptagon_maybe = FGC(heptagon_cocycle_maybe_directed.
↪→filter(max_out_degree=2)).attach_to_ground((2,2,2,2,2,2,2,2))
CPU times: user 2min 12s, sys: 3.66 s, total: 2min 16s
Wall time: 2min 15s
[63]: len(Q_heptagon_maybe)
[63]: 38538
[64]: %time Q_heptagon_maybe_Leibniz = FGC(heptagon_cocycle_maybe_directed.
↪→filter(max_out_degree=3)).attach_to_ground((2,2,2,2,2,2,2,3))
CPU times: user 56min 36s, sys: 20.8 s, total: 56min 57s
Wall time: 56min 42s
[65]: len(Q_heptagon_maybe_Leibniz)
[65]: 1511359
[66]: %time P_Q_heptagon_maybe = wedge.schouten_bracket(Q_heptagon_maybe)
CPU times: user 1h 38min 59s, sys: 30.7 s, total: 1h 39min 30s
Wall time: 1h 38min 57s
[67]: len(P_Q_heptagon_maybe)
[67]: 1040373
[68]: %time Q_heptagon_maybe_Leibniz_expanded = sum(Q_heptagon_maybe_Leibniz.insertion(k,␣
↪→formality_stick, max_out_degree=2) for k in [3,4,5,6,7,8,9,10])
CPU times: user 5h 6min 45s, sys: 1min 9s, total: 5h 7min 55s
Wall time: 5h 6min 34s
[69]: len(Q_heptagon_maybe_Leibniz_expanded)
146 CHAPTER 5. EXAMPLES OF GRAPH COCYCLES
[69]: 1040373
[70]: P_Q_heptagon_maybe == -32*Q_heptagon_maybe_Leibniz_expanded
[70]: True
5.5 The commutator [γ3, γ5]
The commutator [γ3, γ5] of graph cocycles γ3 and γ5 is a sum of graphs, each on 4+6−1 = 9
vertices and 6 + 10 = 16 edges:
[71]: bracket_cocycle = tetrahedron_cocycle.bracket(fivewheel_cocycle)
#bracket_cocycle
[72]: bracket_cocycle.show(ncols=6)
5.5. THE COMMUTATOR [γ3, γ5] 147
Count the number of terms:
[73]: len(bracket_cocycle)
[73]: 68
List the different values of coefficients:
[74]: set([c for (c,g) in bracket_cocycle])
[74]: {-120, -60, 60, 120}
Check that [γ3, γ5] is not a coboundary:
148 CHAPTER 5. EXAMPLES OF GRAPH COCYCLES
[75]: bracket_cocycle.is_coboundary()
[75]: False
Confirm that the cohomology in vertex-edge bi-grading (9, 16) is one-dimensional:
[76]: %time B_9_16 = GC.cohomology_basis(9,16)
CPU times: user 1min 18s, sys: 818 ms, total: 1min 19s
Wall time: 1min 19s
[77]: len(B_9_16)
[77]: 1
The basis element which was found automatically does not equal [γ3, γ5] exactly:
[78]: B_9_16[0] == bracket_cocycle
[78]: False
However, they are proportional in the cohomology:
[79]: (bracket_cocycle - 40*B_9_16[0]).is_coboundary()
[79]: True
Though not as cochains:
[80]: bracket_cocycle == 40*B_9_16[0]
[80]: False
Orient the graph cocycle [γ3, γ5]:
[81]: %time bracket_cocycle_directed = dGC(bracket_cocycle)
CPU times: user 10.7 s, sys: 1.55 s, total: 12.3 s
Wall time: 13.4 s
[82]: len(bracket_cocycle_directed)
[82]: 2752512
In a different way:
[83]: #bracket_directed = dGC(tetrahedron_cocycle).bracket(dGC(fivewheel_cocycle))
The outputs of the calculations in this section are stored externally at https://
rburing.nl/gcaops: for each γ ∈ {γ3, γ5, δ6, γ7} there is a triple of files, consisting of the
flow Qγ built of Kontsevich graphs, the Poisson differential [[P,Qγ]] built of Kontsevich
graphs, and the right-hand side of the Leibniz graph factorization [[P,Qγ]] = ♢γ(P, [[P, P ]]):
• Q_gamma_3.txt and [P,Q_gamma_3].txt and [P,Q_gamma_3]_leibniz.txt
5.5. THE COMMUTATOR [γ3, γ5] 149
• Q_gamma_5.txt and [P,Q_gamma_5].txt and [P,Q_gamma_5]_leibniz.txt
• Q_delta_6.txt and [P,Q_delta_6].txt and [P,Q_delta_6]_leibniz.txt
• Q_gamma_7_maybe.txt and [P,Q_gamma_7_maybe].txt and
[P,Q_gamma_7_maybe]_leibniz.txt

Chapter 6
Graph complex action on Poisson
structures in dimension two
We investigate the action of Kontsevich’s graph complex on Poisson structures in a very
particular case, namely that of Poisson structures on R2. In the following we demonstrate
how to correlate several things: undirected graphs will become directed, (un)directed
graph cocycles will finally be evaluated at copies of Poisson structures and Jacobiators,
trivializing vector fields will be shown to be Hamiltonian with respect to the standard
symplectic structure, the respective Hamiltonian functions will be realized as sums of
graphs, and coboundaries in the graph complex will be mapped to Poisson coboundaries
thanks to identities for the Richardson–Nijenhuis bracket.
First import all necessary functionality from the gcaops package:
[1]: from gcaops.algebra.differential_polynomial_ring import DifferentialPolynomialRing
from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
from gcaops.algebra.homogeneous_polynomial_poisson_complex import PoissonComplex
from gcaops.graph.undirected_graph import UndirectedGraph
from gcaops.graph.undirected_graph_complex import UndirectedGraphComplex
from gcaops.graph.directed_graph import DirectedGraph
from gcaops.graph.directed_graph_complex import DirectedGraphComplex
Let x, y be Cartesian coordinates on R2, u be the coefficient of bi-vector P = u∂x ∧ ∂y;
V0, V1 be components of vector fields on R2, and H be a scalar function (the Hamiltonian)
on R2. Define the superfunction algebra with these coordinates:
[2]: D2 = DifferentialPolynomialRing(QQ, ('u', 'V0', 'V1', 'H'), ('x','y'),␣
↪→max_differential_orders=[8,1,1,1])
x, y = D2.base_variables()
u, V0, V1, H = D2.fibre_variables()
S2 = SuperfunctionAlgebra(D2, [x,y]); S2
[2]: Superfunction algebra over Differential Polynomial Ring in x, y, u, V0, V1, H, u_x,
u_y, u_xx, u_xy, u_yy, u_xxx, u_xxy, u_xyy, u_yyy, u_xxxx, u_xxxy, u_xxyy, u_xyyy,
u_yyyy, u_xxxxx, u_xxxxy, u_xxxyy, u_xxyyy, u_xyyyy, u_yyyyy, u_xxxxxx, u_xxxxxy,
u_xxxxyy, u_xxxyyy, u_xxyyyy, u_xyyyyy, u_yyyyyy, u_xxxxxxx, u_xxxxxxy, u_xxxxxyy,
u_xxxxyyy, u_xxxyyyy, u_xxyyyyy, u_xyyyyyy, u_yyyyyyy, u_xxxxxxxx, u_xxxxxxxy,
u_xxxxxxyy, u_xxxxxyyy, u_xxxxyyyy, u_xxxyyyyy, u_xxyyyyyy, u_xyyyyyyy, u_yyyyyyyy,
V0_x, V0_y, V1_x, V1_y, H_x, H_y over Rational Field with even coordinates [x, y] and
odd coordinates (xi0, xi1)
151
152 CHAPTER 6. THE GC ACTION ON 2D POISSON STRUCTURES
[3]: xi = S2.odd_coordinates(); xi
[3]: (xi0, xi1)
Consider a generic bi-vector field:
[4]: P2 = u*xi[0]*xi[1]; P2
[4]: (u)*xi0*xi1
The Jacobi identity is always satisfied (because the only tri-vector field is zero), so every
bi-vector field on R2 is Poisson:
[5]: P2.bracket(P2)
[5]: 0
Define the graph complexes:
[6]: GC = UndirectedGraphComplex(QQ, connected=True, biconnected=True, min_degree=3,␣
↪→implementation='vector'); GC
[6]: Undirected graph complex over Rational Field with Basis consisting of representatives
of isomorphism classes of undirected graphs (connected, biconnected, of degree at
least 3) with no automorphisms that induce an odd permutation on edges
[7]: dGC = DirectedGraphComplex(QQ, connected=True, biconnected=True, min_degree=3,␣
↪→loops=False, implementation='vector', sparse=True); dGC
[7]: Directed graph complex over Rational Field with Basis consisting of representatives of
isomorphism classes of directed graphs (connected, biconnected, of degree at least 3,
without loops) with no automorphisms that induce an odd permutation on edges
[8]: dfGC = DirectedGraphComplex(QQ, connected=True, implementation='vector', sparse=True);
↪→ dfGC
[8]: Directed graph complex over Rational Field with Basis consisting of representatives of
isomorphism classes of directed graphs (connected) with no automorphisms that induce
an odd permutation on edges
Now at our disposal we have the generic bi-vector field P2, and the graph complexes GC
and dGC. Let us act on the bi-vector field with the graph complexes.
6.1 Tetrahedral γ3 flow
This was the first example by Kontsevich (1996).
Define the tetrahedral flow:
[9]: tetrahedron_graph = UndirectedGraph(4, [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)]);␣
↪→tetrahedron_graph
tetrahedron_cocycle = GC(tetrahedron_graph); tetrahedron_cocycle
[9]: 1*UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
The tetrahedral flow is generally nonzero:
6.1. TETRAHEDRAL γ3 FLOW 153
[10]: tetrahedron_operation2 = S2.graph_operation(tetrahedron_cocycle)
Q_tetra2 = tetrahedron_operation2(P2,P2,P2,P2); Q_tetra2
[10]: (8*u_y^3*u_xxx - 24*u_x*u_y^2*u_xxy + 24*u_x^2*u_y*u_xyy - 8*u_x^3*u_yyy)*xi0*xi1
The tetrahedral flow indeed defines a Poisson 2-cocycle (again because the only tri-vector
field is zero):
[11]: P2.bracket(Q_tetra2)
[11]: 0
We recall that the Poisson cohomology of R2 can be non-trivial, e.g. already for P2 =
xy ∂x ∧ ∂y and an arbitrary (smooth or polynomial) vector field V we have that the bi-
vector field [[P2, V ]] vanishes at the origin, while e.g. the bi-vector ∂x ∧ ∂y does not, hence
this Poisson cocycle is not a coboundary.
However, from [27], the tetrahedral flow is known to be Poisson-trivial,
Qtetra(P2, P2, P2, P2) = [[P2, Xtetra(P2)]] in dimension two.
We find such a vector field. First, let V be an arbitrary vector field:
[12]: V = V0*xi[0] + V1*xi[1]; V
[12]: (V0)*xi0 + (V1)*xi1
Take the Poisson differential:
[13]: PbracketV = P2.bracket(V); PbracketV
[13]: (-V0*u_x - V1*u_y + u*V0_x + u*V1_y)*xi0*xi1
Solve Qtetra(P2, P2, P2, P2) = [[P2, V ]] for V by using homogeneity of degree and differential
weight of differential polynomials to generate an ansatz and solving the arising linear
system:
[14]: from gcaops.algebra.differential_polynomial_solver import solve_homogeneous_diffpoly
[15]: cX_tetra2 = solve_homogeneous_diffpoly(Q_tetra2[0,1], PbracketV[0,1], [V0, V1]);␣
↪→cX_tetra2
[15]: {V0: -16*u_y*u_xy^2 + 16*u_y*u_xx*u_yy + 8*u_y^2*u_xxy - 16*u_x*u_y*u_xyy +
8*u_x^2*u_yyy,
V1: 16*u_x*u_xy^2 - 16*u_x*u_xx*u_yy - 8*u_y^2*u_xxx + 16*u_x*u_y*u_xxy -
8*u_x^2*u_xyy}
[16]: X_tetra2 = cX_tetra2[V0]*xi[0] + cX_tetra2[V1]*xi[1]; X_tetra2
[16]: (-16*u_y*u_xy^2 + 16*u_y*u_xx*u_yy + 8*u_y^2*u_xxy - 16*u_x*u_y*u_xyy +
8*u_x^2*u_yyy)*xi0 + (16*u_x*u_xy^2 - 16*u_x*u_xx*u_yy - 8*u_y^2*u_xxx +
16*u_x*u_y*u_xxy - 8*u_x^2*u_xyy)*xi1
[17]: Q_tetra2 == P2.bracket(X_tetra2)
[17]: True
154 CHAPTER 6. THE GC ACTION ON 2D POISSON STRUCTURES
In fact, the vector field Xtetra(P2) is known to be Hamiltonian with respect to the standard
symplectic structure on R2 [5]. Let us reproduce this result:
[18]: cH_tetra2 = solve_homogeneous_diffpoly(cX_tetra2[V0], diff(H,y), [H]); cH_tetra2
[18]: {H: 8*u_y^2*u_xx - 16*u_x*u_y*u_xy + 8*u_x^2*u_yy}
[19]: cH_tetra2 = solve_homogeneous_diffpoly(cX_tetra2[V1], -diff(H,x), [H]); cH_tetra2
[19]: {H: 8*u_y^2*u_xx - 16*u_x*u_y*u_xy + 8*u_x^2*u_yy}
[20]: H_tetra2 = cH_tetra2[H]; H_tetra2
[20]: 8*u_y^2*u_xx - 16*u_x*u_y*u_xy + 8*u_x^2*u_yy
[21]: diff(H_tetra2, y) == X_tetra2[0]
[21]: True
[22]: -diff(H_tetra2, x) == X_tetra2[1]
[22]: True
The Hamiltonian H itself can be written as a sum of directed graphs:
[23]: ham3 = 8*dfGC(DirectedGraph(3, [(0, 2), (1, 0), (1, 2), (2, 1)])); ham3.show()
[24]: S2.graph_operation(ham3)(P2,P2,P2)[0,1] == H_tetra2
[24]: True
6.2 Five-wheel γ5 flow
Define the five-wheel cocycle:
[25]: fivewheel_graph = UndirectedGraph(6,␣
↪→[(0,1),(1,2),(2,3),(3,4),(0,4),(0,5),(1,5),(2,5),(3,5),(4,5)])
roof_graph = UndirectedGraph(6,␣
↪→[(0,1),(1,2),(2,3),(0,3),(3,4),(0,4),(4,5),(2,5),(1,5),(0,2)])
fivewheel_cocycle = GC(fivewheel_graph) + (5/2)*GC(roof_graph)
Take the 5-wheel cocycle and apply Kontsevich’s formula to evaluate it at six copies of
the Poisson structure:
6.2. FIVE-WHEEL γ5 FLOW 155
[26]: fivewheel_operation2 = S2.graph_operation(fivewheel_cocycle)
%time Q_fivewheel2 = fivewheel_operation2(P2,P2,P2,P2,P2,P2); Q_fivewheel2
CPU times: user 7.66 s, sys: 46.9 ms, total: 7.7 s
Wall time: 7.7 s
[26]: (-10*u_y^3*u_xx*u_yy*u_xxx + 20*u_x*u_y^2*u_xy*u_yy*u_xxx - 10*u_x^2*u_y*u_yy^2*u_xxx
+ 20*u_y^3*u_xx*u_xy*u_xxy - 40*u_x*u_y^2*u_xy^2*u_xxy + 10*u_x*u_y^2*u_xx*u_yy*u_xxy
+ 10*u_x^3*u_yy^2*u_xxy - 10*u_y^3*u_xx^2*u_xyy + 40*u_x^2*u_y*u_xy^2*u_xyy -
10*u_x^2*u_y*u_xx*u_yy*u_xyy - 20*u_x^3*u_xy*u_yy*u_xyy + 10*u_x*u_y^2*u_xx^2*u_yyy -
20*u_x^2*u_y*u_xx*u_xy*u_yyy + 10*u_x^3*u_xx*u_yy*u_yyy - 10*u_y^4*u_xy*u_xxxx +
10*u_x*u_y^3*u_yy*u_xxxx + 10*u_y^4*u_xx*u_xxxy + 20*u_x*u_y^3*u_xy*u_xxxy -
30*u_x^2*u_y^2*u_yy*u_xxxy - 30*u_x*u_y^3*u_xx*u_xxyy + 30*u_x^3*u_y*u_yy*u_xxyy +
30*u_x^2*u_y^2*u_xx*u_xyyy - 20*u_x^3*u_y*u_xy*u_xyyy - 10*u_x^4*u_yy*u_xyyy -
10*u_x^3*u_y*u_xx*u_yyyy + 10*u_x^4*u_xy*u_yyyy - 2*u_y^5*u_xxxxx +
10*u_x*u_y^4*u_xxxxy - 20*u_x^2*u_y^3*u_xxxyy + 20*u_x^3*u_y^2*u_xxyyy -
10*u_x^4*u_y*u_xyyyy + 2*u_x^5*u_yyyyy)*xi0*xi1
Let us do the same in a different way. First let us direct the edges of graphs in all possible
ways; then, filter out all directed graphs with vertices of outgoing degrees greater than
two; finally we place a copy of the Poisson bi-vector into every vertex. The outgoing
edges stand for derivatives of a bi-vector.
[27]: %%time
fivewheel_cocycle_directed = dGC(fivewheel_cocycle)
fivewheel_cocycle_directed_filtered = fivewheel_cocycle_directed.
↪→filter(max_out_degree=2)
fivewheel_operation2_directed = S2.
↪→graph_operation(fivewheel_cocycle_directed_filtered)
Q_fivewheel2 = fivewheel_operation2_directed(P2,P2,P2,P2,P2,P2); Q_fivewheel2
CPU times: user 1.81 s, sys: 3.91 ms, total: 1.81 s
Wall time: 1.82 s
[27]: (-10*u_y^3*u_xx*u_yy*u_xxx + 20*u_x*u_y^2*u_xy*u_yy*u_xxx - 10*u_x^2*u_y*u_yy^2*u_xxx
+ 20*u_y^3*u_xx*u_xy*u_xxy - 40*u_x*u_y^2*u_xy^2*u_xxy + 10*u_x*u_y^2*u_xx*u_yy*u_xxy
+ 10*u_x^3*u_yy^2*u_xxy - 10*u_y^3*u_xx^2*u_xyy + 40*u_x^2*u_y*u_xy^2*u_xyy -
10*u_x^2*u_y*u_xx*u_yy*u_xyy - 20*u_x^3*u_xy*u_yy*u_xyy + 10*u_x*u_y^2*u_xx^2*u_yyy -
20*u_x^2*u_y*u_xx*u_xy*u_yyy + 10*u_x^3*u_xx*u_yy*u_yyy - 10*u_y^4*u_xy*u_xxxx +
10*u_x*u_y^3*u_yy*u_xxxx + 10*u_y^4*u_xx*u_xxxy + 20*u_x*u_y^3*u_xy*u_xxxy -
30*u_x^2*u_y^2*u_yy*u_xxxy - 30*u_x*u_y^3*u_xx*u_xxyy + 30*u_x^3*u_y*u_yy*u_xxyy +
30*u_x^2*u_y^2*u_xx*u_xyyy - 20*u_x^3*u_y*u_xy*u_xyyy - 10*u_x^4*u_yy*u_xyyy -
10*u_x^3*u_y*u_xx*u_yyyy + 10*u_x^4*u_xy*u_yyyy - 2*u_y^5*u_xxxxx +
10*u_x*u_y^4*u_xxxxy - 20*u_x^2*u_y^3*u_xxxyy + 20*u_x^3*u_y^2*u_xxyyy -
10*u_x^4*u_y*u_xyyyy + 2*u_x^5*u_yyyyy)*xi0*xi1
Quite naturally, the outputs of the two cells immediately above coincide, as expected.
The five-wheel flow is a cocycle:
[28]: P2.bracket(Q_fivewheel2)
[28]: 0
The graph flow γ5 is also Poisson-trivial on R2:
156 CHAPTER 6. THE GC ACTION ON 2D POISSON STRUCTURES
[29]: cX_fivewheel2 = solve_homogeneous_diffpoly(Q_fivewheel2[0,1], PbracketV[0,1], [V0,␣
↪→V1]); cX_fivewheel2
[29]: {V0: -12*u_y*u_xy^4 + 24*u_y*u_xx*u_xy^2*u_yy - 12*u_y*u_xx^2*u_yy^2 -
4*u_y^2*u_xy*u_yy*u_xxx + 4*u_x*u_y*u_yy^2*u_xxx + 8*u_y^2*u_xy^2*u_xxy -
6*u_y^2*u_xx*u_yy*u_xxy + 8*u_x*u_y*u_xy*u_yy*u_xxy - 10*u_x^2*u_yy^2*u_xxy +
2*u_y^3*u_xxy^2 + 8*u_y^2*u_xx*u_xy*u_xyy - 32*u_x*u_y*u_xy^2*u_xyy +
4*u_x*u_y*u_xx*u_yy*u_xyy + 20*u_x^2*u_xy*u_yy*u_xyy - 2*u_y^3*u_xxx*u_xyy -
2*u_x*u_y^2*u_xxy*u_xyy + 2*u_x^2*u_y*u_xyy^2 - 6*u_y^2*u_xx^2*u_yyy +
16*u_x*u_y*u_xx*u_xy*u_yyy - 10*u_x^2*u_xx*u_yy*u_yyy + 2*u_x*u_y^2*u_xxx*u_yyy -
2*u_x^2*u_y*u_xxy*u_yyy - 8*u_y^3*u_yy*u_xxxx + 6*u_y^3*u_xy*u_xxxy +
26*u_x*u_y^2*u_yy*u_xxxy + 2*u_y^3*u_xx*u_xxyy - 22*u_x*u_y^2*u_xy*u_xxyy -
28*u_x^2*u_y*u_yy*u_xxyy - 4*u_x*u_y^2*u_xx*u_xyyy + 26*u_x^2*u_y*u_xy*u_xyyy +
10*u_x^3*u_yy*u_xyyy + 2*u_x^2*u_y*u_xx*u_yyyy - 10*u_x^3*u_xy*u_yyyy -
2*u_y^4*u_xxxxy + 8*u_x*u_y^3*u_xxxyy - 12*u_x^2*u_y^2*u_xxyyy + 8*u_x^3*u_y*u_xyyyy -
2*u_x^4*u_yyyyy,
V1: 12*u_x*u_xy^4 - 24*u_x*u_xx*u_xy^2*u_yy + 12*u_x*u_xx^2*u_yy^2 +
10*u_y^2*u_xx*u_yy*u_xxx - 16*u_x*u_y*u_xy*u_yy*u_xxx + 6*u_x^2*u_yy^2*u_xxx -
20*u_y^2*u_xx*u_xy*u_xxy + 32*u_x*u_y*u_xy^2*u_xxy - 4*u_x*u_y*u_xx*u_yy*u_xxy -
8*u_x^2*u_xy*u_yy*u_xxy - 2*u_x*u_y^2*u_xxy^2 + 10*u_y^2*u_xx^2*u_xyy -
8*u_x*u_y*u_xx*u_xy*u_xyy - 8*u_x^2*u_xy^2*u_xyy + 6*u_x^2*u_xx*u_yy*u_xyy +
2*u_x*u_y^2*u_xxx*u_xyy + 2*u_x^2*u_y*u_xxy*u_xyy - 2*u_x^3*u_xyy^2 -
4*u_x*u_y*u_xx^2*u_yyy + 4*u_x^2*u_xx*u_xy*u_yyy - 2*u_x^2*u_y*u_xxx*u_yyy +
2*u_x^3*u_xxy*u_yyy + 10*u_y^3*u_xy*u_xxxx - 2*u_x*u_y^2*u_yy*u_xxxx -
10*u_y^3*u_xx*u_xxxy - 26*u_x*u_y^2*u_xy*u_xxxy + 4*u_x^2*u_y*u_yy*u_xxxy +
28*u_x*u_y^2*u_xx*u_xxyy + 22*u_x^2*u_y*u_xy*u_xxyy - 2*u_x^3*u_yy*u_xxyy -
26*u_x^2*u_y*u_xx*u_xyyy - 6*u_x^3*u_xy*u_xyyy + 8*u_x^3*u_xx*u_yyyy + 2*u_y^4*u_xxxxx
- 8*u_x*u_y^3*u_xxxxy + 12*u_x^2*u_y^2*u_xxxyy - 8*u_x^3*u_y*u_xxyyy +
2*u_x^4*u_xyyyy}
[30]: X_fivewheel2 = cX_fivewheel2[V0]*xi[0] + cX_fivewheel2[V1]*xi[1]
[31]: Q_fivewheel2 == P2.bracket(X_fivewheel2)
[31]: True
This vector field is also Hamiltonian with respect to the standard symplectic structure
on R2.
[32]: cH_fivewheel2 = solve_homogeneous_diffpoly(cX_fivewheel2[V0], diff(H,y), [H]);␣
↪→cH_fivewheel2
[32]: {H: 6*u_y^2*u_xx*u_xy^2 - 12*u_x*u_y*u_xy^3 - 6*u_y^2*u_xx^2*u_yy +
12*u_x*u_y*u_xx*u_xy*u_yy + 6*u_x^2*u_xy^2*u_yy - 6*u_x^2*u_xx*u_yy^2 -
2*u_y^3*u_xy*u_xxx + 2*u_x*u_y^2*u_yy*u_xxx + 2*u_y^3*u_xx*u_xxy +
2*u_x*u_y^2*u_xy*u_xxy - 4*u_x^2*u_y*u_yy*u_xxy - 4*u_x*u_y^2*u_xx*u_xyy +
2*u_x^2*u_y*u_xy*u_xyy + 2*u_x^3*u_yy*u_xyy + 2*u_x^2*u_y*u_xx*u_yyy -
2*u_x^3*u_xy*u_yyy - 2*u_y^4*u_xxxx + 8*u_x*u_y^3*u_xxxy - 12*u_x^2*u_y^2*u_xxyy +
8*u_x^3*u_y*u_xyyy - 2*u_x^4*u_yyyy}
[33]: cH_fivewheel2 = solve_homogeneous_diffpoly(cX_fivewheel2[V1], -diff(H,x), [H]);␣
↪→cH_fivewheel2
[33]: {H: 6*u_y^2*u_xx*u_xy^2 - 12*u_x*u_y*u_xy^3 - 6*u_y^2*u_xx^2*u_yy +
12*u_x*u_y*u_xx*u_xy*u_yy + 6*u_x^2*u_xy^2*u_yy - 6*u_x^2*u_xx*u_yy^2 -
2*u_y^3*u_xy*u_xxx + 2*u_x*u_y^2*u_yy*u_xxx + 2*u_y^3*u_xx*u_xxy +
6.2. FIVE-WHEEL γ5 FLOW 157
2*u_x*u_y^2*u_xy*u_xxy - 4*u_x^2*u_y*u_yy*u_xxy - 4*u_x*u_y^2*u_xx*u_xyy +
2*u_x^2*u_y*u_xy*u_xyy + 2*u_x^3*u_yy*u_xyy + 2*u_x^2*u_y*u_xx*u_yyy -
2*u_x^3*u_xy*u_yyy - 2*u_y^4*u_xxxx + 8*u_x*u_y^3*u_xxxy - 12*u_x^2*u_y^2*u_xxyy +
8*u_x^3*u_y*u_xyyy - 2*u_x^4*u_yyyy}
[34]: H_fivewheel2 = cH_fivewheel2[H]; H_fivewheel2
[34]: 6*u_y^2*u_xx*u_xy^2 - 12*u_x*u_y*u_xy^3 - 6*u_y^2*u_xx^2*u_yy +
12*u_x*u_y*u_xx*u_xy*u_yy + 6*u_x^2*u_xy^2*u_yy - 6*u_x^2*u_xx*u_yy^2 -
2*u_y^3*u_xy*u_xxx + 2*u_x*u_y^2*u_yy*u_xxx + 2*u_y^3*u_xx*u_xxy +
2*u_x*u_y^2*u_xy*u_xxy - 4*u_x^2*u_y*u_yy*u_xxy - 4*u_x*u_y^2*u_xx*u_xyy +
2*u_x^2*u_y*u_xy*u_xyy + 2*u_x^3*u_yy*u_xyy + 2*u_x^2*u_y*u_xx*u_yyy -
2*u_x^3*u_xy*u_yyy - 2*u_y^4*u_xxxx + 8*u_x*u_y^3*u_xxxy - 12*u_x^2*u_y^2*u_xxyy +
8*u_x^3*u_y*u_xyyy - 2*u_x^4*u_yyyy
[35]: diff(H_fivewheel2, y) == X_fivewheel2[0]
[35]: True
[36]: -diff(H_fivewheel2, x) == X_fivewheel2[1]
[36]: True
The Hamiltonian of the found trivializing vector field for the γ5 flow can be written as a
sum of directed graphs:
[37]: ham5 = (6)*dfGC(DirectedGraph(5, [(0, 4), (1, 2), (1, 3), (2, 3), (3, 0), (3, 4), (4,␣
↪→1), (4, 2)])) + \
(-2)*dfGC(DirectedGraph(5, [(0, 3), (0, 4), (1, 3), (2, 0), (2, 4), (3, 2),␣
↪→(3, 4), (4, 1)])) + \
(-2)*dfGC(DirectedGraph(5, [(0, 3), (0, 4), (1, 2), (1, 4), (2, 0), (2, 4),␣
↪→(3, 1), (3, 4)]))
ham5.show()
Actually you get a bi-vector on R2, and the Hamiltonian is its coefficient.
[38]: S2.graph_operation(ham5)(P2,P2,P2,P2,P2)[0,1] == H_fivewheel2
[38]: True
158 CHAPTER 6. THE GC ACTION ON 2D POISSON STRUCTURES
6.3 Graph coboundary δ6 = d(β6) and the
Poisson-trivial flow
The flow associated with δ6 is Poisson-trivial in all dimensions (and particularly on R2).
We first define what will be the trivializing vector field Xδ6 = Or(β )(P⊗66 2 ):
[39]: delta6_primitive_graph = UndirectedGraph(6,␣
↪→[(0,2),(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)])
delta6_primitive = GC(delta6_primitive_graph)
delta6_cocycle = delta6_primitive.differential(); delta6_cocycle
[39]: (-4)*UndirectedGraph(7, [(0, 1), (0, 4), (0, 6), (1, 2), (1, 6), (2, 3), (2, 5), (3,
5), (3, 6), (4, 5), (4, 6), (5, 6)]) + (4)*UndirectedGraph(7, [(0, 1), (0, 2), (0, 5),
(1, 4), (1, 5), (2, 4), (2, 6), (3, 4), (3, 5), (3, 6), (4, 6), (5, 6)]) +
(4)*UndirectedGraph(7, [(0, 3), (0, 4), (0, 5), (1, 2), (1, 4), (1, 5), (2, 4), (2,
6), (3, 5), (3, 6), (4, 6), (5, 6)]) + (2)*UndirectedGraph(7, [(0, 1), (0, 2), (0, 6),
(1, 2), (1, 5), (2, 4), (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6)])
[40]: delta6_primitive_operation2 = S2.graph_operation(delta6_primitive);␣
↪→delta6_primitive_operation2
[40]: Symmetric operation of arity 6 and degree -11 on Superfunction algebra over
Differential Polynomial Ring in x, y, u, V0, V1, H, u_x, u_y, u_xx, u_xy, u_yy, u_xxx,
u_xxy, u_xyy, u_yyy, u_xxxx, u_xxxy, u_xxyy, u_xyyy, u_yyyy, u_xxxxx, u_xxxxy,
u_xxxyy, u_xxyyy, u_xyyyy, u_yyyyy, u_xxxxxx, u_xxxxxy, u_xxxxyy, u_xxxyyy, u_xxyyyy,
u_xyyyyy, u_yyyyyy, u_xxxxxxx, u_xxxxxxy, u_xxxxxyy, u_xxxxyyy, u_xxxyyyy, u_xxyyyyy,
u_xyyyyyy, u_yyyyyyy, u_xxxxxxxx, u_xxxxxxxy, u_xxxxxxyy, u_xxxxxyyy, u_xxxxyyyy,
u_xxxyyyyy, u_xxyyyyyy, u_xyyyyyyy, u_yyyyyyyy, V0_x, V0_y, V1_x, V1_y, H_x, H_y over
Rational Field with even coordinates [x, y] and odd coordinates (xi0, xi1)
Here is the vector field Xδ6 :
[41]: %time X_delta6_2 = delta6_primitive_operation2(P2,P2,P2,P2,P2,P2); X_delta6_2
CPU times: user 7.75 s, sys: 35.7 ms, total: 7.79 s
Wall time: 7.78 s
[41]: (4*u_y^2*u_xy^2*u_yy*u_xxx - 8*u_x*u_y*u_xy*u_yy^2*u_xxx + 4*u_x^2*u_yy^3*u_xxx +
4*u_y^2*u_xy^3*u_xxy - 16*u_y^2*u_xx*u_xy*u_yy*u_xxy + 8*u_x*u_y*u_xy^2*u_yy*u_xxy +
16*u_x*u_y*u_xx*u_yy^2*u_xxy - 12*u_x^2*u_xy*u_yy^2*u_xxy - 8*u_y^3*u_xy*u_xxy^2 +
8*u_x*u_y^2*u_yy*u_xxy^2 - 8*u_x*u_y*u_xy^3*u_xyy + 12*u_y^2*u_xx^2*u_yy*u_xyy -
16*u_x*u_y*u_xx*u_xy*u_yy*u_xyy + 20*u_x^2*u_xy^2*u_yy*u_xyy -
8*u_x^2*u_xx*u_yy^2*u_xyy + 8*u_y^3*u_xy*u_xxx*u_xyy - 8*u_x*u_y^2*u_yy*u_xxx*u_xyy +
8*u_x*u_y^2*u_xy*u_xxy*u_xyy - 8*u_x^2*u_y*u_yy*u_xxy*u_xyy - 8*u_x^2*u_y*u_xy*u_xyy^2
+ 8*u_x^3*u_yy*u_xyy^2 - 4*u_y^2*u_xx^2*u_xy*u_yyy + 16*u_x*u_y*u_xx*u_xy^2*u_yyy -
12*u_x^2*u_xy^3*u_yyy - 8*u_x*u_y*u_xx^2*u_yy*u_yyy + 8*u_x^2*u_xx*u_xy*u_yy*u_yyy -
8*u_x*u_y^2*u_xy*u_xxx*u_yyy + 8*u_x^2*u_y*u_yy*u_xxx*u_yyy +
8*u_x^2*u_y*u_xy*u_xxy*u_yyy - 8*u_x^3*u_yy*u_xxy*u_yyy + 4*u_y^3*u_xy^2*u_xxxy -
8*u_x*u_y^2*u_xy*u_yy*u_xxxy + 4*u_x^2*u_y*u_yy^2*u_xxxy - 8*u_y^3*u_xx*u_xy*u_xxyy +
4*u_x*u_y^2*u_xy^2*u_xxyy + 8*u_x*u_y^2*u_xx*u_yy*u_xxyy - 4*u_x^3*u_yy^2*u_xxyy +
4*u_y^3*u_xx^2*u_xyyy - 4*u_x^2*u_y*u_xy^2*u_xyyy - 8*u_x^2*u_y*u_xx*u_yy*u_xyyy +
8*u_x^3*u_xy*u_yy*u_xyyy - 4*u_x*u_y^2*u_xx^2*u_yyyy + 8*u_x^2*u_y*u_xx*u_xy*u_yyyy -
4*u_x^3*u_xy^2*u_yyyy)*xi0 + (-12*u_y^2*u_xy^3*u_xxx + 8*u_y^2*u_xx*u_xy*u_yy*u_xxx +
16*u_x*u_y*u_xy^2*u_yy*u_xxx - 8*u_x*u_y*u_xx*u_yy^2*u_xxx - 4*u_x^2*u_xy*u_yy^2*u_xxx
6.3. GRAPH COBOUNDARY δ6 = D(β6) AND THE POISSON-TRIVIAL FLOW 159
+ 20*u_y^2*u_xx*u_xy^2*u_xxy - 8*u_x*u_y*u_xy^3*u_xxy - 8*u_y^2*u_xx^2*u_yy*u_xxy -
16*u_x*u_y*u_xx*u_xy*u_yy*u_xxy + 12*u_x^2*u_xx*u_yy^2*u_xxy + 8*u_y^3*u_xx*u_xxy^2 -
8*u_x*u_y^2*u_xy*u_xxy^2 - 12*u_y^2*u_xx^2*u_xy*u_xyy + 8*u_x*u_y*u_xx*u_xy^2*u_xyy +
4*u_x^2*u_xy^3*u_xyy + 16*u_x*u_y*u_xx^2*u_yy*u_xyy - 16*u_x^2*u_xx*u_xy*u_yy*u_xyy -
8*u_y^3*u_xx*u_xxx*u_xyy + 8*u_x*u_y^2*u_xy*u_xxx*u_xyy - 8*u_x*u_y^2*u_xx*u_xxy*u_xyy
+ 8*u_x^2*u_y*u_xy*u_xxy*u_xyy + 8*u_x^2*u_y*u_xx*u_xyy^2 - 8*u_x^3*u_xy*u_xyy^2 +
4*u_y^2*u_xx^3*u_yyy - 8*u_x*u_y*u_xx^2*u_xy*u_yyy + 4*u_x^2*u_xx*u_xy^2*u_yyy +
8*u_x*u_y^2*u_xx*u_xxx*u_yyy - 8*u_x^2*u_y*u_xy*u_xxx*u_yyy -
8*u_x^2*u_y*u_xx*u_xxy*u_yyy + 8*u_x^3*u_xy*u_xxy*u_yyy - 4*u_y^3*u_xy^2*u_xxxx +
8*u_x*u_y^2*u_xy*u_yy*u_xxxx - 4*u_x^2*u_y*u_yy^2*u_xxxx + 8*u_y^3*u_xx*u_xy*u_xxxy -
4*u_x*u_y^2*u_xy^2*u_xxxy - 8*u_x*u_y^2*u_xx*u_yy*u_xxxy + 4*u_x^3*u_yy^2*u_xxxy -
4*u_y^3*u_xx^2*u_xxyy + 4*u_x^2*u_y*u_xy^2*u_xxyy + 8*u_x^2*u_y*u_xx*u_yy*u_xxyy -
8*u_x^3*u_xy*u_yy*u_xxyy + 4*u_x*u_y^2*u_xx^2*u_xyyy - 8*u_x^2*u_y*u_xx*u_xy*u_xyyy +
4*u_x^3*u_xy^2*u_xyyy)*xi1
[42]: delta6_operation2 = S2.graph_operation(delta6_cocycle); delta6_operation2
[42]: Symmetric operation of arity 7 and degree -12 on Superfunction algebra over
Differential Polynomial Ring in x, y, u, V0, V1, H, u_x, u_y, u_xx, u_xy, u_yy, u_xxx,
u_xxy, u_xyy, u_yyy, u_xxxx, u_xxxy, u_xxyy, u_xyyy, u_yyyy, u_xxxxx, u_xxxxy,
u_xxxyy, u_xxyyy, u_xyyyy, u_yyyyy, u_xxxxxx, u_xxxxxy, u_xxxxyy, u_xxxyyy, u_xxyyyy,
u_xyyyyy, u_yyyyyy, u_xxxxxxx, u_xxxxxxy, u_xxxxxyy, u_xxxxyyy, u_xxxyyyy, u_xxyyyyy,
u_xyyyyyy, u_yyyyyyy, u_xxxxxxxx, u_xxxxxxxy, u_xxxxxxyy, u_xxxxxyyy, u_xxxxyyyy,
u_xxxyyyyy, u_xxyyyyyy, u_xyyyyyyy, u_yyyyyyyy, V0_x, V0_y, V1_x, V1_y, H_x, H_y over
Rational Field with even coordinates [x, y] and odd coordinates (xi0, xi1)
Here is the flow Ṗ ⊗72 = Or(δ6)(P2 ):
[43]: %time Q_delta6_2 = delta6_operation2(P2,P2,P2,P2,P2,P2,P2); Q_delta6_2
CPU times: user 2min 9s, sys: 752 ms, total: 2min 10s
Wall time: 2min 10s
[43]: (24*u_y^3*u_xy^3*u_xxx - 16*u_y^3*u_xx*u_xy*u_yy*u_xxx -
40*u_x*u_y^2*u_xy^2*u_yy*u_xxx + 16*u_x*u_y^2*u_xx*u_yy^2*u_xxx +
24*u_x^2*u_y*u_xy*u_yy^2*u_xxx - 8*u_x^3*u_yy^3*u_xxx - 40*u_y^3*u_xx*u_xy^2*u_xxy +
8*u_x*u_y^2*u_xy^3*u_xxy + 16*u_y^3*u_xx^2*u_yy*u_xxy +
64*u_x*u_y^2*u_xx*u_xy*u_yy*u_xxy - 16*u_x^2*u_y*u_xy^2*u_yy*u_xxy -
56*u_x^2*u_y*u_xx*u_yy^2*u_xxy + 24*u_x^3*u_xy*u_yy^2*u_xxy - 16*u_y^4*u_xx*u_xxy^2 +
32*u_x*u_y^3*u_xy*u_xxy^2 - 16*u_x^2*u_y^2*u_yy*u_xxy^2 + 24*u_y^3*u_xx^2*u_xy*u_xyy -
16*u_x*u_y^2*u_xx*u_xy^2*u_xyy + 8*u_x^2*u_y*u_xy^3*u_xyy -
56*u_x*u_y^2*u_xx^2*u_yy*u_xyy + 64*u_x^2*u_y*u_xx*u_xy*u_yy*u_xyy -
40*u_x^3*u_xy^2*u_yy*u_xyy + 16*u_x^3*u_xx*u_yy^2*u_xyy + 16*u_y^4*u_xx*u_xxx*u_xyy -
32*u_x*u_y^3*u_xy*u_xxx*u_xyy + 16*u_x^2*u_y^2*u_yy*u_xxx*u_xyy +
16*u_x*u_y^3*u_xx*u_xxy*u_xyy - 32*u_x^2*u_y^2*u_xy*u_xxy*u_xyy +
16*u_x^3*u_y*u_yy*u_xxy*u_xyy - 16*u_x^2*u_y^2*u_xx*u_xyy^2 +
32*u_x^3*u_y*u_xy*u_xyy^2 - 16*u_x^4*u_yy*u_xyy^2 - 8*u_y^3*u_xx^3*u_yyy +
24*u_x*u_y^2*u_xx^2*u_xy*u_yyy - 40*u_x^2*u_y*u_xx*u_xy^2*u_yyy +
24*u_x^3*u_xy^3*u_yyy + 16*u_x^2*u_y*u_xx^2*u_yy*u_yyy - 16*u_x^3*u_xx*u_xy*u_yy*u_yyy
- 16*u_x*u_y^3*u_xx*u_xxx*u_yyy + 32*u_x^2*u_y^2*u_xy*u_xxx*u_yyy -
16*u_x^3*u_y*u_yy*u_xxx*u_yyy + 16*u_x^2*u_y^2*u_xx*u_xxy*u_yyy -
32*u_x^3*u_y*u_xy*u_xxy*u_yyy + 16*u_x^4*u_yy*u_xxy*u_yyy + 8*u_y^4*u_xy^2*u_xxxx -
16*u_x*u_y^3*u_xy*u_yy*u_xxxx + 8*u_x^2*u_y^2*u_yy^2*u_xxxx -
16*u_y^4*u_xx*u_xy*u_xxxy + 16*u_x*u_y^3*u_xx*u_yy*u_xxxy +
16*u_x^2*u_y^2*u_xy*u_yy*u_xxxy - 16*u_x^3*u_y*u_yy^2*u_xxxy + 8*u_y^4*u_xx^2*u_xxyy +
16*u_x*u_y^3*u_xx*u_xy*u_xxyy - 16*u_x^2*u_y^2*u_xy^2*u_xxyy -
160 CHAPTER 6. THE GC ACTION ON 2D POISSON STRUCTURES
32*u_x^2*u_y^2*u_xx*u_yy*u_xxyy + 16*u_x^3*u_y*u_xy*u_yy*u_xxyy +
8*u_x^4*u_yy^2*u_xxyy - 16*u_x*u_y^3*u_xx^2*u_xyyy + 16*u_x^2*u_y^2*u_xx*u_xy*u_xyyy +
16*u_x^3*u_y*u_xx*u_yy*u_xyyy - 16*u_x^4*u_xy*u_yy*u_xyyy +
8*u_x^2*u_y^2*u_xx^2*u_yyyy - 16*u_x^3*u_y*u_xx*u_xy*u_yyyy +
8*u_x^4*u_xy^2*u_yyyy)*xi0*xi1
The δ6 flow is Poisson trivial:
[44]: schouten2 = S2.schouten_bracket(); schouten2
[44]: Schouten bracket on Superfunction algebra over Differential Polynomial Ring in x, y,
u, V0, V1, H, u_x, u_y, u_xx, u_xy, u_yy, u_xxx, u_xxy, u_xyy, u_yyy, u_xxxx, u_xxxy,
u_xxyy, u_xyyy, u_yyyy, u_xxxxx, u_xxxxy, u_xxxyy, u_xxyyy, u_xyyyy, u_yyyyy,
u_xxxxxx, u_xxxxxy, u_xxxxyy, u_xxxyyy, u_xxyyyy, u_xyyyyy, u_yyyyyy, u_xxxxxxx,
u_xxxxxxy, u_xxxxxyy, u_xxxxyyy, u_xxxyyyy, u_xxyyyyy, u_xyyyyyy, u_yyyyyyy,
u_xxxxxxxx, u_xxxxxxxy, u_xxxxxxyy, u_xxxxxyyy, u_xxxxyyyy, u_xxxyyyyy, u_xxyyyyyy,
u_xyyyyyyy, u_yyyyyyyy, V0_x, V0_y, V1_x, V1_y, H_x, H_y over Rational Field with even
coordinates [x, y] and odd coordinates (xi0, xi1)
[45]: Q_delta6_2 == schouten2(P2,-2*X_delta6_2)
[45]: True
Using the Nijenhuis–Richardson bracket [πS,Or(β ⊗76)]NR(P2 ) = −(7/2)Or(δ6)(P2), we
get an independent verification that the δ6 flow is Poisson trivial:
[46]: schouten_delta6_primitive_operation2 = schouten2.bracket(delta6_primitive_operation2)
[47]: schouten_delta6_primitive_operation2
[47]: Symmetric operation of arity 7 and degree -12 on Superfunction algebra over
Differential Polynomial Ring in x, y, u, V0, V1, H, u_x, u_y, u_xx, u_xy, u_yy, u_xxx,
u_xxy, u_xyy, u_yyy, u_xxxx, u_xxxy, u_xxyy, u_xyyy, u_yyyy, u_xxxxx, u_xxxxy,
u_xxxyy, u_xxyyy, u_xyyyy, u_yyyyy, u_xxxxxx, u_xxxxxy, u_xxxxyy, u_xxxyyy, u_xxyyyy,
u_xyyyyy, u_yyyyyy, u_xxxxxxx, u_xxxxxxy, u_xxxxxyy, u_xxxxyyy, u_xxxyyyy, u_xxyyyyy,
u_xyyyyyy, u_yyyyyyy, u_xxxxxxxx, u_xxxxxxxy, u_xxxxxxyy, u_xxxxxyyy, u_xxxxyyyy,
u_xxxyyyyy, u_xxyyyyyy, u_xyyyyyyy, u_yyyyyyyy, V0_x, V0_y, V1_x, V1_y, H_x, H_y over
Rational Field with even coordinates [x, y] and odd coordinates (xi0, xi1) given by
the Nijenhuis-Richardson bracket of two symmetric operations
[48]: %time schouten_delta6_primitive_operation2(P2,P2,P2,P2,P2,P2,P2) == -(7/2)*Q_delta6_2
CPU times: user 1min 37s, sys: 296 ms, total: 1min 38s
Wall time: 1min 38s
[48]: True
The trivializing vector field is Hamiltonian with respect to the standard symplectic struc-
ture:
[49]: cH_delta6_2 = solve_homogeneous_diffpoly(X_delta6_2[0], diff(H,y), [H]); cH_delta6_2
[49]: {H: 4*u_y^3*u_xy^2*u_xxx - 8*u_x*u_y^2*u_xy*u_yy*u_xxx + 4*u_x^2*u_y*u_yy^2*u_xxx -
8*u_y^3*u_xx*u_xy*u_xxy + 4*u_x*u_y^2*u_xy^2*u_xxy + 8*u_x*u_y^2*u_xx*u_yy*u_xxy -
4*u_x^3*u_yy^2*u_xxy + 4*u_y^3*u_xx^2*u_xyy - 4*u_x^2*u_y*u_xy^2*u_xyy -
6.4. HEPTAGON-WHEEL γ7 FLOW 161
8*u_x^2*u_y*u_xx*u_yy*u_xyy + 8*u_x^3*u_xy*u_yy*u_xyy - 4*u_x*u_y^2*u_xx^2*u_yyy +
8*u_x^2*u_y*u_xx*u_xy*u_yyy - 4*u_x^3*u_xy^2*u_yyy}
[50]: solve_homogeneous_diffpoly(X_delta6_2[1], -diff(H,x), [H])
[50]: {H: 4*u_y^3*u_xy^2*u_xxx - 8*u_x*u_y^2*u_xy*u_yy*u_xxx + 4*u_x^2*u_y*u_yy^2*u_xxx -
8*u_y^3*u_xx*u_xy*u_xxy + 4*u_x*u_y^2*u_xy^2*u_xxy + 8*u_x*u_y^2*u_xx*u_yy*u_xxy -
4*u_x^3*u_yy^2*u_xxy + 4*u_y^3*u_xx^2*u_xyy - 4*u_x^2*u_y*u_xy^2*u_xyy -
8*u_x^2*u_y*u_xx*u_yy*u_xyy + 8*u_x^3*u_xy*u_yy*u_xyy - 4*u_x*u_y^2*u_xx^2*u_yyy +
8*u_x^2*u_y*u_xx*u_xy*u_yyy - 4*u_x^3*u_xy^2*u_yyy}
[51]: H_delta6_2 = cH_delta6_2[H]
[52]: diff(H_delta6_2, y) == X_delta6_2[0]
[52]: True
[53]: -diff(H_delta6_2, x) == X_delta6_2[1]
[53]: True
The Hamiltonian can be realized as a sum of graphs:
[54]: ham6_1 = (4)*dfGC(DirectedGraph(6, [(0, 4), (0, 5), (1, 3), (2, 0), (2, 5), (3, 2),␣
↪→(3, 4), (4, 5), (5, 1), (5, 3)])); ham6_1.show()
[55]: S2.graph_operation(ham6_1)(P2,P2,P2,P2,P2,P2)[0,1] == H_delta6_2
[55]: True
6.4 Heptagon-wheel γ7 flow
Get a representative of the cohomology class for γ7:
[56]: heptagon_cocycle_maybe = GC.cohomology_basis(8,14)[0]
Orient it:
[57]: %time heptagon_cocycle_maybe_directed = dGC(heptagon_cocycle_maybe)
162 CHAPTER 6. THE GC ACTION ON 2D POISSON STRUCTURES
CPU times: user 2.53 s, sys: 44 ms, total: 2.58 s
Wall time: 2.58 s
[58]: len(heptagon_cocycle_maybe_directed)
[58]: 595476
Filter out graphs that will produce zero when evaluated only at bi-vector fields.
[59]: heptagon_cocycle_maybe_directed_filtered = heptagon_cocycle_maybe_directed.
↪→filter(max_out_degree=2)
[60]: len(heptagon_cocycle_maybe_directed_filtered)
[60]: 20422
Define the operation Or(γ7) on multi-vectors on R2.
[61]: heptagon_operation2 = S2.graph_operation(heptagon_cocycle_maybe_directed_filtered)
[62]: heptagon_operation2
[62]: Symmetric operation of arity 8 and degree -14 on Superfunction algebra over
Differential Polynomial Ring in x, y, u, V0, V1, H, u_x, u_y, u_xx, u_xy, u_yy, u_xxx,
u_xxy, u_xyy, u_yyy, u_xxxx, u_xxxy, u_xxyy, u_xyyy, u_yyyy, u_xxxxx, u_xxxxy,
u_xxxyy, u_xxyyy, u_xyyyy, u_yyyyy, u_xxxxxx, u_xxxxxy, u_xxxxyy, u_xxxyyy, u_xxyyyy,
u_xyyyyy, u_yyyyyy, u_xxxxxxx, u_xxxxxxy, u_xxxxxyy, u_xxxxyyy, u_xxxyyyy, u_xxyyyyy,
u_xyyyyyy, u_yyyyyyy, u_xxxxxxxx, u_xxxxxxxy, u_xxxxxxyy, u_xxxxxyyy, u_xxxxyyyy,
u_xxxyyyyy, u_xxyyyyyy, u_xyyyyyyy, u_yyyyyyyy, V0_x, V0_y, V1_x, V1_y, H_x, H_y over
Rational Field with even coordinates [x, y] and odd coordinates (xi0, xi1)
To get the Kontsevich γ7 flow, evaluate the operation Or(γ7) at P⊗82 :
[63]: %time Q_heptagon2 = heptagon_operation2(P2,P2,P2,P2,P2,P2,P2,P2)
CPU times: user 1h 43min 48s, sys: 875 ms, total: 1h 43min 49s
Wall time: 1h 43min 50s
[64]: P2.bracket(Q_heptagon2)
[64]: 0
Next we establish that in dimension two, the γ7 flow is Poisson trivial.
[65]: %time cX_hepta2 = solve_homogeneous_diffpoly(Q_heptagon2[0,1], PbracketV[0,1], [V0,␣
↪→V1]) #; cX_hepta2
CPU times: user 15min 58s, sys: 4.17 s, total: 16min 2s
Wall time: 16min 2s
[66]: X_hepta2 = cX_hepta2[V0]*xi[0] + cX_hepta2[V1]*xi[1]
[67]: Q_heptagon2 == P2.bracket(X_hepta2)
6.4. HEPTAGON-WHEEL γ7 FLOW 163
[67]: True
The trivializing vector field is Hamiltonian with respect to the standard symplectic struc-
ture:
[68]: cH_hepta2 = solve_homogeneous_diffpoly(cX_hepta2[V0], diff(H,y), [H]); cH_hepta2
[68]: {H: 199/4*u_y^2*u_xx*u_xy^4 - 199/2*u_x*u_y*u_xy^5 - 199/2*u_y^2*u_xx^2*u_xy^2*u_yy +
199*u_x*u_y*u_xx*u_xy^3*u_yy + 199/4*u_x^2*u_xy^4*u_yy + 199/4*u_y^2*u_xx^3*u_yy^2 -
199/2*u_x*u_y*u_xx^2*u_xy*u_yy^2 - 199/2*u_x^2*u_xx*u_xy^2*u_yy^2 +
199/4*u_x^2*u_xx^2*u_yy^3 - 9*u_y^3*u_xy^3*u_xxx + 23/4*u_y^3*u_xx*u_xy*u_yy*u_xxx +
31/2*u_x*u_y^2*u_xy^2*u_yy*u_xxx - 23/4*u_x*u_y^2*u_xx*u_yy^2*u_xxx -
39/4*u_x^2*u_y*u_xy*u_yy^2*u_xxx + 13/4*u_x^3*u_yy^3*u_xxx + 6*u_y^4*u_yy*u_xxx^2 +
31/2*u_y^3*u_xx*u_xy^2*u_xxy - 4*u_x*u_y^2*u_xy^3*u_xxy - 23/4*u_y^3*u_xx^2*u_yy*u_xxy
- 101/4*u_x*u_y^2*u_xx*u_xy*u_yy*u_xxy + 8*u_x^2*u_y*u_xy^2*u_yy*u_xxy +
85/4*u_x^2*u_y*u_xx*u_yy^2*u_xxy - 39/4*u_x^3*u_xy*u_yy^2*u_xxy -
12*u_y^4*u_xy*u_xxx*u_xxy - 24*u_x*u_y^3*u_yy*u_xxx*u_xxy - 9*u_y^4*u_xx*u_xxy^2 +
54*u_x*u_y^3*u_xy*u_xxy^2 + 9*u_x^2*u_y^2*u_yy*u_xxy^2 - 39/4*u_y^3*u_xx^2*u_xy*u_xyy
+ 8*u_x*u_y^2*u_xx*u_xy^2*u_xyy - 4*u_x^2*u_y*u_xy^3*u_xyy +
85/4*u_x*u_y^2*u_xx^2*u_yy*u_xyy - 101/4*u_x^2*u_y*u_xx*u_xy*u_yy*u_xyy +
31/2*u_x^3*u_xy^2*u_yy*u_xyy - 23/4*u_x^3*u_xx*u_yy^2*u_xyy +
15*u_y^4*u_xx*u_xxx*u_xyy - 6*u_x*u_y^3*u_xy*u_xxx*u_xyy +
27*u_x^2*u_y^2*u_yy*u_xxx*u_xyy - 9*u_x*u_y^3*u_xx*u_xxy*u_xyy -
90*u_x^2*u_y^2*u_xy*u_xxy*u_xyy - 9*u_x^3*u_y*u_yy*u_xxy*u_xyy +
9*u_x^2*u_y^2*u_xx*u_xyy^2 + 54*u_x^3*u_y*u_xy*u_xyy^2 - 9*u_x^4*u_yy*u_xyy^2 +
13/4*u_y^3*u_xx^3*u_yyy - 39/4*u_x*u_y^2*u_xx^2*u_xy*u_yyy +
31/2*u_x^2*u_y*u_xx*u_xy^2*u_yyy - 9*u_x^3*u_xy^3*u_yyy -
23/4*u_x^2*u_y*u_xx^2*u_yy*u_yyy + 23/4*u_x^3*u_xx*u_xy*u_yy*u_yyy -
15*u_x*u_y^3*u_xx*u_xxx*u_yyy + 18*u_x^2*u_y^2*u_xy*u_xxx*u_yyy -
15*u_x^3*u_y*u_yy*u_xxx*u_yyy + 27*u_x^2*u_y^2*u_xx*u_xxy*u_yyy -
6*u_x^3*u_y*u_xy*u_xxy*u_yyy + 15*u_x^4*u_yy*u_xxy*u_yyy -
24*u_x^3*u_y*u_xx*u_xyy*u_yyy - 12*u_x^4*u_xy*u_xyy*u_yyy + 6*u_x^4*u_xx*u_yyy^2 +
18*u_y^4*u_xy^2*u_xxxx + 16*u_y^4*u_xx*u_yy*u_xxxx - 68*u_x*u_y^3*u_xy*u_yy*u_xxxx +
34*u_x^2*u_y^2*u_yy^2*u_xxxx + 12*u_y^5*u_xxy*u_xxxx - 24*u_x*u_y^4*u_xyy*u_xxxx +
12*u_x^2*u_y^3*u_yyy*u_xxxx - 68*u_y^4*u_xx*u_xy*u_xxxy + 64*u_x*u_y^3*u_xy^2*u_xxxy +
4*u_x*u_y^3*u_xx*u_yy*u_xxxy + 68*u_x^2*u_y^2*u_xy*u_yy*u_xxxy -
68*u_x^3*u_y*u_yy^2*u_xxxy - 12*u_y^5*u_xxx*u_xxxy - 12*u_x*u_y^4*u_xxy*u_xxxy +
60*u_x^2*u_y^3*u_xyy*u_xxxy - 36*u_x^3*u_y^2*u_yyy*u_xxxy + 34*u_y^4*u_xx^2*u_xxyy +
68*u_x*u_y^3*u_xx*u_xy*u_xxyy - 164*u_x^2*u_y^2*u_xy^2*u_xxyy -
40*u_x^2*u_y^2*u_xx*u_yy*u_xxyy + 68*u_x^3*u_y*u_xy*u_yy*u_xxyy +
34*u_x^4*u_yy^2*u_xxyy + 36*u_x*u_y^4*u_xxx*u_xxyy - 36*u_x^2*u_y^3*u_xxy*u_xxyy -
36*u_x^3*u_y^2*u_xyy*u_xxyy + 36*u_x^4*u_y*u_yyy*u_xxyy - 68*u_x*u_y^3*u_xx^2*u_xyyy +
68*u_x^2*u_y^2*u_xx*u_xy*u_xyyy + 64*u_x^3*u_y*u_xy^2*u_xyyy +
4*u_x^3*u_y*u_xx*u_yy*u_xyyy - 68*u_x^4*u_xy*u_yy*u_xyyy - 36*u_x^2*u_y^3*u_xxx*u_xyyy
+ 60*u_x^3*u_y^2*u_xxy*u_xyyy - 12*u_x^4*u_y*u_xyy*u_xyyy - 12*u_x^5*u_yyy*u_xyyy +
34*u_x^2*u_y^2*u_xx^2*u_yyyy - 68*u_x^3*u_y*u_xx*u_xy*u_yyyy + 18*u_x^4*u_xy^2*u_yyyy
+ 16*u_x^4*u_xx*u_yy*u_yyyy + 12*u_x^3*u_y^2*u_xxx*u_yyyy - 24*u_x^4*u_y*u_xxy*u_yyyy
+ 12*u_x^5*u_xyy*u_yyyy + 16*u_y^5*u_xy*u_xxxxx - 16*u_x*u_y^4*u_yy*u_xxxxx -
16*u_y^5*u_xx*u_xxxxy - 48*u_x*u_y^4*u_xy*u_xxxxy + 64*u_x^2*u_y^3*u_yy*u_xxxxy +
64*u_x*u_y^4*u_xx*u_xxxyy + 32*u_x^2*u_y^3*u_xy*u_xxxyy - 96*u_x^3*u_y^2*u_yy*u_xxxyy
- 96*u_x^2*u_y^3*u_xx*u_xxyyy + 32*u_x^3*u_y^2*u_xy*u_xxyyy +
64*u_x^4*u_y*u_yy*u_xxyyy + 64*u_x^3*u_y^2*u_xx*u_xyyyy - 48*u_x^4*u_y*u_xy*u_xyyyy -
16*u_x^5*u_yy*u_xyyyy - 16*u_x^4*u_y*u_xx*u_yyyyy + 16*u_x^5*u_xy*u_yyyyy +
2*u_y^6*u_xxxxxx - 12*u_x*u_y^5*u_xxxxxy + 30*u_x^2*u_y^4*u_xxxxyy -
40*u_x^3*u_y^3*u_xxxyyy + 30*u_x^4*u_y^2*u_xxyyyy - 12*u_x^5*u_y*u_xyyyyy +
2*u_x^6*u_yyyyyy}
164 CHAPTER 6. THE GC ACTION ON 2D POISSON STRUCTURES
[69]: H_hepta2 = cH_hepta2[H]
[70]: diff(H_hepta2,y) == X_hepta2[0]
[70]: True
[71]: -diff(H_hepta2,x) == X_hepta2[1]
[71]: True
The Hamiltonian H of the trivializing vector field for the γ7 flow on R2 can be realized
as a sum of graphs:
[72]: ham7_1 = (2)*dfGC(DirectedGraph(7, [(0, 5), (0, 6), (1, 2), (1, 6), (2, 4), (2, 6),␣
↪→(3, 1), (3, 6), (4, 0), (4, 6), (5, 3), (5, 6)])) + (16)*dfGC(DirectedGraph(7, [(0,␣
↪→5), (0, 6), (1, 3), (1, 6), (2, 1), (2, 6), (3, 2), (3, 5), (4, 0), (4, 6), (5, 4),␣
↪→(5, 6)])) + (-12)*dfGC(DirectedGraph(7, [(0, 5), (0, 6), (1, 3), (1, 6), (2, 1),␣
↪→(2, 6), (3, 2), (3, 5), (4, 0), (4, 6), (5, 4), (6, 5)])) +␣
↪→(34)*dfGC(DirectedGraph(7, [(0, 5), (0, 6), (1, 4), (1, 6), (2, 1), (2, 6), (3, 0),␣
↪→(3, 4), (4, 2), (4, 5), (5, 3), (5, 6)])) + (16)*dfGC(DirectedGraph(7, [(0, 4), (0,␣
↪→6), (1, 4), (1, 5), (2, 0), (2, 6), (3, 1), (3, 5), (4, 3), (4, 6), (5, 2), (5,␣
↪→6)])) + (15)*dfGC(DirectedGraph(7, [(0, 5), (0, 6), (1, 4), (1, 5), (2, 1), (2, 4),␣
↪→(3, 0), (3, 6), (4, 3), (4, 6), (5, 2), (6, 5)])) + (-6)*dfGC(DirectedGraph(7, [(0,␣
↪→4), (0, 6), (1, 4), (1, 5), (2, 0), (2, 6), (3, 1), (3, 5), (4, 3), (4, 6), (5, 2),␣
↪→(6, 5)])) + (-9)*dfGC(DirectedGraph(7, [(0, 3), (0, 6), (1, 3), (1, 4), (2, 0), (2,␣
↪→6), (3, 5), (3, 6), (4, 2), (4, 5), (5, 1), (6, 4)])) + (13/
↪→4)*dfGC(DirectedGraph(7, [(0, 4), (0, 6), (1, 3), (1, 6), (2, 1), (2, 3), (3, 4),␣
↪→(3, 6), (4, 2), (4, 5), (5, 0), (6, 5)])) + (-199/4)*dfGC(DirectedGraph(7, [(0, 1),␣
↪→(0, 6), (1, 5), (1, 6), (2, 4), (3, 2), (4, 0), (4, 3), (5, 2), (5, 3), (6, 4), (6,␣
↪→5)])); ham7_1.show()
6.4. HEPTAGON-WHEEL γ7 FLOW 165
[73]: S2.graph_operation(ham7_1)(P2,P2,P2,P2,P2,P2,P2)[0,1] == H_hepta2
[73]: True
But we discover an alternative realization of the same Hamiltonian on R2 by using Kon-
tsevich’s directed graphs.
[74]: ham7_2 = (2)*dfGC(DirectedGraph(7, [(0, 5), (0, 6), (1, 3), (1, 6), (2, 1), (2, 6),␣
↪→(3, 2), (3, 6), (4, 0), (4, 6), (5, 4), (5, 6)])) + (16)*dfGC(DirectedGraph(7, [(0,␣
↪→5), (0, 6), (1, 3), (1, 6), (2, 1), (2, 6), (3, 2), (3, 5), (4, 0), (4, 6), (5, 4),␣
↪→(5, 6)])) + (-12)*dfGC(DirectedGraph(7, [(0, 5), (0, 6), (1, 3), (1, 6), (2, 1),␣
↪→(2, 6), (3, 2), (3, 5), (4, 0), (4, 6), (5, 4), (6, 5)])) +␣
↪→(34)*dfGC(DirectedGraph(7, [(0, 5), (0, 6), (1, 4), (1, 6), (2, 1), (2, 6), (3, 0),␣
↪→(3, 4), (4, 2), (4, 5), (5, 3), (5, 6)])) + (16)*dfGC(DirectedGraph(7, [(0, 4), (0,␣
↪→6), (1, 4), (1, 5), (2, 0), (2, 6), (3, 1), (3, 5), (4, 3), (4, 6), (5, 2), (5,␣
↪→6)])) + (15)*dfGC(DirectedGraph(7, [(0, 5), (0, 6), (1, 4), (1, 5), (2, 1), (2, 4),␣
↪→(3, 0), (3, 6), (4, 3), (4, 6), (5, 2), (6, 5)])) + (-6)*dfGC(DirectedGraph(7, [(0,␣
↪→4), (0, 6), (1, 4), (1, 5), (2, 0), (2, 6), (3, 1), (3, 5), (4, 3), (4, 6), (5, 2),␣
↪→(6, 5)])) + (-9)*dfGC(DirectedGraph(7, [(0, 3), (0, 6), (1, 3), (1, 4), (2, 0), (2,␣
↪→6), (3, 5), (3, 6), (4, 2), (4, 5), (5, 1), (6, 4)])) + (13/
↪→4)*dfGC(DirectedGraph(7, [(0, 4), (0, 6), (1, 3), (1, 6), (2, 1), (2, 3), (3, 4),␣
↪→(3, 6), (4, 2), (4, 5), (5, 0), (6, 5)])) + (-199/4)*dfGC(DirectedGraph(7, [(0, 1),␣
↪→(0, 6), (1, 5), (1, 6), (2, 4), (3, 2), (4, 0), (4, 3), (5, 2), (5, 3), (6, 4), (6,␣
↪→5)]))
166 CHAPTER 6. THE GC ACTION ON 2D POISSON STRUCTURES
[75]: S2.graph_operation(ham7_2)(P2,P2,P2,P2,P2,P2,P2)[0,1] == H_hepta2
[75]: True
Let us inspect the difference of two graph realizations for the Hamiltonian Hγ7 .
[76]: ham7_1 - ham7_2
[76]: (2)*DirectedGraph(7, [(0, 5), (0, 6), (1, 2), (1, 6), (2, 4), (2, 6), (3, 1), (3, 6),
(4, 0), (4, 6), (5, 3), (5, 6)]) + (-2)*DirectedGraph(7, [(0, 5), (0, 6), (1, 3), (1,
6), (2, 1), (2, 6), (3, 2), (3, 6), (4, 0), (4, 6), (5, 4), (5, 6)])
[77]: (ham7_1 - ham7_2).show()
Let us keep in mind that however that for generic Poisson structures all of these formulas
are valid only in dimension two: the trivializing vector field X, its Hamiltonian H, and
its graph realization.
6.5 Commutator [γ3, γ5] flow
[39]: bracket_cocycle = tetrahedron_cocycle.bracket(fivewheel_cocycle) #; bracket_cocycle.
↪→show()
[40]: len(bracket_cocycle)
[40]: 68
[ ]: %time bracket_cocycle_directed = dGC(bracket_cocycle)
[81]: len(bracket_cocycle_directed)
[81]: 2752512
[82]: %time bracket_cocycle_directed_filtered = bracket_cocycle_directed.
↪→filter(max_out_degree=2)
CPU times: user 4min 43s, sys: 14.4 s, total: 4min 58s
Wall time: 4min 58s
[83]: len(bracket_cocycle_directed_filtered)
6.5. COMMUTATOR [γ3, γ5] FLOW 167
[83]: 42252
Let us inspect the maximal in-degree of vertices in the directed graphs.
[84]: %time max(max(DiGraph(g.edges()).in_degree(k) for k in range(9)) for (c,g) in␣
↪→bracket_cocycle_directed_filtered)
CPU times: user 12.5 s, sys: 252 ms, total: 12.8 s
Wall time: 12.8 s
[84]: 7
Now, with sufficient computational power one could in principle continue, to calculate
the oriented graphs on two sinks for the Poisson flow:
[ ]: #FGC = FormalityGraphComplex(QQ, lazy=True)
[ ]: #Q_bracket = FGC(bracket_cocycle_directed_filtered).
↪→attach_to_ground((2,2,2,2,2,2,2,2,2))
[ ]: #len(Q_bracket)
One would evaluate it at P⊗92 :
[85]: #bracket_cocycle_operation2 = S2.graph_operation(bracket_cocycle_directed_filtered)
[86]: #%time Q_bracket2 = bracket_cocycle_operation2(P2,P2,P2,P2,P2,P2,P2,P2,P2)
And one would try to find a trivializing vector field:
[87]: #cX_bracket2 = solve_homogeneous_diffpoly(Q_bracket2[0,1], PbracketV[0,1], [V0, V1]);␣
↪→cX_bracket2
The trivializing vector field could then probably be expressed in terms of graphs, like
the others above. In fact, perhaps it would be more efficient to try to find the graphs
first. Finally, it would be interesting to compare [Or(γ3),Or(γ5)] with Or([γ3, γ5]), also
in terms of graphs.

Chapter 7
Graph complex action on rank two
rescaled Nambu–Poisson structures
We consider a class of Poisson brackets whose coefficients are differential polyno-
mial in the functional parameters ρ(x, y, z) and a(x, y, z) on R3 or ρ(x, y, z, w) and
a0(x, y, z, w, ), a1(x, y, z, w) on R4.
7.1 Tetrahedral flow on rescaled Nambu–Poisson
structures on R3  
ax ay az
By construction, {f, g} = ρ(x, y, z) · detfx fy f z , where the smooth function a is
gx gy gz
the global Casimir of the Poisson bracket.
We import the relevant functionality from the gcaops package:
[1]: from gcaops.graph.undirected_graph import UndirectedGraph
from gcaops.graph.undirected_graph_complex import UndirectedGraphComplex
from gcaops.graph.directed_graph_complex import DirectedGraphComplex
Define the graph complexes:
[2]: GC = UndirectedGraphComplex(QQ); GC
[2]: Undirected graph complex over Rational Field with Basis consisting of representatives
of isomorphism classes of undirected graphs with no automorphisms that induce an odd
permutation on edges
[3]: dGC = DirectedGraphComplex(QQ, implementation='vector'); dGC
[3]: Directed graph complex over Rational Field with Basis consisting of representatives of
isomorphism classes of directed graphs with no automorphisms that induce an odd
permutation on edges
Define the tetrahedron cocycle γ3:
[4]: tetrahedron_graph = UndirectedGraph(4, [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)])
tetrahedron = GC(tetrahedron_graph)
169
170 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
[5]: tetrahedron_oriented = dGC(tetrahedron); tetrahedron_oriented
[5]: (24)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]) +
(-8)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 3), (2, 1), (3, 2)]) +
(-24)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 2), (2, 3), (3, 1)]) +
(-8)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 3), (2, 1), (2, 3)])
[6]: tetrahedron_oriented_filtered = tetrahedron_oriented.filter(max_out_degree=2);␣
↪→tetrahedron_oriented_filtered
[6]: (-24)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 2), (2, 3), (3, 1)]) +
(-8)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 3), (2, 1), (2, 3)])
7.1.1 Superfunction algebra
Define the superfunction algebra:
[7]: from gcaops.algebra.differential_polynomial_ring import DifferentialPolynomialRing
[8]: D3 = DifferentialPolynomialRing(QQ, ('rho','a'), ('x','y','z'),␣
↪→max_differential_orders=[3+1,4+1]) #; D3
[9]: from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
[10]: S3.<xi0,xi1,xi2> = SuperfunctionAlgebra(D3, D3.base_variables()) #; S3
[11]: rho, a = D3.fibre_variables()
[12]: xi = S3.gens()
[13]: X = S3.even_coordinates()
Consider a rescaled Nambu–Poisson bi-vector field:
[14]: P = rho*sum(sigma.sign()*diff(a,X[sigma(1)-1])*xi[sigma(2)-1]*xi[sigma(3)-1] for␣
↪→sigma in Permutations(3))/2; P
[14]: (rho*a_x)*xi1*xi2 + (-rho*a_y)*xi0*xi2 + (rho*a_z)*xi0*xi1
The Jacobi identity is satisfied:
[15]: P.bracket(P)
[15]: 0
The bracket is a “derived” bracket P = [[ρξ1ξ2ξ3, a]]:
[16]: P == (rho*xi[0]*xi[1]*xi[2]).bracket(a)
[16]: True
7.1.2 Tetrahedral flow
First example by Kontsevich (1996, revised 2017). Define the tetrahedral flow:
7.1. THE γ3 FLOW ON 3D RESCALED NAMBU–POISSON STRUCTURES 171
[17]: # NOTE: filtered out graphs with out degree > 2, because we won't pass in any␣
↪→3-vectors
tetrahedron_operation3 = S3.graph_operation(tetrahedron_oriented_filtered);␣
↪→tetrahedron_operation3
[17]: Symmetric operation of arity 4 and degree -6 on Superfunction algebra over
Differential Polynomial Ring in x, y, z, rho, a, rho_x, rho_y, rho_z, rho_xx, rho_xy,
rho_xz, rho_yy, rho_yz, rho_zz, rho_xxx, rho_xxy, rho_xxz, rho_xyy, rho_xyz, rho_xzz,
rho_yyy, rho_yyz, rho_yzz, rho_zzz, rho_xxxx, rho_xxxy, rho_xxxz, rho_xxyy, rho_xxyz,
rho_xxzz, rho_xyyy, rho_xyyz, rho_xyzz, rho_xzzz, rho_yyyy, rho_yyyz, rho_yyzz,
rho_yzzz, rho_zzzz, a_x, a_y, a_z, a_xx, a_xy, a_xz, a_yy, a_yz, a_zz, a_xxx, a_xxy,
a_xxz, a_xyy, a_xyz, a_xzz, a_yyy, a_yyz, a_yzz, a_zzz, a_xxxx, a_xxxy, a_xxxz,
a_xxyy, a_xxyz, a_xxzz, a_xyyy, a_xyyz, a_xyzz, a_xzzz, a_yyyy, a_yyyz, a_yyzz,
a_yzzz, a_zzzz, a_xxxxx, a_xxxxy, a_xxxxz, a_xxxyy, a_xxxyz, a_xxxzz, a_xxyyy,
a_xxyyz, a_xxyzz, a_xxzzz, a_xyyyy, a_xyyyz, a_xyyzz, a_xyzzz, a_xzzzz, a_yyyyy,
a_yyyyz, a_yyyzz, a_yyzzz, a_yzzzz, a_zzzzz over Rational Field with even coordinates
(x, y, z) and odd coordinates (xi0, xi1, xi2)
[18]: %time Q_tetra3 = tetrahedron_operation3(P,P,P,P)
CPU times: user 297 ms, sys: 63 µs, total: 297 ms
Wall time: 296 ms
Both terms in the tetrahedral flow are generally nonzero:
[19]: #Q_tetra3
[20]: len(Q_tetra3[0,1].monomials()), len(Q_tetra3[0,2].monomials()), len(Q_tetra3[1,2].
↪→monomials())
[20]: (1504, 1504, 1504)
The tetrahedral flow indeed defines a Poisson 2-cocycle:
[21]: P.bracket(Q_tetra3)
[21]: 0
7.1.3 The induced flow
We have Q (P [ρ, a]⊗4tetra ) = P [ρ̇, a] + P [ρ, ȧ] for differential polynomials ρ̇ and ȧ.
In fact ȧ = 4 ·Qtetra(P, P, P, a).
[22]: %time adot = 4 * tetrahedron_operation3(P,P,P,a)
CPU times: user 95 ms, sys: 3.91 ms, total: 98.9 ms
Wall time: 97.9 ms
[23]: #adot
[24]: len(adot[tuple()].monomials())
172 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
[24]: 228
[25]: P0 = rho*sum(sigma.sign()*diff(adot,X[sigma(1)-1])*xi[sigma(2)-1]*xi[sigma(3)-1] for␣
↪→sigma in Permutations(3))/2
[26]: Q_remainder = Q_tetra3 - P0
[27]: len(Q_remainder[0,1].monomials()), len(Q_remainder[0,2].monomials()),␣
↪→len(Q_remainder[1,2].monomials())
[27]: (426, 426, 426)
[28]: P_withoutprefactor = sum(sigma.
↪→sign()*diff(a,X[sigma(1)-1])*xi[sigma(2)-1]*xi[sigma(3)-1] for sigma in␣
↪→Permutations(3))/2; P_withoutprefactor
[28]: (a_x)*xi1*xi2 + (-a_y)*xi0*xi2 + (a_z)*xi0*xi1
Get ρ̇ by (differential) polynomial division:
[29]: Q_remainder[0,1] % P_withoutprefactor[0,1] == 0
[29]: True
[30]: rhodot = Q_remainder[0,1] // P_withoutprefactor[0,1]
[31]: len(rhodot.monomials())
[31]: 426
[32]: rhodot * P_withoutprefactor == Q_remainder
[32]: True
[33]: Q_tetra3 == rhodot * P_withoutprefactor + P0
[33]: True
[34]: #rhodot
[35]: #%time sol = solve_homogeneous_diffpoly(D4('rhodot')*D4(P_withoutprefactor[0,1]),␣
↪→jQ_remainder01, [rhodot])
7.1.4 Symmetry
Sum over permutations:
[36]: adot_maybe = 0
from itertools import product
for sigma, tau, zeta in product(SymmetricGroup(3),repeat=3):
u1,v1,w1 = X[sigma(1)-1], X[sigma(2)-1], X[sigma(3)-1]
u2,v2,w2 = X[tau(1)-1], X[tau(2)-1], X[tau(3)-1]
u3,v3,w3 = X[zeta(1)-1], X[zeta(2)-1], X[zeta(3)-1]
adot_maybe += sigma.sign() * tau.sign() * zeta.sign() * (\
7.1. THE γ3 FLOW ON 3D RESCALED NAMBU–POISSON STRUCTURES 173
2*diff(a,u1) * diff(a,u2) * diff(a,u3) * diff(rho,w1) * diff(rho,w2) *␣
↪→diff(rho,w3) * diff(a,v1,v2,v3) + \
-6*rho * diff(a,u1,v2) * diff(a,u2) * diff(a,u3) * diff(rho,w1) *␣
↪→diff(rho,w3) * diff(a,v1,v3,w2) + \
-6*rho * rho*diff(a,u1) * diff(a,u2,u3) * diff(a,v1,v2) * diff(rho,w3) *␣
↪→diff(a,v3,w1,w2)
)
adot_maybe == adot/4
[36]: True
[37]: rhodot_maybe = 0
from itertools import product
for sigma, tau, zeta in product(SymmetricGroup(3),repeat=3):
u1,v1,w1 = X[sigma(1)-1], X[sigma(2)-1], X[sigma(3)-1]
u2,v2,w2 = X[tau(1)-1], X[tau(2)-1], X[tau(3)-1]
u3,v3,w3 = X[zeta(1)-1], X[zeta(2)-1], X[zeta(3)-1]
rhodot_maybe += sigma.sign() * tau.sign() * zeta.sign() * (\
-2*diff(a,u1) * diff(a,u2) * diff(a,u3) * diff(rho,v1) * diff(rho,v2) *␣
↪→diff(rho,v3) * diff(rho,w1,w2,w3) + \
6 * diff(a,u1,v2) * diff(a,u2) * diff(a,u3) * diff(rho,v1) * diff(rho,v3) *␣
↪→diff(rho,w2) * diff(rho,w1,w3) + \
-12*rho * diff(a,u1) * diff(a,u2,u3) * diff(a,v1,v2) * diff(rho,v3) *␣
↪→diff(rho,w1) * diff(rho,w2,w3) + \
-6*rho * diff(a,u1,v2) * diff(a,u2) * diff(a,u3) * diff(rho,v1) *␣
↪→diff(rho,v3) * diff(rho,w1,w2,w3) + \
6*rho * rho*diff(a,u1) * diff(a,u2,u3) * diff(a,v1,v2) * diff(rho,v3) *␣
↪→diff(rho,w1,w2,w3)
)
rhodot_maybe == rhodot/4
[37]: True
7.1.5 Differential polynomial triviality
The Poisson cohomology of R3 is generally non-trivial.
Is the tetrahedral flow non-trivial, for the rescaled Nambu–Poisson bracket P [ρ, a] on R3?
First, let V be an arbitrary vector field:
[38]: D3V = DifferentialPolynomialRing(QQ, ('rho','a','V0','V1','V2','H'), ('x','y','z'),␣
↪→max_differential_orders=[3+1,4+1,1,1,1,1]) #; D3V
[39]: V0, V1, V2, H = D3V.fibre_variables()[2:]
[40]: S3V = SuperfunctionAlgebra(D3V, D3V.base_variables(), names='xi0,xi1,xi2') #; S3V
[41]: V = V0*S3V(xi0) + V1*S3V(xi1) + V2*S3V(xi2); V
[41]: (V0)*xi0 + (V1)*xi1 + (V2)*xi2
Take the Poisson differential:
[42]: PbracketV = S3V(P).bracket(V); PbracketV
174 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
[42]: (V0*rho_x*a_y + V1*rho_y*a_y + V2*rho_z*a_y + rho*V0*a_xy + rho*V1*a_yy + rho*V2*a_yz
- rho*a_y*V0_x + rho*a_x*V0_y + rho*a_z*V2_y - rho*a_y*V2_z)*xi0*xi2 + (-V0*rho_x*a_x
- V1*rho_y*a_x - V2*rho_z*a_x - rho*V0*a_xx - rho*V1*a_xy - rho*V2*a_xz - rho*a_y*V1_x
+ rho*a_x*V1_y - rho*a_z*V2_x + rho*a_x*V2_z)*xi1*xi2 + (-V0*rho_x*a_z - V1*rho_y*a_z
- V2*rho_z*a_z - rho*V0*a_xz - rho*V1*a_yz - rho*V2*a_zz + rho*a_z*V0_x - rho*a_x*V0_z
+ rho*a_z*V1_y - rho*a_y*V1_z)*xi0*xi1
Solve Qtetra(P, P, P, P ) = [[P, V ]] for V with differential polynomial coefficients by using
homogeneity:
[43]: from gcaops.algebra.differential_polynomial_solver import solve_homogeneous_diffpoly
[44]: set_verbose(1)
[45]: %time sol = solve_homogeneous_diffpoly(S3V(Q_tetra3)[0,1], PbracketV[0,1], [V0,V1,V2])
verbose 1 (12: differential_polynomial_solver.py, solve_homogeneous_diffpoly) target
degrees: (4, 4, 0, 0, 0, 0)
verbose 1 (12: differential_polynomial_solver.py, solve_homogeneous_diffpoly) target
weights: (3, 3, 4)
verbose 1 (12: differential_polynomial_solver.py, solve_homogeneous_diffpoly) ansatz
degrees: {V1: {(3, 3, 0, 0, 0, 0)}, V0: {(3, 3, 0, 0, 0, 0)}, V2: {(3, 3, 0, 0, 0,
0)}}
verbose 1 (12: differential_polynomial_solver.py, solve_homogeneous_diffpoly) ansatz
weights: {V1: {(3, 2, 3)}, V0: {(2, 3, 3)}, V2: {(3, 3, 2)}}
verbose 1 (12: differential_polynomial_solver.py, solve_homogeneous_diffpoly) ansatz
#monomials: {V0: 2843, V1: 2843, V2: 2843}
verbose 1 (12: differential_polynomial_solver.py, solve_homogeneous_diffpoly)
len(target_basis) == 17085
verbose 1 (12: differential_polynomial_solver.py, solve_homogeneous_diffpoly)
len(ansatz_basis) == 7477
verbose 1 (12: differential_polynomial_solver.py, multimod echelon) Multimodular
echelon algorithm on 17085 x 7477 matrix
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, multimod echelon) Multimodular
echelon algorithm on 7332 x 17085 matrix
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, multimod echelon) Multimodular
echelon algorithm on 7332 x 7332 matrix
7.1. THE γ3 FLOW ON 3D RESCALED NAMBU–POISSON STRUCTURES 175
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, multimod echelon) done: the echelon
form mod p is the identity matrix and possibly some 0 rows
verbose 1 (12: differential_polynomial_solver.py, multimod echelon) Multimodular
echelon algorithm on 7332 x 7333 matrix
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
verbose 1 (12: differential_polynomial_solver.py, sparse_matrix_pyx matrix_modint
echelon)
CPU times: user 25min 29s, sys: 1.93 s, total: 25min 30s
Wall time: 25min 29s
[46]: V_tetra3 = D3(sol[V0])*xi0 + D3(sol[V1])*xi1 + D3(sol[V2])*xi2 #; V_tetra3
[47]: tuple(len(V_tetra3[i].monomials()) for i in range(3))
[47]: (352, 347, 339)
[48]: P.bracket(V_tetra3) == Q_tetra3
[48]: True
So, the tetrahedral flow of a rescaled Nambu–Poisson structure on R3 is Poisson-trivial!
Moreover, by running the script several times one can get many different such trivializ-
ing vector fields — evidently, they are defined modulo Poisson coboundaries [[P,H]] for
arbitrary Hamiltonian functions H (in particular homogeneous differential polynomial
ones).
7.1.6 Total skew-symmetry of the trivializing vector field
The trivializing vector field is expressed as a sum (of 11 terms) over permutations with
signs plus a Po∑isson exact-term given by a Hamiltonian function:
V = (−)σ(−)τ (−)ζ (σ ⊗ τ ⊗ ζ)(11 terms in vector field) + [[P,Hγ3 ]].
σ,τ,ζ∈S3
Each of the permutations σ, τ, ζ acts on its own triple of base variables, σ : (∑x, y, z) →7
(σ(x), σ(y), σ(z)) etc., thus reproducing three Civita symbols and three sums ~εi~i ∂i1 ⊗
∂i2 ⊗ ∂i3 .
[49]: V_skew = 0
from itertools import product
for sigma, tau, zeta in product(SymmetricGroup(3),repeat=3):
i_0,i_1,i_2 = sigma(1)-1, sigma(2)-1, sigma(3)-1
i_3,i_4,i_5 = tau(1)-1, tau(2)-1, tau(3)-1
i_6,i_7,i_8 = zeta(1)-1, zeta(2)-1, zeta(3)-1
V_skew += sigma.sign() * tau.sign() * zeta.sign() * (\
176 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
12 * diff(rho, X[i_0], X[i_3]) * diff(rho, X[i_7]) * rho * diff(a, X[i_1],␣
↪→X[i_4]) * diff(a, X[i_2], X[i_5]) * diff(a, X[i_8]) * xi[i_6] + \
48 * diff(rho, X[i_0], X[i_3]) * diff(rho, X[i_5]) * rho * diff(a, X[i_1],␣
↪→X[i_4]) * diff(a, X[i_2], X[i_6]) * diff(a, X[i_8]) * xi[i_7] + \
8 * diff(rho, X[i_0], X[i_6]) * diff(rho, X[i_1], X[i_7]) * diff(rho, X[i_4])␣
↪→* diff(a, X[i_2]) * diff(a, X[i_5]) * diff(a, X[i_8]) * xi[i_3] + \
-40 * diff(rho, X[i_0], X[i_6]) * diff(rho, X[i_2]) * diff(rho, X[i_4]) *␣
↪→diff(a, X[i_1], X[i_7]) * diff(a, X[i_5]) * diff(a, X[i_8]) * xi[i_3] + \
8 * diff(rho, X[i_2]) * diff(rho, X[i_4]) * diff(rho, X[i_8]) * diff(a,␣
↪→X[i_0], X[i_6]) * diff(a, X[i_1], X[i_7]) * diff(a, X[i_5]) * xi[i_3] + \
24 * diff(rho, X[i_0], X[i_6]) * diff(rho, X[i_4]) * diff(rho, X[i_8]) *␣
↪→diff(a, X[i_2]) * diff(a, X[i_3], X[i_7]) * diff(a, X[i_5]) * xi[i_1] + \
-12 * diff(rho, X[i_7]) * rho * rho * diff(a, X[i_0], X[i_3]) * diff(a,␣
↪→X[i_1], X[i_4]) * diff(a, X[i_2], X[i_5], X[i_8]) * xi[i_6] + \
24 * diff(rho, X[i_4]) * diff(rho, X[i_6]) * rho * diff(a, X[i_0], X[i_3]) *␣
↪→diff(a, X[i_7]) * diff(a, X[i_2], X[i_5], X[i_8]) * xi[i_1] + \
-36 * diff(rho, X[i_1]) * diff(rho, X[i_4]) * rho * diff(a, X[i_0], X[i_3]) *␣
↪→diff(a, X[i_7]) * diff(a, X[i_2], X[i_5], X[i_8]) * xi[i_6] + \
8 * diff(rho, X[i_1]) * diff(rho, X[i_3]) * diff(rho, X[i_6]) * diff(a,␣
↪→X[i_4]) * diff(a, X[i_7]) * diff(a, X[i_2], X[i_5], X[i_8]) * xi[i_0] + \
-8 * diff(rho, X[i_3]) * diff(rho, X[i_6]) * diff(rho, X[i_2], X[i_5],␣
↪→X[i_8]) * diff(a, X[i_1]) * diff(a, X[i_4]) * diff(a, X[i_7]) * xi[i_0]
)
#V_skew
[50]: tuple(len(V_skew[i].monomials()) for i in range(3))
[50]: (324, 324, 324)
[51]: Q_tetra3 == P.bracket(V_skew)
[51]: True
Here is an example of the Hamiltonian Hγ3 (consisting of 20 monomials) which was
obtained for the trivializing vector field V from long ago: V − Vskew = [[P,Hγ3 ]]:
[52]: #H_tetra3 = D3('12*rho_z^2*a_xy^2 - 24*rho*rho_zz*a_xy^2 - 48*rho_y*rho_xz*a_y*a_xz -␣
↪→48*rho_y*rho_z*a_xy*a_xz + 24*rho_y^2*a_xz^2 - 12*rho_z^2*a_xx*a_yy +␣
↪→48*rho_x*rho_z*a_xz*a_yy + 48*rho_y*rho_z*a_xx*a_yz - 48*rho_x*rho_z*a_xy*a_yz -␣
↪→48*rho_x*rho_y*a_xz*a_yz + 36*rho_x^2*a_yz^2 - 24*rho_y^2*a_xx*a_zz +␣
↪→24*rho*rho_yy*a_xx*a_zz + 48*rho_x*rho_y*a_xy*a_zz - 24*rho_x^2*a_yy*a_zz -␣
↪→12*rho*rho_y*a_zz*a_xxy + 24*rho*rho_yz*a_y*a_xxz - 24*rho*rho_z*a_yy*a_xxz -␣
↪→24*rho_y^2*a_x*a_xzz - 24*rho*rho_z*a_xx*a_yyz')
But as the representative of [V mod [[P, ·]]] typically changes every time the solver is
invoked, we shall find another Hγ3 in the next step, for the representative V which we
currently have.
[53]: set_verbose(0)
[97]: H_tetra3 = solve_homogeneous_diffpoly(S3V(V_tetra3 - V_skew)[0], S3V(P).
↪→bracket(H)[0], [H])[H]
[98]: H_tetra3
7.2. THE γ3 FLOW ON 4D NAMBU–POISSON STRUCTURES 177
[98]: 24*rho_z^2*a_xy^2 - 96*rho_y*rho_xy*a_z*a_xz - 48*rho_y*rho_z*a_xy*a_xz +
48*rho*rho_yz*a_xy*a_xz + 12*rho_y^2*a_xz^2 - 24*rho_z^2*a_xx*a_yy +
48*rho_y*rho_z*a_xx*a_yz - 48*rho_x*rho_z*a_xy*a_yz + 48*rho*rho_xz*a_xy*a_yz +
96*rho_x*rho_y*a_xz*a_yz + 24*rho_x^2*a_yz^2 - 12*rho_y^2*a_xx*a_zz +
48*rho_x*rho_y*a_xy*a_zz - 12*rho_x^2*a_yy*a_zz + 12*rho_z^2*a_y*a_xxy +
48*rho_y^2*a_z*a_xxz + 24*rho*rho_z*a_yy*a_xxz + 12*rho_z^2*a_x*a_xyy +
48*rho_x*rho_z*a_z*a_xyy - 48*rho*rho_xz*a_z*a_xyy + 24*rho_x^2*a_z*a_yyz -
24*rho*rho_z*a_xx*a_yyz
[99]: len(H_tetra3.monomials())
[99]: 22
In contrast with the previous run Qtetra → V → H (see the #H_tetra3 = ... cell above),
this new Hamiltonian –for the new representative V – of trivializing vector field contains
only 22 monomials.
[57]: V_tetra3 == V_skew + S3V(P).bracket(H_tetra3)
[57]: True
Our crucial remark is that the marker monomials for the velocity ȧ of the Casimir,
now induced by ȧ = −V (a) with a totally skew-symmetric Vskew-part, cannot change for
different representatives of [V mod [[P, ·]]].
[58]: -V_skew.bracket(a) == adot
[58]: True
[59]: -V_tetra3.bracket(a) == adot
[59]: True
We see that the expression −Vskew(a) equals ȧ.
Moreover, ρ̇ξ1ξ2ξ3 = −[[Vskew, ρξ1ξ2ξ3]]:
[60]: -V_skew.bracket(rho*xi[0]*xi[1]*xi[2])[0,1,2] == rhodot
[60]: True
7.2 Tetrahedral flow on Nambu–Poisson structures
on R4
We have ( )
{ } det ∂(a0, a1, f, g)f, g = .
∂(x, y, z, w)
We import the relevant functionality from the gcaops package:
[1]: from gcaops.graph.undirected_graph import UndirectedGraph
from gcaops.graph.undirected_graph_complex import UndirectedGraphComplex
178 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
[2]: GC = UndirectedGraphComplex(QQ); GC
[2]: Undirected graph complex over Rational Field with Basis consisting of representatives
of isomorphism classes of undirected graphs with no automorphisms that induce an odd
permutation on edges
Define the tetrahedron cocycle γ3:
[3]: tetrahedron_graph = UndirectedGraph(4, [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)]);␣
↪→tetrahedron_graph
[3]: UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
[4]: tetrahedron = GC(tetrahedron_graph); tetrahedron
[4]: 1*UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
[5]: from gcaops.graph.directed_graph_complex import DirectedGraphComplex
[6]: dGC = DirectedGraphComplex(QQ, implementation='vector')
[7]: tetrahedron_oriented = dGC(tetrahedron); tetrahedron_oriented
[7]: (24)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]) +
(-8)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 3), (2, 1), (3, 2)]) +
(-24)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 2), (2, 3), (3, 1)]) +
(-8)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 3), (2, 1), (2, 3)])
[8]: tetrahedron_oriented_filtered = tetrahedron_oriented.filter(max_out_degree=2);␣
↪→tetrahedron_oriented_filtered
[8]: (-24)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 2), (2, 3), (3, 1)]) +
(-8)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 3), (2, 1), (2, 3)])
The differential polynomial ring with fibre variables a0, a1 and base variables x, y, z, w:
[9]: from gcaops.algebra.differential_polynomial_ring import DifferentialPolynomialRing
[10]: D4 = DifferentialPolynomialRing(QQ, ('a0','a1'), ('x','y','z','w'),␣
↪→max_differential_orders=[4+1,4+1]) #; D4
[13]: a0, a1 = D4.fibre_variables()
[14]: a = [a0,a1]
The superfunction algebra:
[11]: from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
[12]: S4.<xi0,xi1,xi2,xi3> = SuperfunctionAlgebra(D4, D4.base_variables()) #; S4
[15]: xi = S4.gens()
[16]: X = S4.even_coordinates()
The Nambu–Poisson structure:
7.2. THE γ3 FLOW ON 4D NAMBU–POISSON STRUCTURES 179
[17]: P = sum(sigma.sign()*diff(a[0],X[sigma(1)-1])*diff(a[1],␣
↪→X[sigma(2)-1])*xi[sigma(3)-1]*xi[sigma(4)-1] for sigma in Permutations(4))/2; P
[17]: (-a0_y*a1_x + a0_x*a1_y)*xi2*xi3 + (a0_z*a1_x - a0_x*a1_z)*xi1*xi3 + (-a0_w*a1_x +
a0_x*a1_w)*xi1*xi2 + (-a0_z*a1_y + a0_y*a1_z)*xi0*xi3 + (a0_w*a1_y -
a0_y*a1_w)*xi0*xi2 + (-a0_w*a1_z + a0_z*a1_w)*xi0*xi1
In fact the Poisson bracket is a “derived” bracket P = −[[[[∂x ∧ ∂y ∧ ∂z ∧ ∂w, a0]], a1]]:
[18]: P == -(xi0*xi1*xi2*xi3).bracket(a0).bracket(a1)
[18]: True
[19]: P.bracket(P)
[19]: 0
[20]: # NOTE: filtered out graphs with out degree > 2, because we won't pass in any␣
↪→3-vectors
tetrahedron_operation4 = S4.graph_operation(tetrahedron_oriented_filtered) #;␣
↪→tetrahedron_operation4
The evolutions ȧ0 and ȧ1 of a0 and a1:
[21]: %time adot = [4*tetrahedron_operation4(P,P,P,a[i]) for i in range(2)]
CPU times: user 12.8 s, sys: 58 ms, total: 12.9 s
Wall time: 12.9 s
[22]: #adot[0]
[23]: #adot[1]
[24]: len(adot[0][tuple()].monomials()), len(adot[1][tuple()].monomials())
[24]: (9024, 9024)
The flow Qtetra(P [a0, a1]):
[25]: %time Q_tetra4 = tetrahedron_operation4(P,P,P,P)
CPU times: user 7min 45s, sys: 1min 20s, total: 9min 6s
Wall time: 9min 6s
[26]: [len(Q_tetra4[i,j].monomials()) for i,j in [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)]]
[26]: [161040, 161040, 161040, 161040, 161040, 161040]
We have the equality Qtetra(P [a0, a1]) = P [ȧ0, a1] + P [a0, ȧ1]:
[27]: P0 = sum(sigma.sign()*diff(adot[0],X[sigma(1)-1])*diff(a[1],␣
↪→X[sigma(2)-1])*xi[sigma(3)-1]*xi[sigma(4)-1] for sigma in Permutations(4))/2
180 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
[28]: P1 = sum(sigma.sign()*diff(a[0],X[sigma(1)-1])*diff(adot[1],␣
↪→X[sigma(2)-1])*xi[sigma(3)-1]*xi[sigma(4)-1] for sigma in Permutations(4))/2
[29]: Q_remainder = Q_tetra4 - P0 - P1; Q_remainder
[29]: 0
[30]: Q_tetra4 == P0 + P1
[30]: True
7.2.1 Symmetry
The evolutions ȧ0 and ȧ1 are induced by the tetrahedral flow Q ⊗4γ3(P ) on the class
P [a0, a1] of Nambu–Poisson brackets on R4 with the pre-factor ρ = 1. We represent
either velocity by using three Civita symbols (with four indices each).
[31]: a0dot_maybe = 0
from itertools import product
for sigma, tau, zeta in product(SymmetricGroup(4),repeat=3):
s1,t1,u1,v1 = X[sigma(1)-1], X[sigma(2)-1], X[sigma(3)-1], X[sigma(4)-1]
s2,t2,u2,v2 = X[tau(1)-1], X[tau(2)-1], X[tau(3)-1], X[tau(4)-1]
s3,t3,u3,v3 = X[zeta(1)-1], X[zeta(2)-1], X[zeta(3)-1], X[zeta(4)-1]
a0dot_maybe += sigma.sign() * tau.sign() * zeta.sign() * (\
3*diff(a0,s1,u2,u3) * diff(a0,t1,t2) * diff(a1,s2) * diff(a1,s3,v1) *␣
↪→diff(a1,t3,u1) * diff(a0,v2) * diff(a0,v3) + \
+6*diff(a0,s1,u2) * diff(a0,t1) * diff(a0,t2,v3) * diff(a0,u3,v1,v2) *␣
↪→diff(a1,t3,u1) * diff(a1,s2) * diff(a1,s3)
)
a0dot_maybe == adot[0]/4
[31]: True
[32]: a1dot_maybe = 0
from itertools import product
for sigma, tau, zeta in product(SymmetricGroup(4),repeat=3):
s1,t1,u1,v1 = X[sigma(1)-1], X[sigma(2)-1], X[sigma(3)-1], X[sigma(4)-1]
s2,t2,u2,v2 = X[tau(1)-1], X[tau(2)-1], X[tau(3)-1], X[tau(4)-1]
s3,t3,u3,v3 = X[zeta(1)-1], X[zeta(2)-1], X[zeta(3)-1], X[zeta(4)-1]
a1dot_maybe += sigma.sign() * tau.sign() * zeta.sign() * (\
3*diff(a0,s1) * diff(a1,t1,u2) * diff(a1,u1,u3,v2) * diff(a0,s2,t3) *␣
↪→diff(a0,s3,t2) * diff(a1,v1) * diff(a1,v3) + \
-3*diff(a0,s1,t2) * diff(a1,u1) * diff(a1,u2,u3,v1) * diff(a0,t1) *␣
↪→diff(a0,t3) * diff(a1,s2,v3) * diff(a1,s3,v2)
)
a1dot_maybe == adot[1]/4
[32]: True
7.2.2 Differential polynomial (non)triviality
One could try to solve the equation Qtetra(P [a0, a1]) = [[P, V [a0, a1]]] for a vector field
V [a0, a1] as follows:
7.3. THE γ3 FLOW ON 4D RESCALED NAMBU–POISSON STRUCTURES 181
[33]: D4V = DifferentialPolynomialRing(QQ, ('a0','a1', 'V0','V1','V2','V3'),␣
↪→('x','y','z','w'), max_differential_orders=[4+1,4+1,1,1,1,1]) #; D4V
[34]: V0, V1, V2, V3 = D4V.fibre_variables()[2:]
[35]: S4V = SuperfunctionAlgebra(D4V, D4V.base_variables(), names='xi0,xi1,xi2,xi3') #; S4V
[36]: V = V0*S4V(xi0) + V1*S4V(xi1) + V2*S4V(xi2) + V3*S4V(xi3); V
[36]: (V0)*xi0 + (V1)*xi1 + (V2)*xi2 + (V3)*xi3
[37]: PbracketV = S4V(P).bracket(V)
[38]: from gcaops.algebra.differential_polynomial_solver import solve_homogeneous_diffpoly
[39]: set_verbose(1)
[ ]: %time sol = solve_homogeneous_diffpoly(S4V(Q_tetra4)[0,1], PbracketV[0,1],␣
↪→[V0,V1,V2,V3])
So far, the (non)triviality of the flow Qtetra(P [a0, a1]) remains an open problem.
7.3 Tetrahedral flow on rescaled Nambu–Poisson
structures on R4
We have ( )
{ } ∂(a0, a1, f, g)f, g = ρ(x, y, z, w) · det .
∂(x, y, z, w)
We import the relevant functionality from the gcaops package:
[1]: from gcaops.graph.undirected_graph import UndirectedGraph
from gcaops.graph.undirected_graph_complex import UndirectedGraphComplex
[2]: GC = UndirectedGraphComplex(QQ); GC
[2]: Undirected graph complex over Rational Field with Basis consisting of representatives
of isomorphism classes of undirected graphs with no automorphisms that induce an odd
permutation on edges
Define the tetrahedron cocycle γ3:
[3]: tetrahedron_graph = UndirectedGraph(4, [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)]);␣
↪→tetrahedron_graph
[3]: UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
[4]: tetrahedron = GC(tetrahedron_graph); tetrahedron
[4]: 1*UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
[5]: from gcaops.graph.directed_graph_complex import DirectedGraphComplex
182 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
[6]: dGC = DirectedGraphComplex(QQ, implementation='vector')
[7]: tetrahedron_oriented = dGC(tetrahedron); tetrahedron_oriented
[7]: (24)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]) +
(-8)*DirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 3), (2, 1), (3, 2)]) +
(-24)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 2), (2, 3), (3, 1)]) +
(-8)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 3), (2, 1), (2, 3)])
[8]: tetrahedron_oriented_filtered = tetrahedron_oriented.filter(max_out_degree=2);␣
↪→tetrahedron_oriented_filtered
[8]: (-24)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 2), (2, 3), (3, 1)]) +
(-8)*DirectedGraph(4, [(0, 2), (0, 3), (1, 0), (1, 3), (2, 1), (2, 3)])
The differential polynomial ring with fibre variables ρ, a0, a1 and base variables x, y, z, w:
[9]: from gcaops.algebra.differential_polynomial_ring import DifferentialPolynomialRing
[10]: D4 = DifferentialPolynomialRing(QQ, ('rho','a0','a1'), ('x','y','z','w'),␣
↪→max_differential_orders=[3+1,4+1,4+1]) #; D4
[13]: rho, a0, a1 = D4.fibre_variables()
[14]: a = [a0,a1]
The superfunction algebra:
[11]: from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
[12]: S4.<xi0,xi1,xi2,xi3> = SuperfunctionAlgebra(D4, D4.base_variables()) #; S4
[15]: xi = S4.gens()
[16]: X = S4.even_coordinates()
The Nambu–Poisson structure:
[17]: P = rho*sum(sigma.sign()*diff(a[0],X[sigma(1)-1])*diff(a[1],␣
↪→X[sigma(2)-1])*xi[sigma(3)-1]*xi[sigma(4)-1] for sigma in Permutations(4))/2; P
[17]: (-rho*a0_y*a1_x + rho*a0_x*a1_y)*xi2*xi3 + (rho*a0_z*a1_x - rho*a0_x*a1_z)*xi1*xi3 +
(-rho*a0_w*a1_x + rho*a0_x*a1_w)*xi1*xi2 + (-rho*a0_z*a1_y + rho*a0_y*a1_z)*xi0*xi3 +
(rho*a0_w*a1_y - rho*a0_y*a1_w)*xi0*xi2 + (-rho*a0_w*a1_z + rho*a0_z*a1_w)*xi0*xi1
In fact the Poisson bracket is a “derived” bracket: P = −[[[[ρ ∂x ∧ ∂y ∧ ∂z ∧ ∂w]], a0]], a1]]:
[18]: P == -(rho*xi0*xi1*xi2*xi3).bracket(a0).bracket(a1)
[18]: True
[19]: P.bracket(P)
[19]: 0
7.3. THE γ3 FLOW ON 4D RESCALED NAMBU–POISSON STRUCTURES 183
[20]: # NOTE: filtered out graphs with out degree > 2, because we won't pass in any␣
↪→3-vectors
tetrahedron_operation4 = S4.graph_operation(tetrahedron_oriented_filtered) #;␣
↪→tetrahedron_operation4
The evolutions ȧ0 and ȧ1 of a0 and a1:
[21]: %time adot = [4*tetrahedron_operation4(P,P,P,a[i]) for i in range(2)]
CPU times: user 57.3 s, sys: 71.3 ms, total: 57.4 s
Wall time: 57.4 s
[22]: #adot[0]
[23]: #len(adot[0].monomials())
[24]: #adot[1]
[25]: #len(adot[1].monomials())
The tetrahedral flow Qtetra(P [ρ, a0, a1]):
[26]: %time Q_tetra4 = tetrahedron_operation4(P,P,P,P)
CPU times: user 1h 23min 10s, sys: 45min 6s, total: 2h 8min 17s
Wall time: 2h 8min 16s
[27]: len(str(Q_tetra4))
[27]: 318982343
We verify the equality Qtetra(P [ρ, a0, a1]) = P [ρ̇, a0, a1] + P [ρ, ȧ0, a1] + P [ρ, a0, ȧ1]:
[28]: P0 = rho*sum(sigma.sign()*diff(adot[0],X[sigma(1)-1])*diff(a[1],␣
↪→X[sigma(2)-1])*xi[sigma(3)-1]*xi[sigma(4)-1] for sigma in Permutations(4))/2
[29]: P1 = rho*sum(sigma.sign()*diff(a[0],X[sigma(1)-1])*diff(adot[1],␣
↪→X[sigma(2)-1])*xi[sigma(3)-1]*xi[sigma(4)-1] for sigma in Permutations(4))/2
[30]: Q_remainder = Q_tetra4 - P0 - P1
[31]: len(str(Q_remainder))
[31]: 62481479
[32]: P_withoutprefactor = sum(sigma.sign()*diff(a[0],X[sigma(1)-1])*diff(a[1],␣
↪→X[sigma(2)-1])*xi[sigma(3)-1]*xi[sigma(4)-1] for sigma in Permutations(4))/2;␣
↪→P_withoutprefactor
[32]: (-a0_w*a1_z + a0_z*a1_w)*xi0*xi1 + (a0_w*a1_y - a0_y*a1_w)*xi0*xi2 + (-a0_z*a1_y +
a0_y*a1_z)*xi0*xi3 + (-a0_w*a1_x + a0_x*a1_w)*xi1*xi2 + (a0_z*a1_x -
a0_x*a1_z)*xi1*xi3 + (-a0_y*a1_x + a0_x*a1_y)*xi2*xi3
184 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
[33]: Q_remainder01 = Q_remainder[0,1]
[34]: Q_remainder01 % P_withoutprefactor[0,1] == 0
[34]: True
[35]: rhodot = Q_remainder01 // P_withoutprefactor[0,1] #; rhodot
[40]: len(rhodot.monomials())
[40]: 90024
[37]: rhodot * P_withoutprefactor == Q_remainder
[37]: True
[38]: Q_tetra4 == rhodot*P_withoutprefactor + P0 + P1
[38]: True
[39]: from itertools import combinations
for (i,j) in combinations(range(4),2):
print((i,j),
Q_remainder[i,j] % P_withoutprefactor[i,j] == 0,
Q_remainder[i,j] // P_withoutprefactor[i,j] == rhodot)
(0, 1) True True
(0, 2) True True
(0, 3) True True
(1, 2) True True
(1, 3) True True
(2, 3) True True
7.3.1 Symmetry
Now the evolutions ȧ0 and ȧ1 are induced by the tetrahedral flow Q ⊗4γ3(P ) on the class
P [ρ, a0, a1] of generalized Nambu–Poisson brackets on R4 with generic inverse density
ρ(x, y, z, w). We collapse the known formulas of ȧ0 and ȧ1 by using three Civita symbols
(with four indices each). This expression is joint work with D. Lipper (2021). We note
that by setting ρ = 1 one recovers the formulas which were found earlier in that special
case.
[41]: a0dot_maybe = 0
from itertools import product
for sigma, tau, zeta in product(SymmetricGroup(4),repeat=3):
s1,t1,u1,v1 = X[sigma(1)-1], X[sigma(2)-1], X[sigma(3)-1], X[sigma(4)-1]
s2,t2,u2,v2 = X[tau(1)-1], X[tau(2)-1], X[tau(3)-1], X[tau(4)-1]
s3,t3,u3,v3 = X[zeta(1)-1], X[zeta(2)-1], X[zeta(3)-1], X[zeta(4)-1]
a0dot_maybe += sigma.sign() * tau.sign() * zeta.sign() * (\
3 * diff(a0,s1,u2,u3) * diff(a0,t1,t2) * diff(a1,s2) * diff(a1,s3,v1) *␣
↪→diff(a1,t3,u1) * diff(a0,v2) * diff(a0,v3) * rho^3 + \
+6 * diff(a0,s1,u2) * diff(a0,t1) * diff(a0,t2,v3) * diff(a0,u3,v1,v2) *␣
↪→diff(a1,t3,u1) * diff(a1,s2) * diff(a1,s3) * rho^3 + \
7.3. THE γ3 FLOW ON 4D RESCALED NAMBU–POISSON STRUCTURES 185
+3 * diff(a0,v2) * diff(a0,t1,u2,v3) * diff(a1,v1) * diff(rho,s1) *␣
↪→diff(a0,u1) * diff(a0,u3) * diff(a1,s2,t3) * diff(a1,s3,t2) * rho^2 + \
-6 * diff(a0,s1,v3) * diff(a1,s2,t1) * diff(rho,v1) * diff(a0,s3,u1,v2) *␣
↪→diff(a0,u2) * diff(a0,u3) * diff(a1,t2) * diff(a1,t3) * rho^2 + \
-6 * diff(a0,s1,v2,v3) * diff(a0,t1) * diff(a0,u2) * diff(a0,u3,v1) *␣
↪→diff(a1,s2) * diff(a1,s3,u1) * diff(a1,t3) * diff(rho,t2) * rho^2 + \
+6 * diff(a0,s1) * diff(a0,s2,v3) * diff(a0,s3,u1) * diff(a0,t1,t2,t3) *␣
↪→diff(a1,v1) * diff(rho,v2) * diff(a1,u2) * diff(a1,u3) * rho^2 + \
-6 * diff(a0,s1) * diff(a0,t1,u2,u3) * diff(a1,u1,v2) * diff(a0,t2) *␣
↪→diff(a0,t3) * diff(a1,s2) * diff(a1,s3) * diff(rho,v1) * diff(rho,v3) * rho + \
+6 * diff(a0,v2) * diff(a0,s1) * diff(a0,s2,s3,t1) * diff(a0,t2,t3) *␣
↪→diff(rho,v1) * diff(rho,v3) * diff(a1,u1) * diff(a1,u2) * diff(a1,u3) * rho + \
-2 * diff(a0,s1) * diff(a0,t2) * diff(a0,t3,u1,u2) * diff(a0,u3) *␣
↪→diff(a1,t1) * diff(a1,s2) * diff(a1,s3) * diff(rho,v1) * diff(rho,v2) * diff(rho,v3)
)
a0dot_maybe == adot[0]/4
[41]: True
[42]: a1dot_maybe = 0
from itertools import product
for sigma, tau, zeta in product(SymmetricGroup(4),repeat=3):
s1,t1,u1,v1 = X[sigma(1)-1], X[sigma(2)-1], X[sigma(3)-1], X[sigma(4)-1]
s2,t2,u2,v2 = X[tau(1)-1], X[tau(2)-1], X[tau(3)-1], X[tau(4)-1]
s3,t3,u3,v3 = X[zeta(1)-1], X[zeta(2)-1], X[zeta(3)-1], X[zeta(4)-1]
a1dot_maybe += sigma.sign() * tau.sign() * zeta.sign() * (\
+3 * diff(a0,s1) * diff(a1,t1,u2) * diff(a1,u1,u3,v2) * diff(a0,s2,t3) *␣
↪→diff(a0,s3,t2) * diff(a1,v1) * diff(a1,v3) * rho^3 + \
-3 * diff(a0,s1,t2) * diff(a1,u1) * diff(a1,u2,u3,v1) * diff(a0,t1) *␣
↪→diff(a0,t3) * diff(a1,s2,v3) * diff(a1,s3,v2) * rho^3 + \
-6 * diff(a0,u1) * diff(a1,t1,t2,v3) * diff(rho,t3) * diff(a0,u2,v1) *␣
↪→diff(a0,u3,v2) * diff(a1,s1) * diff(a1,s2) * diff(a1,s3) * rho^2 + \
+6 * diff(a0,s1) * diff(a0,t1,t3) * diff(a0,u2) * diff(a1,v2) *␣
↪→diff(a1,s2,v1,v3) * diff(a1,s3,t2) * diff(a1,u1) * diff(rho,u3) * rho^2 + \
+6 * diff(a0,t1,u2) * diff(a1,u1,v2) * diff(rho,u3) * diff(a1,s1,s2,s3) *␣
↪→diff(a0,v1) * diff(a0,v3) * diff(a1,t2) * diff(a1,t3) * rho^2 + \
+3 * diff(a1,v1) * diff(a1,s1,s2,s3) * diff(rho,t1) * diff(a1,t2,v3) *␣
↪→diff(a1,t3,v2) * diff(a0,u1) * diff(a0,u2) * diff(a0,u3) * rho^2 + \
-6 * diff(a0,t1,u2) * diff(a1,u1,u3,v2) * diff(a0,v1) * diff(a0,v3) *␣
↪→diff(rho,t2) * diff(rho,t3) * diff(a1,s1) * diff(a1,s2) * diff(a1,s3) * rho + \
-6 * diff(a1,t1,u2,v3) * diff(a1,u1,u3) * diff(a1,v1) * diff(a1,v2) *␣
↪→diff(rho,t2) * diff(rho,t3) * diff(a0,s1) * diff(a0,s2) * diff(a0,s3) * rho + \
+2 * diff(a1,s1) * diff(a1,s2,s3,t1) * diff(rho,v1) * diff(a1,v2) *␣
↪→diff(a1,v3) * diff(rho,t2) * diff(rho,t3) * diff(a0,u1) * diff(a0,u2) * diff(a0,u3)
)
a1dot_maybe == adot[1]/4
[42]: True
[43]: # TODO: rhodot
7.3.2 Differential polynomial (non)triviality
One could try to solve the equation Qtetra(P [ρ, a0, a1]) = [[P, V [ρ, a0, a1]]] for a vector field
V [ρ, a0, a1] as follows:
186 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
[ ]: D4V = DifferentialPolynomialRing(QQ, ('rho','a0','a1', 'V0','V1','V2','V3'),␣
↪→('x','y','z','w'), max_differential_orders=[3+1,4+1,4+1,1,1,1,1]) #; D4V
[42]: V0, V1, V2, V3 = D4V.fibre_variables()[3:]
[ ]: S4V = SuperfunctionAlgebra(D4V, D4V.base_variables(), names='xi0,xi1,xi2,xi3') #; S4V
[44]: V = V0*S4V(xi0) + V1*S4V(xi1) + V2*S4V(xi2) + V3*S4V(xi3); V
[44]: (V0)*xi0 + (V1)*xi1 + (V2)*xi2 + (V3)*xi3
[46]: PbracketV = S4V(P).bracket(V)
[47]: from gcaops.algebra.differential_polynomial_solver import solve_homogeneous_diffpoly
[48]: set_verbose(1)
[ ]: %time sol = solve_homogeneous_diffpoly(S4V(Q_tetra4)[0,1], PbracketV[0,1],␣
↪→[V0,V1,V2,V3])
verbose 1 (9: differential_polynomial_solver.py, solve_homogeneous_diffpoly) target
degrees: (4, 4, 4, 0, 0, 0, 0)
verbose 1 (9: differential_polynomial_solver.py, solve_homogeneous_diffpoly) target
weights: (3, 3, 4, 4)
verbose 1 (9: differential_polynomial_solver.py, solve_homogeneous_diffpoly) ansatz
degrees: {V0: {(3, 3, 3, 0, 0, 0, 0)}, V3: {(3, 3, 3, 0, 0, 0, 0)}, V1: {(3, 3, 3, 0,
0, 0, 0)}, V2: {(3, 3, 3, 0, 0, 0, 0)}}
verbose 1 (9: differential_polynomial_solver.py, solve_homogeneous_diffpoly) ansatz
weights: {V0: {(2, 3, 3, 3)}, V3: {(3, 3, 3, 2)}, V1: {(3, 2, 3, 3)}, V2: {(3, 3, 2,
3)}}
So far, the (non)triviality of the flow Qtetra(P [ρ, a0, a1]) remains an open problem.
7.4 Five-wheel flow on rescaled Nambu–Poisson
structures on R3
Define the five-wheel graph cocycle γ5:
[2]: from gcaops.graph.undirected_graph import UndirectedGraph
from gcaops.graph.undirected_graph_complex import UndirectedGraphComplex
[3]: GC = UndirectedGraphComplex(QQ); GC
[3]: Undirected graph complex over Rational Field with Basis consisting of representatives
of isomorphism classes of undirected graphs with no automorphisms that induce an odd
permutation on edges
[4]: fivewheel_graph = UndirectedGraph(6,␣
↪→[(0,1),(1,2),(2,3),(3,4),(0,4),(0,5),(1,5),(2,5),(3,5),(4,5)])
[5]: roof_graph = UndirectedGraph(6,␣
↪→[(0,1),(1,2),(2,3),(0,3),(3,4),(0,4),(4,5),(2,5),(1,5),(0,2)])
7.4. THE γ5 FLOW ON 3D RESCALED NAMBU–POISSON STRUCTURES 187
[6]: fivewheel_cocycle = GC(fivewheel_graph) + (5/2)*GC(roof_graph)
[7]: from gcaops.graph.directed_graph_complex import DirectedGraphComplex
[8]: dGC = DirectedGraphComplex(QQ, implementation='vector')
[23]: fivewheel_cocycle_oriented = dGC(fivewheel_cocycle) #; fivewheel_cocycle_oriented
[24]: fivewheel_cocycle_oriented_filtered = fivewheel_cocycle_oriented.
↪→filter(max_out_degree=2) #; fivewheel_cocycle_oriented_filtered
[11]: len(fivewheel_cocycle_oriented_filtered)
[11]: 91
The differential polynomial ring with fibre variables ρ, a and base variables x, y, z:
[12]: from gcaops.algebra.differential_polynomial_ring import DifferentialPolynomialRing
[13]: D3 = DifferentialPolynomialRing(QQ, ('rho','a'), ('x','y','z'),␣
↪→max_differential_orders=[5,6]) #; D3
[16]: rho, a = D3.fibre_variables()
The superfunction algebra:
[14]: from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
[15]: S3.<xi0,xi1,xi2> = SuperfunctionAlgebra(D3, D3.base_variables()) #; S3
[17]: xi = S3.gens()
[18]: X = S3.even_coordinates()
The Nambu–Poisson structure:
[19]: P = rho*sum(sigma.sign()*diff(a,X[sigma(1)-1])*xi[sigma(2)-1]*xi[sigma(3)-1] for␣
↪→sigma in Permutations(3))/2; P
[19]: (rho*a_z)*xi0*xi1 + (-rho*a_y)*xi0*xi2 + (rho*a_x)*xi1*xi2
[20]: P.bracket(P)
[20]: 0
[21]: # NOTE: filtered out graphs with out degree > 2, because we won't pass in any␣
↪→3-vectors
fivewheel_operation3 = S3.graph_operation(fivewheel_cocycle_oriented_filtered) #;␣
↪→fivewheel_operation3
The evolution ȧ of a:
[22]: %time adot = 6 * fivewheel_operation3(P,P,P,P,P,a)
CPU times: user 27min 12s, sys: 1.51 s, total: 27min 13s
Wall time: 27min 13s
188 CHAPTER 7. THE GC ACTS ON RESCALED NAMBU–POISSON BRACKETS
The five-wheel flow Qγ5(P [ρ, a]):
[46]: %time Q_fivewheel3 = fivewheel_operation3(P,P,P,P,P,P)
CPU times: user 14h 27min 1s, sys: 9.58 s, total: 14h 27min 10s
Wall time: 14h 27min 8s
[47]: len(str(Q_fivewheel3))
[47]: 63344567
We have the equality Qγ5(P [ρ, a]) = P [ρ̇, a] + P [ρ, ȧ]:
[66]: P0 = rho*sum(sigma.sign()*diff(adot,X[sigma(1)-1])*xi[sigma(2)-1]*xi[sigma(3)-1] for␣
↪→sigma in Permutations(3))/2
[67]: Q_remainder = Q_fivewheel3 - P0
[68]: len(str(Q_remainder))
[68]: 24690209
[52]: P_withoutprefactor = sum(sigma.
↪→sign()*diff(a[0],X[sigma(1)-1])*xi[sigma(2)-1]*xi[sigma(3)-1] for sigma in␣
↪→Permutations(3))/2; P_withoutprefactor
[52]: (a_z)*xi0*xi1 + (-a_y)*xi0*xi2 + (a_x)*xi1*xi2
[54]: Q_remainder[0,1] % P_withoutprefactor[0,1] == 0
[54]: True
[55]: rhodot = Q_remainder[0,1] // P_withoutprefactor[0,1]
[69]: len(str(rhodot))
[69]: 7863550
[70]: #with open('data/Q_5_3d_rank2_adot_rhodot.txt', 'w') as f:
# f.write('adot = ' + str(adot) + '\n')
# f.write('rhodot = ' + str(rhodot) + '\n')
[71]: rhodot * P_withoutprefactor == Q_remainder
[71]: True
[73]: Q_fivewheel3 == rhodot * P_withoutprefactor + P0
[73]: True
[60]: len(rhodot.monomials())
[60]: 146340
7.4. THE γ5 FLOW ON 3D RESCALED NAMBU–POISSON STRUCTURES 189
[77]: len(list(adot._monomial_coefficients.values())[0][0].monomials())
[77]: 79212
[ ]: from itertools import combinations
for (i,j) in combinations(range(3),2):
print((i,j),
Q_remainder[i,j] % P_withoutprefactor[i,j] == 0,
Q_remainder[i,j] // P_withoutprefactor[i,j] == rhodot)
[ ]: # TODO: differential polynomial (non)triviality

Chapter 8
Graph complex action on R-matrix
Poisson structures
The theory which we use in this chapter originates from the article [31] by Li and Par-
mentier. Those authors recall a method to obtain quadratic and introduce a method to
construct cubic Poisson brackets associated with Lie brackets and R-matrices on asso-
ciative algebras, such as gln(R). Their formulae express Poisson structures in terms of
a Lie bracket, an associative product, a nondegenerate symmetric bilinear form, and an
R-matrix.
We shall evaluate the formulae at specific choices of the arguments: for the Lie algebras
gln(R) with n = 2, 3, for the usual matrix product, for the nondegenerate symmetric
bilinear form 〈A,B〉 = tr(AB), and for various R-matrices. A store of R-matrices is
available from the Appendix within loc. cit.
(By the way, the formula by Li–Parmentier gives us a valid Poisson tensor also for sln(R),
which is not an algebra at all, as it is not closed under the usual product. We demonstrate
this in what follows, by producing the Poisson bracket.)
Definition. The gradient ∇xF ∈ g of F ∈ C∞(g) at x ∈ g is the unique element such
that 〈∇xF, y〉 = (dxF )(y), where dxF is the de Rham differential of F at x.
The gradient exists because the bilinea(r form i)s non-degenerate.
Example 1. For ( x xg = gl2()R) and F : 0 1 7→ x20 we have dxF = 2x0dx0 and hencex2 x3
y y
(dxF )(y) = (d F )
0 1
x = 2x0y0.y2 y3
The nondegenerate bilinear form is (( ) ( ))
〈x, y〉 = tr tr x0 x1 · y0 y((xy) = ) 1x2 x3 y2 y3
tr x0y0 + x1y2 x0y1 + x1y= 3
x2y0 + x3y2 x2y1 + x3y3
= x y
( 0 0
+ x1y2 + x) 2
y1 + x3y3,
and it follows that ∇ 2x0 0xF = .0 0
191
CHAPTER 8. GRAPH COMPLEX ACTION ON R-MATRIX POISSON
192 STRUCTURES
Example 2. The formulas for R-matrix Poisson brackets require only the gradients of
linear coordinate functions. For these we have a specialized implementation:
[1]: from gcaops.algebra.r_matrix_poisson import gradients_of_linear_coordinates
[2]: import itertools
gl2_basis = [matrix(2, lambda i,j: 1 if (i,j) == (a,b) else 0) for (a,b) in itertools.
↪→product(range(2),repeat=2)]
gl2_basis
[2]: [
[1 0] [0 1] [0 0] [0 0]
[0 0], [0 0], [1 0], [0 1]
]
[3]: gradients_of_linear_coordinates(gl2_basis)
[3]: [
[1 0] [0 0] [0 1] [0 0]
[0 0], [1 0], [0 0], [0 1]
] ( )
Indeed, we have e.g. dxx1 = dx1; dx1(y) = y1 and hence ∇
0 0
xx1 = , as can be seen1 0
from the formula for 〈x, y〉 above.
We now import the constructor of R-matrix Poisson structures. For the quadratic and
cubic Poisson structures, it is required that the bilinear form is moreover “associative” in
the sense that 〈X ·Y, Z〉 = 〈X,Y ·Z〉 for all X,Y, Z, and it is assumed that the R-matrix
satisfies the modified Yang–Baxter equation. For the quadratic Poisson structure, it is
further required that the skew part R− of R satisfies the modified Yang-Baxter equation
with the same constant as R itself.
[4]: from gcaops.algebra.r_matrix_poisson import r_matrix_poisson_bivector
To prepare for using this constructor, we define some bases of Lie algebras:
[5]: def gl_basis(field, matrix_dimension):
import itertools
return [matrix(field, matrix_dimension, lambda i,j: 1 if (i,j) == (a,b) else 0)␣
↪→for (a,b) in itertools.product(range(matrix_dimension),repeat=2)]
[6]: gl_basis(QQ, 2)
[6]: [
[1 0] [0 1] [0 0] [0 0]
[0 0], [0 0], [1 0], [0 1]
]
[7]: def so_basis(field, matrix_dimension):
8.1. THE LIE ALGEBRA OF 2× 2 MATRICES 193
return [matrix(field, matrix_dimension, lambda i,j: 1 if (i,j) == (a,b) else -1␣
↪→if (i,j) == (b,a) else 0) for (a,b) in itertools.
↪→combinations(range(matrix_dimension),2)]
[8]: so_basis(QQ, 3)
[8]: [
[ 0 1 0] [ 0 0 1] [ 0 0 0]
[-1 0 0] [ 0 0 0] [ 0 0 1]
[ 0 0 0], [-1 0 0], [ 0 -1 0]
]
[9]: def sl_basis(field, matrix_dimension):
return list(m.matrix() for m in lie_algebras.sl(QQ, matrix_dimension,␣
↪→representation='matrix').basis())
[10]: sl_basis(QQ, 3)
[10]: [
[ 1 0 0] [0 1 0] [0 0 1] [0 0 0] [ 0 0 0] [0 0 0] [0 0 0]
[ 0 0 0] [0 0 0] [0 0 0] [1 0 0] [ 0 1 0] [0 0 1] [0 0 0]
[ 0 0 -1], [0 0 0], [0 0 0], [0 0 0], [ 0 0 -1], [0 0 0], [1 0 0],
[0 0 0]
[0 0 0]
[0 1 0]
]
Moreover, we define some typical R-matrices:
[11]: R_id = lambda X: X
# differences of projections
R_strict = lambda X: matrix(X.nrows(), lambda i,j: X[i,j] if i < j else -X[i,j] if i␣
↪→> j else 0)
R_weak = lambda X: matrix(X.nrows(), lambda i,j: X[i,j] if i <= j else -X[i,j]) # ???
We also define a shorthand function for the tetrahedral flow, which will be used in the
next sections.
[12]: from gcaops.graph.undirected_graph import UndirectedGraph
from gcaops.graph.undirected_graph_complex import UndirectedGraphComplex
GC = UndirectedGraphComplex(QQ)
tetrahedron_graph = UndirectedGraph(4, [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)])
tetrahedron = GC(tetrahedron_graph)
def tetrahedral_flow(P):
S = P.parent()
tetrahedron_operation = S.graph_operation(tetrahedron)
return tetrahedron_operation(P,P,P,P)
In each case in the upcoming sections, we will calculate the tetrahedral flow and check
its (non-)triviality in the respective Poisson cohomology.
8.1 The Lie algebra of 2× 2 matrices
gl2(R) with R = id, quadratic Poisson structure
CHAPTER 8. GRAPH COMPLEX ACTION ON R-MATRIX POISSON
194 STRUCTURES
[13]: P = r_matrix_poisson_bivector(gl_basis(QQ, 2), 2, R_matrix=R_id); P
[13]: (-x0*x1 - x1*x3)*xi0*xi1 + (x0*x2 + x2*x3)*xi0*xi2 + (-x0^2 + x3^2)*xi1*xi2 + (-x0*x1
- x1*x3)*xi1*xi3 + (x0*x2 + x2*x3)*xi2*xi3
[14]: P.bracket(P)
[14]: 0
[15]: Q = tetrahedral_flow(P); Q
[15]: 0
gl2(R) with R = id, cubic Poisson structure
[16]: P = r_matrix_poisson_bivector(gl_basis(QQ, 2), 3, R_matrix=R_id); P
[16]: (x1^2*x2 - x0*x1*x3)*xi0*xi1 + (-x1*x2^2 + x0*x2*x3)*xi0*xi2 + (x0*x1*x2 - x0^2*x3 -
x1*x2*x3 + x0*x3^2)*xi1*xi2 + (x1^2*x2 - x0*x1*x3)*xi1*xi3 + (-x1*x2^2 +
x0*x2*x3)*xi2*xi3
[17]: P.bracket(P)
[17]: 0
[18]: Q = tetrahedral_flow(P); Q
[18]: 0 ( ) ( )
gl2(R) with
x x 0 x
R = 0 1 →7 1− , quadratic Poisson structurex2 x3 x2 0
[19]: P = r_matrix_poisson_bivector(gl_basis(QQ, 2), 2, R_matrix=R_strict); P
[19]: (x0*x1)*xi0*xi1 + (x0*x2)*xi0*xi2 + (2*x1*x2)*xi0*xi3 + (x1*x3)*xi1*xi3 +
(x2*x3)*xi2*xi3
[20]: P.bracket(P)
[20]: 0
[21]: Q = tetrahedral_flow(P); Q
[21]: 0 ( ) ( )
gl (R) with xR = 0 x12 →7
0 x1
− , cubic Poisson structurex2 x3 x2 0
[22]: P = r_matrix_poisson_bivector(gl_basis(QQ, 2), 3, R_matrix=R_strict); P
[22]: (1/2*x0^2*x1 + 1/2*x1^2*x2)*xi0*xi1 + (1/2*x0^2*x2 + 1/2*x1*x2^2)*xi0*xi2 + (x0*x1*x2
+ x1*x2*x3)*xi0*xi3 + (1/2*x1^2*x2 + 1/2*x1*x3^2)*xi1*xi3 + (1/2*x1*x2^2 +
1/2*x2*x3^2)*xi2*xi3
8.1. THE LIE ALGEBRA OF 2× 2 MATRICES 195
[23]: P.bracket(P)
[23]: 0
[24]: Q = tetrahedral_flow(P); Q
[24]: (48*x0^3*x1^2*x2 - 48*x0*x1^3*x2^2 + 96*x0^2*x1^2*x2*x3 + 60*x1^3*x2^2*x3 +
72*x1^2*x2*x3^3 + 12*x1*x3^5)*xi1*xi3 + (48*x0^3*x1*x2^2 - 48*x0*x1^2*x2^3 +
96*x0^2*x1*x2^2*x3 + 60*x1^2*x2^3*x3 + 72*x1*x2^2*x3^3 + 12*x2*x3^5)*xi2*xi3 +
(-12*x0^5*x1 - 72*x0^3*x1^2*x2 - 60*x0*x1^3*x2^2 + 48*x1^3*x2^2*x3 -
96*x0*x1^2*x2*x3^2 - 48*x1^2*x2*x3^3)*xi0*xi1 + (-12*x0^5*x2 - 72*x0^3*x1*x2^2 -
60*x0*x1^2*x2^3 + 48*x1^2*x2^3*x3 - 96*x0*x1*x2^2*x3^2 - 48*x1*x2^2*x3^3)*xi0*xi2 +
(-84*x0^4*x1*x2 - 120*x0^2*x1^2*x2^2 - 144*x0^3*x1*x2*x3 + 120*x1^2*x2^2*x3^2 +
144*x0*x1*x2*x3^3 + 84*x1*x2*x3^4)*xi0*xi3
[25]: from gcaops.algebra.homogeneous_polynomial_poisson_complex import PoissonComplex
[26]: PC = PoissonComplex(P)
[27]: PC(Q).is_coboundary(certificate=True)
[27]: (True,
Poisson cochain (-12*x0^4 + 156*x0*x1*x2*x3 + 48*x1*x2*x3^2)*xi0 + (78*x0^3*x1 +
60*x0*x1^2*x2 - 30*x0^2*x1*x3 + 18*x1^2*x2*x3 + 30*x0*x1*x3^2)*xi1 + (78*x0^3*x2 +
60*x0*x1*x2^2 - 30*x0^2*x2*x3 + 18*x1*x2^2*x3 + 30*x0*x2*x3^2)*xi2 + (108*x0^2*x1*x2 +
12*x3^4)*xi3) ( ) ( )
gl (R) with xR = 0 x1 →7 x0 x12 − , quadratic Poisson structurex2 x3 x2 x3
[28]: P = r_matrix_poisson_bivector(gl_basis(QQ, 2), 2, R_matrix=R_weak); P
[28]: (2*x0*x2)*xi0*xi2 + (2*x1*x2)*xi0*xi3 + (2*x2*x3)*xi2*xi3
[29]: P.bracket(P)
[29]: 0
[30]: Q = tetrahedral_flow(P); Q
[30]: 0 ( ) ( )
gl (R) with x0 xR = 1 7→ x0 x12 − , cubic Poisson structurex2 x3 x2 x3
[31]: P = r_matrix_poisson_bivector(gl_basis(QQ, 2), 3, R_matrix=R_weak); P
[31]: (x1^2*x2)*xi0*xi1 + (x0^2*x2)*xi0*xi2 + (x0*x1*x2 + x1*x2*x3)*xi0*xi3 + (x0*x1*x2 -
x1*x2*x3)*xi1*xi2 + (x1^2*x2)*xi1*xi3 + (x2*x3^2)*xi2*xi3
[32]: P.bracket(P)
[32]: 0
CHAPTER 8. GRAPH COMPLEX ACTION ON R-MATRIX POISSON
196 STRUCTURES
[33]: Q = tetrahedral_flow(P); Q
[33]: (384*x0*x1^2*x2^3 + 384*x0^2*x1*x2^2*x3 - 96*x0*x1*x2^2*x3^2 + 96*x1*x2^2*x3^3 +
96*x2*x3^5)*xi2*xi3 + (-96*x0^4*x1*x2 - 96*x0^2*x1^2*x2^2 - 384*x1^3*x2^3 +
96*x0^3*x1*x2*x3 - 192*x0*x1^2*x2^2*x3 - 96*x0^2*x1*x2*x3^2 - 96*x1^2*x2^2*x3^2 +
96*x0*x1*x2*x3^3 - 96*x1*x2*x3^4)*xi1*xi2 + (-96*x0^5*x2 - 96*x0^3*x1*x2^2 +
96*x0^2*x1*x2^2*x3 - 384*x1^2*x2^3*x3 - 384*x0*x1*x2^2*x3^2)*xi0*xi2 + (-96*x0^4*x1*x2
- 672*x0^2*x1^2*x2^2 - 384*x0^3*x1*x2*x3 + 672*x1^2*x2^2*x3^2 + 384*x0*x1*x2*x3^3 +
96*x1*x2*x3^4)*xi0*xi3 + (-96*x0^3*x1^2*x2 - 672*x0*x1^3*x2^2 - 288*x0^2*x1^2*x2*x3 +
672*x1^3*x2^2*x3 + 288*x0*x1^2*x2*x3^2 + 96*x1^2*x2*x3^3)*xi0*xi1 + (-96*x0^3*x1^2*x2
- 672*x0*x1^3*x2^2 - 288*x0^2*x1^2*x2*x3 + 672*x1^3*x2^2*x3 + 288*x0*x1^2*x2*x3^2 +
96*x1^2*x2*x3^3)*xi1*xi3
[34]: PC = PoissonComplex(P)
[35]: PC(Q).is_coboundary(certificate=True)
[35]: (True,
Poisson cochain (-48*x0^4 + 480*x0*x1*x2*x3)*xi0 + (-48*x0^3*x1 + 480*x0*x1^2*x2 +
48*x0^2*x1*x3 - 48*x0*x1*x3^2 + 48*x1*x3^3)*xi1 + (480*x0^3*x2 + 192*x0*x1*x2^2 -
288*x0^2*x2*x3 - 192*x1*x2^2*x3 + 288*x0*x2*x3^2)*xi2 + (480*x0^2*x1*x2 +
48*x3^4)*xi3)
8.2 The Lie algebra of 3× 3 matrices
gl3(R) with R = id, quadratic Poisson structure
Skew part R− of R is zero, does not satisfy YBE.
gl3(R) with R = id, cubic Poisson structure
[36]: P = r_matrix_poisson_bivector(gl_basis(QQ, 3), 3, R_matrix=R_id); P
[36]: (x1^2*x3 - x0*x1*x4 + x1*x2*x6 - x0*x2*x7)*xi0*xi1 + (x1*x2*x3 - x0*x1*x5 + x2^2*x6 -
x0*x2*x8)*xi0*xi2 + (-x1*x3^2 + x0*x3*x4 - x2*x3*x6 + x0*x5*x6)*xi0*xi3 + (x1*x5*x6 -
x2*x3*x7)*xi0*xi4 + (x2*x3*x4 - x1*x3*x5 + x2*x5*x6 - x2*x3*x8)*xi0*xi5 + (-x1*x3*x6 -
x2*x6^2 + x0*x3*x7 + x0*x6*x8)*xi0*xi6 + (-x1*x4*x6 + x1*x3*x7 - x2*x6*x7 +
x1*x6*x8)*xi0*xi7 + (-x1*x5*x6 + x2*x3*x7)*xi0*xi8 + (x1*x2*x4 - x1^2*x5 + x2^2*x7 -
x1*x2*x8)*xi1*xi2 + (x0*x1*x3 - x0^2*x4 - x1*x3*x4 + x0*x4^2 - x2*x4*x6 +
x0*x5*x7)*xi1*xi3 + (x1^2*x3 - x0*x1*x4 - x2*x4*x7 + x1*x5*x7)*xi1*xi4 + (x1*x2*x3 -
x0*x2*x4 + x2*x4^2 - x1*x4*x5 + x2*x5*x7 - x2*x4*x8)*xi1*xi5 + (x0*x1*x6 - x0^2*x7 -
x1*x3*x7 + x0*x4*x7 - x2*x6*x7 + x0*x7*x8)*xi1*xi6 + (x1^2*x6 - x0*x1*x7 - x2*x7^2 +
x1*x7*x8)*xi1*xi7 + (x1*x2*x6 - x0*x2*x7 + x2*x4*x7 - x1*x5*x7)*xi1*xi8 + (x0*x2*x3 -
x0^2*x5 - x1*x3*x5 + x0*x4*x5 - x2*x5*x6 + x0*x5*x8)*xi2*xi3 + (x1*x2*x3 - x0*x1*x5 -
x2*x5*x7 + x1*x5*x8)*xi2*xi4 + (x2^2*x3 - x0*x2*x5 + x2*x4*x5 - x1*x5^2)*xi2*xi5 +
(x0*x2*x6 + x0*x5*x7 - x0^2*x8 - x1*x3*x8 - x2*x6*x8 + x0*x8^2)*xi2*xi6 + (x1*x2*x6 +
x1*x5*x7 - x0*x1*x8 - x1*x4*x8 - x2*x7*x8 + x1*x8^2)*xi2*xi7 + (x2^2*x6 + x2*x5*x7 -
x0*x2*x8 - x1*x5*x8)*xi2*xi8 + (-x1*x3^2 + x0*x3*x4 + x4*x5*x6 - x3*x5*x7)*xi3*xi4 +
(-x2*x3^2 + x0*x3*x5 + x5^2*x6 - x3*x5*x8)*xi3*xi5 + (-x3*x4*x6 - x5*x6^2 + x3^2*x7 +
x3*x6*x8)*xi3*xi6 + (-x1*x3*x6 + x0*x4*x6 - x4^2*x6 + x3*x4*x7 - x5*x6*x7 +
x4*x6*x8)*xi3*xi7 + (-x2*x3*x6 + x0*x5*x6 - x4*x5*x6 + x3*x5*x7)*xi3*xi8 + (-x2*x3*x4
+ x1*x3*x5 + x5^2*x7 - x4*x5*x8)*xi4*xi5 + (x1*x3*x6 - x0*x3*x7 - x5*x6*x7 +
x3*x7*x8)*xi4*xi6 + (x1*x4*x6 - x1*x3*x7 - x5*x7^2 + x4*x7*x8)*xi4*xi7 + (x1*x5*x6 -
x2*x3*x7)*xi4*xi8 + (x2*x3*x6 + x3*x5*x7 - x0*x3*x8 - x3*x4*x8 - x5*x6*x8 +
x3*x8^2)*xi5*xi6 + (x2*x4*x6 + x4*x5*x7 - x1*x3*x8 - x4^2*x8 - x5*x7*x8 +
8.2. THE LIE ALGEBRA OF 3× 3 MATRICES 197
x4*x8^2)*xi5*xi7 + (x2*x5*x6 + x5^2*x7 - x2*x3*x8 - x4*x5*x8)*xi5*xi8 + (-x1*x6^2 +
x0*x6*x7 - x4*x6*x7 + x3*x7^2)*xi6*xi7 + (-x2*x6^2 - x5*x6*x7 + x0*x6*x8 +
x3*x7*x8)*xi6*xi8 + (-x2*x6*x7 - x5*x7^2 + x1*x6*x8 + x4*x7*x8)*xi7*xi8
[37]: P.bracket(P)
[37]: 0
[38]: #%time Q = tetrahedral_flow(P)
[39]: #PC = PoissonComplex(P)
[40]: #%time PC(Q).is_coboundary(certificate=True) 
x0 x1 x2 0 x1 x2
gl3(R) with R = x x   3 4 x5 7→ −x3 0 x5 , quadratic Poisson structure
x6 x7 x8 −x6 −x7 0
[41]: P = r_matrix_poisson_bivector(gl_basis(QQ, 3), 2, R_matrix=R_strict); P
[41]: (x0*x1)*xi0*xi1 + (x0*x2)*xi0*xi2 + (x0*x3)*xi0*xi3 + (2*x1*x3)*xi0*xi4 +
(2*x2*x3)*xi0*xi5 + (x0*x6)*xi0*xi6 + (2*x1*x6)*xi0*xi7 + (2*x2*x6)*xi0*xi8 +
(x1*x2)*xi1*xi2 + (x1*x4)*xi1*xi4 + (2*x2*x4)*xi1*xi5 + (x1*x7)*xi1*xi7 +
(2*x2*x7)*xi1*xi8 + (x2*x5)*xi2*xi5 + (x2*x8)*xi2*xi8 + (x3*x4)*xi3*xi4 +
(x3*x5)*xi3*xi5 + (x3*x6)*xi3*xi6 + (2*x4*x6)*xi3*xi7 + (2*x5*x6)*xi3*xi8 +
(x4*x5)*xi4*xi5 + (x4*x7)*xi4*xi7 + (2*x5*x7)*xi4*xi8 + (x5*x8)*xi5*xi8 +
(x6*x7)*xi6*xi7 + (x6*x8)*xi6*xi8 + (x7*x8)*xi7*xi8
[42]: P.bracket(P)
[42]: 0
[43]: Q = tetrahedral_flow(P); Q
[43]: 0
gl (R) with R = 
   
x0 x1 x2 0 x1 x2
3 x3 x x   4 5 →7 −x3 0 x5 , cubic Poisson structure
x6 x7 x8 −x6 −x7 0
[44]: P = r_matrix_poisson_bivector(gl_basis(QQ, 3), 3, R_matrix=R_strict); P
[44]: (1/2*x0^2*x1 + 1/2*x1^2*x3)*xi0*xi1 + (1/2*x0^2*x2 + x1*x2*x3 + 1/2*x2^2*x6)*xi0*xi2 +
(1/2*x0^2*x3 + 1/2*x1*x3^2)*xi0*xi3 + (x0*x1*x3 + x1*x3*x4)*xi0*xi4 + (x0*x2*x3 +
x2*x3*x4 + 1/2*x1*x3*x5 + 1/2*x2*x5*x6)*xi0*xi5 + (1/2*x0^2*x6 + x1*x3*x6 +
1/2*x2*x6^2)*xi0*xi6 + (x0*x1*x6 + x1*x4*x6 + 1/2*x1*x3*x7 + 1/2*x2*x6*x7)*xi0*xi7 +
(x0*x2*x6 + x1*x5*x6 + x2*x3*x7 + x2*x6*x8)*xi0*xi8 + (x1*x2*x4 - 1/2*x1^2*x5 +
1/2*x2^2*x7)*xi1*xi2 + (1/2*x1^2*x3 + 1/2*x1*x4^2)*xi1*xi4 + (1/2*x1*x2*x3 + x2*x4^2 +
1/2*x2*x5*x7)*xi1*xi5 + (1/2*x1*x3*x7 + 1/2*x2*x6*x7)*xi1*xi6 + (1/2*x1^2*x6 +
x1*x4*x7 + 1/2*x2*x7^2)*xi1*xi7 + (1/2*x1*x2*x6 + x2*x4*x7 + 1/2*x1*x5*x7 +
x2*x7*x8)*xi1*xi8 + (-1/2*x1*x3*x5 - 1/2*x2*x5*x6)*xi2*xi3 + (1/2*x1*x2*x3 -
1/2*x2*x5*x7)*xi2*xi4 + (1/2*x2^2*x3 + x2*x4*x5 - 1/2*x1*x5^2)*xi2*xi5 + (1/2*x1*x2*x6
+ 1/2*x1*x5*x7)*xi2*xi7 + (1/2*x2^2*x6 + x2*x5*x7 + 1/2*x2*x8^2)*xi2*xi8 +
(1/2*x1*x3^2 + 1/2*x3*x4^2)*xi3*xi4 + (1/2*x2*x3^2 + x3*x4*x5 + 1/2*x5^2*x6)*xi3*xi5 +
CHAPTER 8. GRAPH COMPLEX ACTION ON R-MATRIX POISSON
198 STRUCTURES
(x3*x4*x6 + 1/2*x5*x6^2 - 1/2*x3^2*x7)*xi3*xi6 + (1/2*x1*x3*x6 + x4^2*x6 +
1/2*x5*x6*x7)*xi3*xi7 + (1/2*x2*x3*x6 + x4*x5*x6 + 1/2*x3*x5*x7 + x5*x6*x8)*xi3*xi8 +
(1/2*x4^2*x5 + 1/2*x5^2*x7)*xi4*xi5 + (-1/2*x1*x3*x6 + 1/2*x5*x6*x7)*xi4*xi6 +
(1/2*x4^2*x7 + 1/2*x5*x7^2)*xi4*xi7 + (x4*x5*x7 + x5*x7*x8)*xi4*xi8 + (-1/2*x2*x3*x6 -
1/2*x3*x5*x7)*xi5*xi6 + (1/2*x5^2*x7 + 1/2*x5*x8^2)*xi5*xi8 + (1/2*x1*x6^2 + x4*x6*x7
- 1/2*x3*x7^2)*xi6*xi7 + (1/2*x2*x6^2 + x5*x6*x7 + 1/2*x6*x8^2)*xi6*xi8 + (1/2*x5*x7^2
+ 1/2*x7*x8^2)*xi7*xi8
[45]: P.bracket(P)
[45]: 0
[46]: #%time Q = tetrahedral_flow(P); Q
[47]: #PC = PoissonComplex(P)
[48]: #PC(Q).is_coboundary(certificate=Tru 
e) 
x0 x1 x2 x0 x1 x2
gl3(R) with R = x3 x4 x  7→ 5 −x3 x 4 x5 , quadratic Poisson structure
x6 x7 x8 −x6 −x7 x8
[49]: P = r_matrix_poisson_bivector(gl_basis(QQ, 3), 2, R_matrix=R_weak); P
[49]: (2*x0*x3)*xi0*xi3 + (2*x1*x3)*xi0*xi4 + (2*x2*x3)*xi0*xi5 + (2*x0*x6)*xi0*xi6 +
(2*x1*x6)*xi0*xi7 + (2*x2*x6)*xi0*xi8 + (x1*x2)*xi1*xi2 + (2*x2*x4 - x1*x5)*xi1*xi5 +
(x1*x6)*xi1*xi6 + (x1*x7)*xi1*xi7 + (2*x2*x7)*xi1*xi8 + (x2*x3)*xi2*xi3 +
(x2*x5)*xi2*xi5 + (-x2*x7)*xi2*xi7 + (2*x3*x4)*xi3*xi4 + (x3*x5)*xi3*xi5 +
(x3*x6)*xi3*xi6 + (2*x4*x6 + x3*x7)*xi3*xi7 + (2*x5*x6)*xi3*xi8 + (2*x4*x7)*xi4*xi7 +
(2*x5*x7)*xi4*xi8 + (-x5*x6)*xi5*xi6 + (x6*x7)*xi6*xi7 + (2*x6*x8)*xi6*xi8 +
(2*x7*x8)*xi7*xi8
[50]: P.bracket(P)
[50]: 0
[51]: %time Q = tetrahedral_flow(P); Q
CPU times: user 15.7 s, sys: 51.9 ms, total: 15.8 s
Wall time: 15.8 s
[51]: (-384*x2*x3)*xi0*xi5 + (-384*x1*x3)*xi0*xi4 + (-384*x1*x6)*xi0*xi7 +
(384*x2*x7)*xi1*xi8 + (384*x5*x7)*xi4*xi8 + (384*x5*x6)*xi3*xi8
[52]: PC = PoissonComplex(P)
[53]: PC(Q).is_coboundary(certificate=True)
[53]: (True, Poisson cochain (192*x1)*xi1 + (192*x3)*xi3 + (192*x4)*xi4)
x0 x1 x2
gl (R) with R = x x x 

x0 x1 x2
3 3 4 5 7→ −x3 x4 x 5 , cubic Poisson structure
x6 x7 x8 −x6 −x7 x8
8.3. THE LIE ALGEBRA OF TRACELESS 2× 2 MATRICES 199
[54]: P = r_matrix_poisson_bivector(gl_basis(QQ, 3), 3, R_matrix=R_weak); P
[54]: (x1^2*x3)*xi0*xi1 + (x1*x2*x3 + x2^2*x6)*xi0*xi2 + (x0^2*x3)*xi0*xi3 + (x0*x1*x3 +
x1*x3*x4)*xi0*xi4 + (x0*x2*x3 + x2*x3*x4 + x2*x5*x6)*xi0*xi5 + (x0^2*x6 +
x1*x3*x6)*xi0*xi6 + (x0*x1*x6 + x1*x4*x6 + x1*x3*x7)*xi0*xi7 + (x0*x2*x6 + x1*x5*x6 +
x2*x3*x7 + x2*x6*x8)*xi0*xi8 + (x1*x2*x4 - x1^2*x5 + x2^2*x7)*xi1*xi2 + (x0*x1*x3 -
x1*x3*x4)*xi1*xi3 + (x1^2*x3)*xi1*xi4 + (x1*x2*x3 + x2*x4^2 - x1*x4*x5 +
x2*x5*x7)*xi1*xi5 + (x0*x1*x6)*xi1*xi6 + (x1^2*x6 + x1*x4*x7)*xi1*xi7 + (x1*x2*x6 +
x2*x4*x7 + x2*x7*x8)*xi1*xi8 + (x0*x2*x3 - x1*x3*x5 - x2*x5*x6)*xi2*xi3 + (x1*x2*x3 -
x2*x5*x7)*xi2*xi4 + (x2^2*x3 + x2*x4*x5 - x1*x5^2)*xi2*xi5 + (x0*x2*x6 -
x2*x6*x8)*xi2*xi6 + (x1*x2*x6 + x1*x5*x7 - x2*x7*x8)*xi2*xi7 + (x2^2*x6 +
x2*x5*x7)*xi2*xi8 + (x3*x4^2)*xi3*xi4 + (x3*x4*x5 + x5^2*x6)*xi3*xi5 +
(x3*x4*x6)*xi3*xi6 + (x4^2*x6 + x3*x4*x7)*xi3*xi7 + (x4*x5*x6 + x3*x5*x7 +
x5*x6*x8)*xi3*xi8 + (x5^2*x7)*xi4*xi5 + (x4^2*x7)*xi4*xi7 + (x4*x5*x7 +
x5*x7*x8)*xi4*xi8 + (-x5*x6*x8)*xi5*xi6 + (x4*x5*x7 - x5*x7*x8)*xi5*xi7 +
(x5^2*x7)*xi5*xi8 + (x4*x6*x7)*xi6*xi7 + (x5*x6*x7 + x6*x8^2)*xi6*xi8 +
(x7*x8^2)*xi7*xi8
[55]: P.bracket(P)
[55]: 0
[56]: #%time Q = tetrahedral_flow(P); Q
8.3 The Lie algebra of traceless 2× 2 matrices
N.B. Not an asso(ciative a)lgebr(a, neverth)eless the construction can work:
sl (R) with x xR = 0 12 →7
0 x1 , quadratic Poisson structure ≡ 0
x2 x3 −x2 0
[57]: P = r_matrix_poisson_bivector(sl_basis(QQ, 2), 2, R_matrix=R_strict); P
[57]: 0 ( ) ( )
x x 0 x
sl2(R) with R = 0 1 7→ 1− , cubic Poisson structurex2 x3 x2 0
[58]: P = r_matrix_poisson_bivector(sl_basis(QQ, 2), 3, R_matrix=R_strict); P
[58]: (-1/2*x0^2*x1 - 1/2*x0*x2^2)*xi0*xi2 + (-1/2*x0*x1^2 - 1/2*x1*x2^2)*xi1*xi2
[59]: P.bracket(P)
[59]: 0
[60]: Q = tetrahedral_flow(P); Q
[60]: (60*x0^3*x1^2*x2 + 72*x0^2*x1*x2^3 + 12*x0*x2^5)*xi0*xi2 + (60*x0^2*x1^3*x2 +
72*x0*x1^2*x2^3 + 12*x1*x2^5)*xi1*xi2
[61]: PC = PoissonComplex(P)
CHAPTER 8. GRAPH COMPLEX ACTION ON R-MATRIX POISSON
200 STRUCTURES
[62]: PC(Q).is_coboundary(certificate=True)
[62]: (True,
Poisson cochain (-48*x0^2*x1*x2)*xi0 + (-48*x0*x1^2*x2)*xi1 + (60*x0^2*x1^2 -
12*x2^4)*xi2) ( ) ( )
x x
sl (R) with R = 0 1 →7 x0 x12 − , quadratic Poisson structure ≡ 0x2 x3 x2 x3
[63]: P = r_matrix_poisson_bivector(sl_basis(QQ, 2), 2, R_matrix=R_weak); P
[63]: 0 ( ) ( )
sl2(R) with
x x x x
R = 0 1 7→ 0 1− , cubic Poisson structurex2 x3 x2 x3
[64]: P = r_matrix_poisson_bivector(sl_basis(QQ, 2), 3, R_matrix=R_weak); P
[64]: (2*x0*x1*x2)*xi0*xi1 + (-x0^2*x1)*xi0*xi2 + (-x1*x2^2)*xi1*xi2
[65]: P.bracket(P)
[65]: 0
[66]: Q = tetrahedral_flow(P); Q
[66]: (-384*x0^2*x1^3*x2 + 384*x0*x1^2*x2^3 + 96*x1*x2^5)*xi1*xi2 + (1152*x0^3*x1^2*x2 -
384*x0^2*x1*x2^3)*xi0*xi2 + (-384*x0^3*x1^3 + 384*x0^2*x1^2*x2^2 -
480*x0*x1*x2^4)*xi0*xi1
[67]: PC = PoissonComplex(P)
[68]: PC(Q).is_coboundary(certificate=True)
[68]: (True,
Poisson cochain (384*x0^2*x1*x2 - 192*x0*x2^3)*xi0 + (-384*x0*x1^2*x2)*xi1 +
(384*x0^2*x1^2 - 48*x2^4)*xi2)
8.4 The Lie algebra of traceless 3× 3 matrices
sl3(R) with R = id, cubic Poisson structure
[69]: P = r_matrix_poisson_bivector(sl_basis(QQ, 3), 3, R_matrix=R_id); P
[69]: (x1^2*x3 - x0*x1*x4 + x1*x2*x6 - x0*x2*x7)*xi0*xi1 + (x0^2*x2 + x1*x2*x3 + x0*x2*x4 -
x0*x1*x5 + x2^2*x6)*xi0*xi2 + (-x1*x3^2 + x0*x3*x4 - x2*x3*x6 + x0*x5*x6)*xi0*xi3 +
(x1*x5*x6 - x2*x3*x7)*xi0*xi4 + (x0*x2*x3 + 2*x2*x3*x4 - x1*x3*x5 + x2*x5*x6)*xi0*xi5
+ (-x0^2*x6 - x1*x3*x6 - x0*x4*x6 - x2*x6^2 + x0*x3*x7)*xi0*xi6 + (-x0*x1*x6 -
2*x1*x4*x6 + x1*x3*x7 - x2*x6*x7)*xi0*xi7 + (x0*x1*x2 + 2*x1*x2*x4 - x1^2*x5 +
x2^2*x7)*xi1*xi2 + (x0*x1*x3 - x0^2*x4 - x1*x3*x4 + x0*x4^2 - x2*x4*x6 +
x0*x5*x7)*xi1*xi3 + (x1^2*x3 - x0*x1*x4 - x2*x4*x7 + x1*x5*x7)*xi1*xi4 + (x1*x2*x3 +
2*x2*x4^2 - x1*x4*x5 + x2*x5*x7)*xi1*xi5 + (x0*x1*x6 - 2*x0^2*x7 - x1*x3*x7 -
x2*x6*x7)*xi1*xi6 + (x1^2*x6 - 2*x0*x1*x7 - x1*x4*x7 - x2*x7^2)*xi1*xi7 + (x0*x2*x3 -
8.4. THE LIE ALGEBRA OF TRACELESS 3× 3 MATRICES 201
2*x0^2*x5 - x1*x3*x5 - x2*x5*x6)*xi2*xi3 + (x1*x2*x3 - 2*x0*x1*x5 - x1*x4*x5 -
x2*x5*x7)*xi2*xi4 + (x2^2*x3 - x0*x2*x5 + x2*x4*x5 - x1*x5^2)*xi2*xi5 + (2*x0^3 +
x0*x1*x3 + 3*x0^2*x4 + x1*x3*x4 + x0*x4^2 + 2*x0*x2*x6 + x2*x4*x6 + x0*x5*x7)*xi2*xi6
+ (2*x0^2*x1 + 4*x0*x1*x4 + 2*x1*x4^2 + x1*x2*x6 + x0*x2*x7 + x2*x4*x7 +
x1*x5*x7)*xi2*xi7 + (-x1*x3^2 + x0*x3*x4 + x4*x5*x6 - x3*x5*x7)*xi3*xi4 + (-x2*x3^2 +
2*x0*x3*x5 + x3*x4*x5 + x5^2*x6)*xi3*xi5 + (-x0*x3*x6 - 2*x3*x4*x6 - x5*x6^2 +
x3^2*x7)*xi3*xi6 + (-x1*x3*x6 - 2*x4^2*x6 + x3*x4*x7 - x5*x6*x7)*xi3*xi7 + (-x2*x3*x4
+ x1*x3*x5 + x0*x4*x5 + x4^2*x5 + x5^2*x7)*xi4*xi5 + (x1*x3*x6 - 2*x0*x3*x7 - x3*x4*x7
- x5*x6*x7)*xi4*xi6 + (x1*x4*x6 - x1*x3*x7 - x0*x4*x7 - x4^2*x7 - x5*x7^2)*xi4*xi7 +
(2*x0^2*x3 + 4*x0*x3*x4 + 2*x3*x4^2 + x2*x3*x6 + x0*x5*x6 + x4*x5*x6 +
x3*x5*x7)*xi5*xi6 + (x0*x1*x3 + x0^2*x4 + x1*x3*x4 + 3*x0*x4^2 + 2*x4^3 + x2*x4*x6 +
x0*x5*x7 + 2*x4*x5*x7)*xi5*xi7 + (-x1*x6^2 + x0*x6*x7 - x4*x6*x7 + x3*x7^2)*xi6*xi7
[70]: P.bracket(P)
[70]: 0
[71]: #%time Q = tetrahedral_flow(P); Q

Chapter 9
Graph complex action on star
products
In this chapter we bring everything together. We use star products and gauge transfor-
mations from Chapters 1 and 3, Poisson structures from Chapter 2, graph cocycles from
Chapters 4 and 5, and flows from Chapters 6, 7, and 8.
Finally, let us bring together, compare and contrast two types of deformations: Kontse-
vich’s universal construction of deformations of Poisson brackets (i.e. Licherowicz–Poisson
classes) and gauge transformations of star products.
[1]: from gcaops.graph.formality_graph import FormalityGraph
from gcaops.graph.formality_graph_basis import KontsevichGraphBasis
from gcaops.graph.formality_graph_complex import FormalityGraphComplex
from gcaops.graph.formality_graph_operator import FormalityGraphOperator
set_verbose(-1)
Define the quotient ring SR[h̄, ε]/(h̄5, ε2), i.e. the symbolic ring SR with the variables h̄
and ε  —conveniently satisfying h̄5 = 0 and ε2 = 0, respectively — adjoined to it:
[2]: R = PolynomialRing(SR, 2, names='hbar,eps')
Q.<hbar,eps> = R.quotient(R.ideal(R.gen(0)^5, R.gen(1)^2))
Q.element_class.derivative = lambda self, x: self.parent()(self.lift().derivative(x))
We recall the Proposition from §1.3: gauge transformations T = id+h̄kTk mod ō(h̄k) con-
centrated in degree k act on ⋆-products modulo ō(h̄k) by adding the Hochschild cobound-
aries h̄ dH(Tk) mod ō(h̄k).
9.1 Poisson-trivial deformations and gauge
transformations
Now let us take a Poisson-trivial deformation P 7→ P + ε[[P,X]] mod ō(ε) of a given
Poisson structure, and induce a deformation of the Kontsevich star product. This results
not in adding a Hochschild coboundary (that is, not a Gerstenhaber bracket with the
usual multiplication) but this results in adding to ⋆ the Gerstenhaber bracket [h̄P,X]G
of the Poisson bi-vector h̄P (viewed as a bi-differential operator) with the vector field
X in the trivial deformation. Hence, the deformation of the star product is produced
by a gauge transformation T = id+h̄k · const ·X; here the derivation X is given by the
203
204 CHAPTER 9. GRAPH COMPLEX ACTION ON STAR PRODUCTS
sunflower graph, k = 3, the star-product is affected at order 4, and all structures are
taken modulo ō(h̄4).
Example. Let P = x2y ξ1ξ2 be a Poisson structure (on R2 with coordinates x, y):
[3]: from gcaops.algebra.superfunction_algebra import SuperfunctionAlgebra
from gcaops.algebra.polydifferential_operator import PolyDifferentialOperatorAlgebra
SA.<xi1,xi2> = SuperfunctionAlgebra(Q, var('x,y'))
PA.<ddx,ddy> = PolyDifferentialOperatorAlgebra(Q, var('x,y'))
[4]: P = y*x^2*xi1*xi2; P
[4]: (x^2*y)*xi1*xi2
Calculate the Kontsevich star product expansion mod ō(h̄4) for the Poisson structure P :
[5]: FGC = FormalityGraphComplex(SR, lazy=True); FGC
[5]: Formality graph complex over Symbolic Ring with Basis consisting of representatives of
isomorphism classes of formality graphs with no automorphisms that induce an odd
permutation on edges
[6]: star4_txt = open('data/star4.txt').read().rstrip()
star4 = FGC.element_from_kgs_encoding(star4_txt) #; star4
[7]: star4_operator = FormalityGraphOperator(SA, PA, star4)
%time star4_op = star4_operator.value_at_copies_of(hbar*P) #; star4_op
CPU times: user 1min 52s, sys: 74.9 ms, total: 1min 52s
Wall time: 1min 52s
Deform the Poisson structure P by using the sunflower graph at ε (cf. [6]):
[8]: KGB = KontsevichGraphBasis()
[9]: sunflower = list(KGB.graphs(1,3))[-1]; sunflower.show()
9.1. POISSON-TRIVIAL DEFORMATIONS AND GAUGE TRANSFORMATIONS205
[10]: X = FGC(sunflower); X
[10]: 1*FormalityGraph(1, 3, [(1, 2), (1, 3), (2, 1), (2, 3), (3, 0), (3, 2)])
The value of this graph at three copies of h̄P is a unary differential operator:
[11]: X_operator = FormalityGraphOperator(SA, PA, X)
X_op = X_operator.value_at_copies_of(hbar*P); X_op
[11]: ((-4*x^4)*hbar^3)*ddx + (8*x^3*y*hbar^3)*ddy
This unary first-order differential operator is naturally a vector field (i.e. a superfunction
linear in ξ’s):
[12]: X_vec = SA(X_op); X_vec
[12]: ((-4*x^4)*hbar^3)*xi1 + (8*x^3*y*hbar^3)*xi2
So, let us construct a Poisson-trivial deformation P 7→ P + ε[[P,X]] + ō(ε) + ō(h̄3).
[13]: P2 = P + eps*P.bracket(X_vec); P2
[13]: ((-8*x^5*y)*hbar^3*eps + x^2*y)*xi1*xi2
Calculate the star product expansion ⋆′ mod ō(h̄4) and mod ō(ε) for the deformed Poisson
structure P2:
[14]: %time star4_deformed_op = star4_operator.value_at_copies_of(hbar*P2) #;␣
↪→star4_deformed_op
206 CHAPTER 9. GRAPH COMPLEX ACTION ON STAR PRODUCTS
CPU times: user 3min 38s, sys: 216 ms, total: 3min 39s
Wall time: 3min 39s
The difference ⋆′ − ⋆ between the two star product expansions modulo ō(h̄4) and ō(ε) is
the following bi-differential operator, naturally given by the bi-vector h̄ε[[P,X]]:
[15]: star4_deformed_op - star4_op
[15]: ((-8*x^5*y)*hbar^4*eps)*(ddx ⊗ ddy) + (8*x^5*y*hbar^4*eps)*(ddy ⊗ ddx)
The difference is a Hochschild cocycle because it is a derivation in each argument:
[16]: (star4_deformed_op - star4_op).hochschild_differential()
[16]: 0
Remark. The difference is not a Hochschild coboundary, yet the difference is equal to ε
times the Gerstenhaber bracket of the Poisson bi-vector h̄P and the “sunflower” vector
field X:
[17]: hP_op = PA(hbar*P); hP_op
[17]: (x^2*y*hbar)*(ddx ⊗ ddy) + ((-x^2*y)*hbar)*(ddy ⊗ ddx)
The above is a bi-derivation. Let us take its Gerstenhaber bracket with the derivation
X.
[18]: eps*hP_op.bracket(X_op)
[18]: ((-8*x^5*y)*hbar^4*eps)*(ddx ⊗ ddy) + (8*x^5*y*hbar^4*eps)*(ddy ⊗ ddx)
This justifies our remark.
The sunflower graph is a derivation, hence its Hochschild differential vanishes and a gauge
transformation T = id+εX with this graph at h̄3 does not affect the star-product at h̄3,
but it will affect ⋆ at order h̄4 — indeed, by adding the Gerstenhaber bracket of X not
with the usual multiplication at h̄0 but with the Poisson structure P in the next order
h̄1. This is how the deformation is realized by a gauge transformation:
[19]: T_op = PA.identity_operator() + eps*X_op
[20]: T_inverse_op = PA.identity_operator() - eps*X_op
[21]: %time star4_gauged_op = T_inverse_op.insertion(0, star4_op.insertion(0, T_op).
↪→insertion(1, T_op))
CPU times: user 15.7 s, sys: 40 ms, total: 15.7 s
Wall time: 15.7 s
[22]: star4_gauged_op - star4_op
[22]: ((-8*x^5*y)*hbar^4*eps)*(ddx ⊗ ddy) + (8*x^5*y*hbar^4*eps)*(ddy ⊗ ddx)
Three outputs in this subsection are identical.
9.2. POISSON-TRIVIAL DEFORMATION AND THE GAUGE TRANSFORM IN
TERMS OF GRAPHS 207
9.2 Poisson-trivial deformation and the gauge
transform in terms of graphs
Claim. The gauge transformation of the star-product from the previous subsection (in
the above, restricted to a particular Poisson structure P ) is produced by using Kontsevich
graphs (likewise, containing a copy of this Poisson structure P in every aerial vertex).
Now, the task is to solve —for the gauge and Leibniz graph coefficients— the equation,
⋆′ − def⋆ = ε[[h̄P,X((h̄P )⊗3)]] = ε[h̄P, T3((h̄P )⊗3)]G + ♢(P, [[P, P ]]) mod ō(h̄4) mod ō(ε).
First, let us generate the gauge graphs:
[23]: len(KGB.graphs(1,3))
[23]: 4
Gauge transformation in terms of graphs:
[24]: G = FGC(FormalityGraph(1,0,[])) + FGC([(var('g13_{}'.format(k)),g) for k,g in␣
↪→enumerate(KGB.graphs(1,3))])
[25]: #G.show()
[26]: G_inverse = FGC(FormalityGraph(1,0,[])) - G.homogeneous_part(1, 3, 6)
Let us check that G_inverse is the inverse of gauge transformation G modulo ō(h̄4).
[27]: G.insertion(0, G_inverse, max_num_aerial=4)
[27]: 1*FormalityGraph(1, 0, [])
[28]: G_inverse.insertion(0, G, max_num_aerial=4)
[28]: 1*FormalityGraph(1, 0, [])
[29]: %time star4_gauged = G_inverse.insertion(0, star4.insertion(0, G, max_num_aerial=4).
↪→insertion(1, G, max_num_aerial=4), max_num_aerial=4)
CPU times: user 1.25 s, sys: 4 ms, total: 1.26 s
Wall time: 1.26 s
The gauge transformation adds 72 terms to the star-product modulo ō(h̄4):
[30]: len(star4_gauged - star4)
[30]: 72
At h̄3 the gauge transformation amounts to adding the Hochschild differential of the
terms in G at h̄3:
[31]: (star4_gauged - star4).homogeneous_part(2,3,6) == G.homogeneous_part(1,3,6).
↪→hochschild_differential()
208 CHAPTER 9. GRAPH COMPLEX ACTION ON STAR PRODUCTS
[31]: True
At h̄4 the gauge transformation amounts to adding the Hochschild differential of the
terms in G at h̄4 and the Gerstenhaber bracket [wedge, terms in G at h̄3], where wedge
stands at h̄1 in ⋆:
[32]: wedge = star4.homogeneous_part(2,1,2); wedge
[32]: 1*FormalityGraph(2, 1, [(2, 0), (2, 1)])
[33]: (star4_gauged - star4).homogeneous_part(2,4,8) == G.homogeneous_part(1,4,8).
↪→hochschild_differential() + wedge.gerstenhaber_bracket(G.homogeneous_part(1,3,6))
[33]: True
Now, before generating any Leibniz graphs for ♢, and actually instead of doing that, let
us inspect that the equation [[P,X]] = [P, T3]G has a solution T3 in terms of gauge graphs:
[34]: obstruction = wedge.schouten_bracket(FGC(sunflower)) - (star4_gauged - star4)
eqns = [c == 0 for c,g in obstruction]
solve(eqns, [g for (g,_) in G.homogeneous_part(1,3,6)])
[34]: [[g13_0 == 0, g13_1 == 0, g13_2 == 0, g13_3 == -2]]
But what is the gauge graph (on 3 aerial vertices and 1 sink) whose coefficient is g13_3?
[35]: KGB.graphs(1,3)[3]
[35]: FormalityGraph(1, 3, [(1, 2), (1, 3), (2, 1), (2, 3), (3, 0), (3, 2)])
We observe that the sought-for gauge tranformation T = id+h̄3T mod ō(h̄43 ) of the star-
product is completely determined by the “sunflower” graph that gave us the vector field
X for a Poisson-trivial deformation of the bracket P inside the star-product.
In particular, the formula of gauge transformation for the specific Poisson bi-vector P =
x2y ξ1ξ2 (see the Example in §9.1) is obtained by evaluating the sunflower graph at a
copy of P in each aerial vertex.
9.3 How the tetrahedral flow deforms the
star-product
Last but not least, we recall the gauge construction from the previous paragraph, now
taking the Kontsvich tetrahedral flow to deform the Poisson bracket. We keep in mind
that there is no universal mechanism (over all affine manifolds in all dimensions at once)
for the Kontsevich tetrahedral flow to be Poisson-trivial in terms of graphs. So the graph
equation to solve (for the gauge and Leibniz graph coefficients) is this:
′− def⋆ ⋆ = εQ ⊗4 3 3tetra((h̄P ) ) = εh̄ dH(T3)+εh̄ [h̄P, T3]G+εh̄4dH(T4)+♢(P, [[P, P ]]) mod ō(h̄4) mod ō(ε).
First let us define the tetrahedral flow in terms of Kontsevich graphs (see [27] and [6]):
9.3. HOW THE TETRAHEDRAL FLOW DEFORMS THE STAR-PRODUCT 209
[36]: Q_tetra = FGC.element_from_kgs_encoding("""h^4:
2 4 1 0 1 2 4 2 5 2 3 1
2 4 1 0 3 1 4 2 5 2 3 -3
2 4 1 0 3 4 5 1 2 2 4 -3""")
[37]: Q_tetra.show()
[38]: Q_tetra
[38]: (-1)*FormalityGraph(2, 4, [(2, 4), (2, 5), (3, 2), (3, 5), (4, 3), (4, 5), (5, 0), (5,
1)]) + (-3)*FormalityGraph(2, 4, [(2, 3), (2, 5), (3, 4), (3, 5), (4, 1), (4, 2), (5,
0), (5, 4)]) + (-3)*FormalityGraph(2, 4, [(2, 3), (2, 4), (3, 4), (3, 5), (4, 1), (4,
5), (5, 0), (5, 2)])
[39]: star4_tetra_deformed = star4 + Q_tetra
To generate the gauge transformation, we remember the four gauge graphs for T3 on three
aerial vertices, and we generate 60 new gauge graphs on four aerial vertices:
[40]: len(KGB.graphs(1,4))
[40]: 60
[41]: G4 = FGC(FormalityGraph(1,0,[])) + FGC([(var('g13_{}'.format(k)),g) for k,g in␣
↪→enumerate(KGB.graphs(1,3))]) + FGC([(var('g14_{}'.format(k)),g) for k,g in␣
↪→enumerate(KGB.graphs(1,4))])
[42]: G4_inverse = FGC(FormalityGraph(1,0,[])) - G.homogeneous_part(1,3,6) - G.
↪→homogeneous_part(1,4,8)
[43]: %time star4_gauged4 = G4_inverse.insertion(0, star4.insertion(0, G4,␣
↪→max_num_aerial=4).insertion(1, G4, max_num_aerial=4), max_num_aerial=4)
CPU times: user 1.66 s, sys: 4 ms, total: 1.66 s
Wall time: 1.66 s
Generate Leibniz graphs on 2 sinks and 3 aerial vertices:
[44]: from gcaops.graph.formality_graph_basis import LeibnizGraphBasis
LGB = LeibnizGraphBasis(positive_differential_order=True)
210 CHAPTER 9. GRAPH COMPLEX ACTION ON STAR PRODUCTS
[45]: L23 = FGC([(var('l{}'.format(k)),g) for k, g in enumerate(LGB.graphs(2,3))])
[46]: len(L23)
[46]: 60
Expand Leibniz graphs into Kontsevich graphs built of wedges:
[47]: stick = FGC(FormalityGraph(0,2,[(0,1)])); stick
[47]: 1*FormalityGraph(0, 2, [(0, 1)])
[48]: L23_expanded = sum(L23.insertion(k,stick,max_out_degree=2) for k in [2,3,4])
[49]: len(L23_expanded)
[49]: 235
Try solving the system of linear algebraic equations for the coefficients of all the gauge
and Leibniz graphs:
[50]: obstruction = Q_tetra - ((star4_gauged4 - star4) + L23_expanded)
eqns3 = [c for c,g in obstruction.homogeneous_part(2,3,6)]
eqns4 = [c for c,g in obstruction.homogeneous_part(2,4,8)]
solve(eqns3 + eqns4,
[l for (l,_) in L23] + \
[g for (g,_) in G.homogeneous_part(1,3,6)] + \
[g for (g,_) in G.homogeneous_part(1,4,8)])
[50]: []
No solution. Indeed, on the one hand the “tetrahedron on top of wedge” graph Γ in
Qtetra can only be affected by the “tetrahedron with one sink” gauge graph γ at h̄3 with
coefficient g13_0 (and Γ cannot be affected by any Leibniz graph, because the contraction
of any edge in Γ results in a zero graph); hence the coefficient g13_0 of γ must be −1:
[51]: eqns4[0]
[51]: -g13_0 - 1
But on the other hand the Hochschild differential of the “tetrahedron with one sink”
graph γ is nonzero, and shows up at h̄3, which forces g13_0 to be zero:
[52]: solve(eqns3, [g for (g,_) in G.homogeneous_part(1,3,6)])
[52]: [[g13_0 == 0, g13_1 == 0, g13_2 == 0, g13_3 == r1]]
So the deformation of the star product modulo ō(h̄4) given by the tetrahedron cannot be
realized as a gauge transformation in terms of graphs.
List of references
[1] Peter Banks, Erik Panzer, and Brent Pym. Multiple zeta values in deformation
quantization. Invent. Math., 222(1):79–159, 2020.
[2] François Bayen, Moshé Flato, Christian Frønsdal, André Lichnerowicz, and Daniel
Sternheimer. Deformation theory and quantization. I. Deformations of symplectic
structures. Ann. Physics, 111(1):61–110, 1978.
[3] Nabiha Ben Amar. K-star products on dual of Lie algebras. J. Lie Theory, 13(2):329–
357, 2003.
[4] Nabiha Ben Amar. A comparison between Rieffel’s and Kontsevich’s deformation
quantizations for linear Poisson tensors. Pacific J. Math., 229(1):1–24, 2007.
[5] Anass Bouisaghouane. The Kontsevich tetrahedral flow in 2D: a toy model. Preprint
arXiv:1702.06044 [math.DG] — 6 -p., 2017.
[6] Anass Bouisaghouane, Ricardo Buring, and Arthemy V. Kiselev. The Kontse-
vich tetrahedral flow revisited. J. Geom. Phys., 119:272–285, 2017. Preprint
arXiv:1608.01710 [q-alg] — 29 p.
[7] Francis Brown. Mixed Tate motives over Z. Ann. of Math. (2), 175(2):949–976,
2012.
[8] Ricardo Buring and Arthemy V. Kiselev. Universal cocycles and the graph complex
action on homogeneous Poisson brackets by diffeomorphisms. Physics of Particles
and Nuclei Letters, 17(5):707–713, 2020. (Proc. International workshop SQS’19 on
Supersymmetries and Quantum Symmetries, 26–31 August 2019, Yerevan, Armenia)
Preprint arXiv:1912.12664 [math.SG] — 8 p.
[9] Ricardo Buring and Arthemy V. Kiselev. The expansion ⋆ mod ō(h̄4) and computer-
assisted proof schemes in the Kontsevich deformation quantization. Exp. Math. (in
press), 31(3 or 4), 2022. Preprint arXiv:1702.00681 [math.CO] — 77 p.
[10] Ricardo Buring, Arthemy V. Kiselev, and Nina J. Rutten. Infinitesimal deformations
of Poisson bi-vectors using the Kontsevich graph calculus. J. Phys.: Conf. Ser., 965,
Paper 012010, 2018. Proc. XXV Int. conf. ‘Integrable Systems & Quantum Symme-
tries’ (6–10 June 2017, CVUT Prague, Czech Republic). Preprint arXiv:1710.02405
[math.CO] — 12 p.
[11] Ricardo Buring, Arthemy V. Kiselev, and Nina J. Rutten. Poisson brackets symme-
try from the pentagon-wheel cocycle in the graph complex. Physics of Particles and
211
212 LIST OF REFERENCES
Nuclei, 49(5):924–928, 2018. (Proc. International workshop SQS’17 on Supersym-
metries and quantum symmetries, July 31 – August 5, 2017, JINR Dubna, Russia)
Preprint arXiv:1712.05259 [math-ph] — 4 p.
[12] Alberto S. Cattaneo. Formality and star products. In Poisson geometry, deformation
quantisation and group representations, volume 323 of London Math. Soc. Lecture
Note Ser., pages 79–144. Cambridge Univ. Press, Cambridge, 2005. Lecture notes
taken by D. Indelicato.
[13] Willem A. de Kok. The relation between Feynman diagrams and Kontsevich graphs
in non-commutative star products. Master’s thesis, University of Groningen, 2021.
[14] Marc De Wilde and Pierre B.A. Lecomte. Formal deformations of the Poisson Lie
algebra of a symplectic manifold and star-products. Existence, equivalence, deriva-
tions. In Deformation theory of algebras and structures and applications (Il Ciocco,
1986), volume 247 of NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., pages 897–960.
Kluwer Acad. Publ., Dordrecht, 1988.
[15] The Sage developers. SageMath, the Sage Mathematics Software System (Version
9.2), 2021. https://www.sagemath.org.
[16] Vasily A. Dolgushev. Stable formality quasi-isomorphisms for Hochschild cochains.
Mém. Soc. Math. Fr. (N.S.), (168):vi + 108 pages, 2021.
[17] Vasily. A. Dolgushev, Christopher L. Rogers, and Thomas H. Willwacher. Kontse-
vich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector
fields. Ann. of Math. (2), 182(3):855–943, 2015.
[18] Chiara Esposito. Formality theory: From Poisson structures to deformation quanti-
zation, volume 2 of SpringerBriefs in Mathematical Physics. Springer, Cham, 2015.
[19] Boris V. Fedosov. A simple geometrical construction of deformation quantization.
J. Differential Geom., 40(2):213–238, 1994.
[20] Giovanni Felder and Boris Shoikhet. Deformation quantization with traces. Lett.
Math. Phys., 53(1):75–86, 2000. Preprint arXiv:math/0002057 [math.QA] — 14 p.
[21] Giovanni Felder and Thomas Willwacher. On the (ir)rationality of Kontsevich
weights. Int. Math. Res. Not. IMRN, (4):701–716, 2010. Preprint arXiv:0808.2762
[math.QA] — 11 p.
[22] Murray Gerstenhaber. The cohomology structure of an associative ring. Ann. of
Math. (2), 78:267–288, 1963.
[23] Murray Gerstenhaber. On the deformation of rings and algebras. Ann. of Math. (2),
79:59–103, 1964.
[24] Hilbrand J. Groenewold. On the principles of elementary quantum mechanics. Phys-
ica, 12:405–460, 1946.
[25] Gerhard P. Hochschild. On the cohomology groups of an associative algebra. Ann.
of Math. (2), 46:58–67, 1945.
LIST OF REFERENCES 213
[26] Maxim Kontsevich. Formal (non)commutative symplectic geometry. In The Gelfand
Mathematical Seminars, 1990–1992, pages 173–187. Birkhäuser Boston, Boston,
MA, 1993.
[27] Maxim Kontsevich. Formality conjecture. In Deformation theory and symplectic
geometry (Ascona, 1996), volume 20 of Math. Phys. Stud., pages 139–156. Kluwer
Acad. Publ., Dordrecht, 1997.
[28] Maxim Kontsevich. Deformation quantization of Poisson manifolds. Lett. Math.
Phys., 66(3):157–216, 2003.
[29] Maxim Kontsevich. XI Solomon Lefschetz Memorial Lecture Series: Hodge struc-
tures in non-commutative geometry. Morfismos, 11(2):1–32, 2007. Notes by Ernesto
Lupercio.
[30] Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke. Poisson structures,
volume 347 of Grundlehren der mathematischen Wissenschaften. Springer, Heidel-
berg, 2013.
[31] Luen-Chau Li and Serge Parmentier. Nonlinear Poisson structures and r-matrices.
Comm. Math. Phys., 125(4):545–563, 1989.
[32] Brendan McKay and Dror Bar-Natan. Graph Cohomology - An Overview and
Some Computations. Preprint https://www.math.toronto.edu/~drorbn/papers/
GCOC/GCOC.ps — 13 p., 2001.
[33] Nina Rutten and Arthemy V. Kiselev. The defining properties of the Kontsevich
unoriented graph complex. Journal of Physics: Conference Series, 1194:1–10, 2019.
Paper 012095.
[34] Hermann Weyl. Gruppentheorie und Quantenmechanik. Wissenschaftliche Buchge-
sellschaft, Darmstadt, second edition, 1977.
[35] Thomas Willwacher. M. Kontsevich’s graph complex and the Grothendieck-
Teichmüller Lie algebra. Invent. Math., 200(3):671–760, 2015.
[36] Thomas Willwacher. The homotopy braces formality morphism. Duke Math. J.,
165(10):1815–1964, 2016.
[37] Thomas Willwacher and Carlo A. Rossi. P. Etingof’s conjecture about Drinfeld
associators. Preprint arXiv:1404.2047 [math.QA] — 47 p., 2014.
[38] Thomas Willwacher and Marko Živković. Multiple edges in M. Kontsevich’s graph
complexes and computations of the dimensions and Euler characteristics. Adv. Math.,
272:553–578, 2015.

Part II
Research articles
Abstract
This part consists of eleven articles that can be read independently. Each article
is introduced with a commentary that discusses its relation to Part I of the dis-
sertation.
Contents of Part II
10 On the Kontsevich ⋆-product associativity mechanism . . . . . . . . . . . . . . . . . . . . . . 217
11 The expansion ⋆ mod ō(h̄4) and computer-assisted proof schemes in the Kon-
tsevich deformation quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12 Formality morphism as the mechanism of ⋆-product associativity: how it works 303
13 The heptagon-wheel cocycle in the Kontsevich graph complex . . . . . . . . . . . . . . 321
14 Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph
calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
15 The Kontsevich tetrahedral flow revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
16 Poisson brackets symmetry from the pentagon-wheel cocycle in the graph com-
plex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
17 The orientation morphism: from graph cocycles to deformations of Poisson
structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
18 The Kontsevich graph orientation morphism revisited . . . . . . . . . . . . . . . . . . . . . . 415
19 Universal cocycles and the graph complex action on homogeneous Poisson
brackets by diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
20 The hidden symmetry of Kontsevich’s graph flows on the spaces of Nambu-
determinant Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Chapter 10
On the Kontsevich ⋆-product
associativity mechanism
This chapter is based on the peer-reviewed journal publication R. Buring and A.V.
Kiselev, Physics of Particles and Nuclei Letters, 14(2), 403–407, 2017. (Preprint
arXiv:1602.09036 [q-alg] – 4 p.)
Commentary. In reference to Part I of the dissertation, the material of this chapter is
used in Chapter 1. The typo in the definition of If is corrected in Chapter 12. The
associativity mechanism from this chapter is explained in Chapter 12.
217
On the Kontsevich ⋆-product associativity mechanism
R. Buring∗, A. V. Kiselev∗,§
Abstract
The deformation quantization by Kontsevich is a way to construct an associative non-
commutative star-product ⋆ = ×+ℏ { , }P + ō(ℏ) in the algebra of formal power series
in ℏ on a given finite-dimensional affine Poisson manifold: here × is the usual multi-
plication, { , }P 6= 0 is the Poisson bracket, and ℏ is the deformation parameter. The
product ⋆ is assembled at all powers ℏk⩾0 via summation over a certain set of weighted
graphs with k+2 vertices; for each k>0, every such graph connects the two co-multi-
ples of ⋆ using k copies of { , }P . Cattaneo and Felder interpreted these topological
portraits as genuine Feynman diagrams in the Ikeda–Izawa model for quantum gravity.
By expanding the star-product up to ō(ℏ3), i.e., with respect to graphs with at
most five vertices but possibly containing loops, we illustrate the mechanism Assoc =
♦ (Poisson) that converts the Jacobi identity for the bracket { , }P into the associativity
of ⋆.
Denote by × the multiplication in the commutative associative unital algebra C∞(Nn → R)
of scalar functions on a smooth n-dimensional real manifold Nn. Suppose first that a non-
commutative deformation ⋆ = ×+O(ℏ) of × is still unital (f ⋆1 = f = 1⋆f) and associative,
(f ⋆ g) ⋆ h = f ⋆ (g ⋆ h) for f, g(, h ∈ C
∞(Nn))∣[[ℏ]]. By taking 3! = 6 copies of the associati-vity equation for the star-product ⋆, we infer that the skew-symmetric part of the leading
deformation term, {f, g}⋆ := 1ℏ f ⋆ g − g ⋆ f ∣ℏ , is a Poisson bracket.1:=0
Now the other way round: can the multiplication × on a Poisson manifold Nn be de-
formed using the bracket { , }P such that the k[[ℏ]]-linear star-product ⋆ = ×+ℏ { , }P+ ō(ℏ)
stays associative? Kontsevich proved [1] that on finite-dimensional affine2 Poisson manifolds,
this is always possible: from { , }P one obtains the∑bi-differential terms Bk(·, ·) at all powersof ℏk⩾0 in the formal series for ⋆. This associative unital ⋆-product was constructed in [1]
using a pictorial language: the operators B = Γk {Γ}w(Γ) × Bk (·, ·) are encoded by the
weighted oriented graphs Γ with k + 2 vertices and 2k edges but without tadpoles or multi-
ple edges; in every such Γ, there are k internal vertices (each of them is a tail for two edges)
and 2 sinks (no issued edges). The Poisson bracket { , }P w
i j ∑ith coefficients P ij(u) at u ∈ Nnprovides the “building block” ∧ = ←−− • −−−→ in which n
Left Right i,j=1
is implicit and the vertex
∗Johann Bernoulli Institute for Mathematic∑s & Computer Science, University of Groningen, P.O. Box 407,9700 AK Groningen, The Netherlands. § Partially supported by JBI RUG project 103511 (Groningen).1The left-hand side of the Jacobi identity ⟳{{f, g}⋆, h}⋆ = 0 is an obstruction to the associativity of
the star-product: whenever the Jacobi identity is violated, one cannot have that (f ⋆ g) ⋆ h = f ⋆ (g ⋆ h).
2On affine manifolds Nn, the only shape of coordinate changes is ũ = A ·u+ c⃗. Yet no loss of generality
occurs if the space Nn is the fibre in an affine bundle π of physical fields {u = ϕ(x)} over the space-
time Mm 3 x; the Jacobians ∂ũ/∂u = A(x) are then constant over Nn. (The arguments of ⋆ are local
functionals of sections, ϕ ∈ Γ(π)→ k; the ⋆-product is marked by the variational Poisson brackets { , }P on
the jet space J∞(π).) The deformation quantization from [1] is lifted to the gauge field set-up in [2].
218
contains P ij(u). To indicate the ordering of indexes in P ij = −Pji, the out-going edges
are ordered by Left ≺ Right. The edges carry the derivatives ∂i ≡ ∂/∂ui and ∂j ≡ ∂/∂uj,
respectively. Every such derivation acts on the content of the vertex at the arrowhead via
the Leibniz rule (and it does so independently from the other in-coming arrows, if any).3
The weights4 w(Γ) ∈ R of such graphs Γ are given by the integrals over configuration
spaces of k distinct points in the hyperbolic plane H2 (e.g., in its upper half-plane model).5
∑ The associativity postulate for ⋆ yields the infinite system of quadratic algebraic equationsfor the weights w(Γ) of graphs.6 Kontsevich shows [1] that the left-hand side JacP(·, ·, ·) :=
⟳{{·, ·}P , ·}P of the Jacobi identity for { , }P is the only obstruction to the balance As-
soc (f, g, h) := (f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) = 0 at all powers ℏk of the deformation parameter at
once.7 The core question that we address in this note is how the mechanism Assoc = ♦ (Pois-
son) works explicitly, making the star-product ⋆ = ×+ ℏ { , }P + ō(ℏ) associative by virtue
of Jacobi identity for the Poisson bracket { , }P . Expanding the Kontsevich ⋆-product in ℏ
up to ō(ℏ3) and with respect to all the graphs Γi∣ such that w∣(Γi) =6 0, we obtain
8
3For example, ←− −→{f, g} i jP(u) = f ←−− • −−−→ g = (f)∂ ∣ ij ∣i · P (u) · ∂j (g), see (1) above.
Left Right u u
4Willwacher and Felder (2010) conjecture that the weights can be irrational numbers for some graphs.
5The wedge factors within the integrand in the formula for w(Γ) are copies of the kernel of the singular
linear integral operator (d ∗ d)−1 in t(he hy)perbolic geometry of H
2, see [3]. Cattaneo and Felder also
showed that the ⋆-product of two functions f, g ∈∫C∞(Nn → C) a(moun)ts to(the F)eynma(n pa(th integ)ra)l
calculation of the correlation function, f ⋆g (u) = ∞ DXDη f X(0) ×g X(1) × exp
i
ℏS P, [X, η] ,X( )=u
in the Ikeda–Izawa topological open string model on a disk D ' H2 with boundary ∂D 3 0, 1,∞; here
X : D → Nn and η : D → T ∗D ⊗ X∗(T ∗Nn). All details and further references are found in [3, 4]; still let
us remember that within the Ikeda–Izawa model, the perturbative expansions in ℏ run, in particular, over
the graphs with tadpoles (which must be regularized by hand) but at the same time, those path integral
calculations reproduce only the weighted oriented graphs without “eyes” (e.g., as in ← ·⇄· →, see Eq. (1)
above). Because, to the best of our knowledge, the eye-containing graphs Γi such that w(Γi) 6= 0 cannot all
at once be eliminated from the star-product ⋆ via gauge transformations of its arguments and of its output,
see Remark 1 on p. 221 and [1], many graphs in the original construction of ⋆ were not recovered in [3].
Hence there is an open problem to extend or modify the Ikeda–Izawa Poisson σ-model such that in the new
set-up, the correlation functions would expand with respect to all the Kontsevich graphs Γi with w(Γi) 6= 0.
6That system solution is not claimed unique: one is provided by the Kontsevich integrals. Number-theo-
retic properties of those weights were explored by Kontsevich in the context of motives and by Willwacher–
Felder and Garay–van Straten in the context of Riemann ζ-function and Euler Γ-function, respectively.
7Ensuring the associativity Assoc (f, g, h) = 0, the tri-vector JacP(·, ·, ·) is not necessarily (indeed, far not
always! ) evaluated at the three arguments f, g, h of the associator for ⋆.
8Balancing the associativity of a star-product order-by-order up to ō(ℏ3), Penkava and Vanhaecke (1998)
derived a set of weights for the (k + 2)-vertex Kontsevich graphs without loops. Yet no loops are destroyed
in either of the copies of ⋆ when the composition ⋆ ◦ ⋆ is taken; the associativity of loopless star-products is
only a part of the full claim for ⋆. So, we integrate over the configuration spaces of k ⩽ 3 points in H2 for
all the Kontsevich graphs (e.g., with loops).
219
r r r
r r r r r B ( ) 	 “eye”ℏ1 r 
A⋆ = + 
 AU r ℏ2 r  rB+ 
/SwBN r ℏ2 r@Rr r	 r rℏ2 + ?	@Rr + r	@Rg r? + r?L R r?+f f g r 1! f g 2! f g 3 f g f g 6 f g
( rC rrCB r	

J C Ĵ r r r r r
r


	L
ℏ3 r BCr r  r r Rr r
	r @Rr rA r )
 @
+ 
/SwBNCW + ?L R?+ 	?@R r?+r ?r
R?r
	@Rr r r?	+ 	@Rr r
@RAUr 
r	

?+Rr?	@Rr + r	@Rr?
 +
6 f g f g f g f g f g g( r r ) ( rf r rf
g
r rQ  H 	Qsr r	+ r Hj  )
ℏ3 @Rr r	 ℏ3  
+ 	r?Hr@Rjr + r	Hr@Rjr? + Ur?L R r?+ r?L R r?
 + r r	r @Rr?	@R?+ ?r	@Rr? + o(ℏ3). (1)
3 f g f g 6 f g f g f g f g
In every composition ⋆ ◦ ⋆ the sums of graphs act on sums of graphs by linearity; each
incoming edge acts via the Leibniz rule (see above). The mechanism for Assoc (f, g, h) to
vanish is two-step: first, the sums in ⋆◦⋆ are reduced using the antisymmetry of the Poisson
bi-vector P . The output is then redrucred modulo thre r(consequencr esr of) Jacobi identity,
9
r L	@ 	
R @RJacP(f, g, h) = 	@Rr @Rr − rH
r
 Hjr − 	r 	r@Rr = 0. (2)
f g h f g h f g h
For ⋆ given by (1), the associator contains 6 terms at ℏ, 38 terms ∼ ℏ2, and 218 terms ∼ ℏ3.
After the use of P ij = −Pji, we infer that Assoc (f, g, h) starts at ℏ2 with 2/3 times (2).
Next, there are 39 terms at ℏ3; we now examine how their sum A vanishes by virtue of (2)
and its differential consequences.10 O(f trhemr, three wrhich arre the erasierst to)recognize are11
L
2P ij 2Jac (∂ r	@ 	
R @R
3 P if, ∂jg, h) = · r@ r @ r −W r3 	?R R ? W
rHr r
 H −jr 	r = 0, (3)? W	r@Rr
as well as 2P ij JacP(f, ∂ig, ∂jh) = 0 and 2P ij JacP(∂if, g, ∂jh) = 0. So, there remain 303 3
terms which vanish via (2) in a way rm#ore inrtricat er t#han (3).r It isr c#lear thatHAj r  Hj Hj riP ij r 	r@ r iASf := ∂j JacP(∂if, g, h) =A"UA	@R @!−AR "AU r
r
Lr	
 AR r i− r r@RH r
 Hj!A"UA	 	@Rr!= 0. (4)
Working out the Leibniz rule in (4), we collect the graphs according to the number of
derivatives falling on each of (f, g, h). The edge j- provides the differential orders12 (3, 1, 1),
(2, 2, 1), (2, 1, 2), and (2, 1, 1) twice.∑Likewise, we see (1, 1, 1) in (2) and (2, 2, 1) in (3).
Lemma. A tri-differential operator IJK|I|,|J |,|K|⩾0 c ∂I ⊗∂∑J ⊗∂K vanishes identically iff allits coefficients vanish: cIJK = 0 for every triple (I, J,K) of multi-indices; here ∂ = ∂α1L 1 ◦· · ·◦
∂αnn for a multi-index L = (α1, . . . , αn). Moreover, the sums IJK|I|=i,|J |=j,|K|=k c ∂I⊗∂J⊗∂K
are then zero for all (i, j, k); in a vanishing sum X of graphs, we denote by Xijk its vanishing
restriction13 to a fixed differential order (i, j, k).
9By default, the L ≺ R edge ordering equals the left ≺ right direction in which edges start on these pages.
10Within the variational geometry of Poisson field models (cf. [2]), a tiny leak of the associativity for ⋆
may occur, if it does at all, only at orders ℏ⩾4 because at most one arrow falls on JacP(·, ·, ·) in the balance
Assoc (f, g, h) = ō(ℏ3). But unlike the always vanishing first variation of a homologically trivial functional
JacP(·, ·, ·) ∼= 0, its higher-order variations can be nonzero.
11We use the Einstein summation convention; a sum over all indices is also implicit in the graph notation.
12In fact, the double edge to f contributes with zero at (3, 1, 1) due to the skew-symmetry Pij = −Pji.
13For example, relation (3) is the consequence of (4) at order (2, 2, 1); restriction of (4) to (2, 1, 1) yields
220
The Poisson(bi-vector comp)onents#P ij carn also se#rve as arrgu me#nts ofr the J acobiator:14
I := ∂ Jac (P ijP , g, h) ∂ f = r r	@ − r	
R − @Rrf j i " = 0. 	r@( ) Rr
@R
j r
?
 !"rH r ( 
r

 Hjr!"	r 	r@Rj r? j   ?r ) !
Likewise, I ijg := ∂i JacP(f,P , h) ∂ g = 0 and I := ∂ Jac (f, g,P ijj h i P ) ∂jh = 0. It is the
expansion of If , Ig, Ih via the Leibniz rule that produces the graphs with “eyes”. It also
yields an order (1, 1, 1) differential operator on (f, g, h) which cannot be obtained from (4).
Claim. The sum A of 39 terms at ℏ3 in Assoc (f, g, h) vanishes by virtue of restriction of
Sf , Sg, Sh and If , Ig, Ih to the orders (i, j, k) that are present in . Indeed, we have15
[3]
A A =
2 [3](S 2f )221, A122(= (Sg)122, and A) 221[3]212=− 2(Sh)212, see (3). Finally, we( deduce that )[8]A 13 3 3 111= (I6 f −[9]
Ih)
1 1 1
111, A112= If + Ig − Sh ,
[4]
A =1121 (If − Ih)121, and
[9]
A = 1S − 1211 f I − 1I . The6 6 3 112 3 3 6 g 6 h 211
total number of terms which we thus eliminate equals (3 + 3 + 3) + 8 + 9 + 4 + 9 = 39. □
Remark 1. The deformation quantization is a gauge t(heory: each argument • of ⋆marks)its
gauge class [•] und[er thr e linr ear mrapsr t : • →7 r[•] = •+ℏr I
∅ ∂ ∂ (P ij)≡0ו+I⟳ ∂ P ij ∂ (•) +
(0) r r i j* Y * r r i rj ]+ (1) sr ≡0 (2) sr (3) r

Swr≡0 (4)2 3 r

Swr (5) +r/SSwr≡0 (6) Z~j (7) 
ℏ I A 3t B r r S
 + ℏ I 6Swt9 + I Swt9 + I A t
 + I A t
 + I A 3
 + I B Y
 + I r
 wSrAU
 ? AU
 AU
 AUt
 BN t?
 JĴ?t


(α)
+o(ℏ3), where the constants I ∈ k can be arbitrary16 and t is fo(rmally inve)rtible over k[[ℏ]].
In turn, the star-products are gauged17 by using t: f ⋆′ g := t−1 t(f) ⋆ t(g) . This degree of
freedom extends the uniqueness problem for Kontsevich’s solution ⋆ of Assoc (f, g, h) = 0.
Namely, not the exact balance of power series but an equ[ivalence][=] of gauge c[lasses (u]p to
unrelated transformations at all steps) can be sought in [f ] ⋆ [g] ⋆ [h] [=] [f ] ⋆ [g] ⋆ [h] .
Remark 2. Each graph Γ in (1) encodes the polydifferential operator of scalar arguments
in a coordinate-free way. The Jacobians ∂u/∂ũ of affine mappings appear on the edges
but then they join the content ℏP ij of internal vertices at the arrowtails,18 forming P̃αβ
from P ij( .)Independent from u ∈ N
n, these Jacobians stay invisible to all in-coming arrows
(if any). So, the operator given by a graph Γ with ℏP(u) in its vertices is equal to the one for
ℏP̃ ũ(u) there. This reasoning works for the variational Poisson brackets { , }P on J∞(π)
for affine bundles π with fibre Nn over points x ∈ Mm, see [2]. The graphs Γ then yield
local variational polydifferential operators yet the pictorial language of [1] is the same.19
(r r r Xrrz r ) ( Xr rXzr r r ) ( rXz r r )XzRr 	r@ r r  	r@ r r 
 r	
 r r 
AUA r	
 R R @Rr X
r RX
C +  − 
 H + 
 rH r − r r r+ C r XXz@Rr = 0.CW	@R @R 
	@R @R 
 

 Hj 
 

 Hj 
	 	@R CW	 	r@Rr
Similarly, we have S := Pij∂ Jac (f, ∂ g, h) = 0 and S := Pijg j P i h ∂j JacP(f, g, ∂ih) = 0.
14The three tadpoles produce JacP(∂ Piji , g, h) ∂jf = 0, which plays its rôle in A111 (see the claim below).
15By using the symbol [m]= we indicate the number m of terms that are eliminated at each step.
16The view [3] on ⋆-products as ℏ-expansions of path integrals shows that the graphs Γi in (1) are genuine
Feynman diagrams for the channel marked by P. The weights w(Γi) integrate over the energy of each
intermediate vertex. Quite naturally, a particle • shares its energy-mass with the interaction carriers P as
it gets coated by them. But no object • can spend more energy on growing its gauge tail than the amount
it actually has; hence every set [•] is bounded in the space of parameters I.
17 +For ex←−ample∣∣, t−→he loop graph at∣∣ℏ2/6 in (1) is gauged∣out by ℏ2p pt(•) = •+ 123Uq , see [1] for further(details.18 )E.g., ′ αβ ′ ←−∂ P̃ ∂ = ∂ ∂ui P̃αβ ∂uj −→ ←− −→α (β i ∂ũα) β ∂j = ∂ · P
ij∣ · ∂ so that {f, g} (u) = {f, g} ũ(u) .
ũ ũ(u) ∂ũ i u j P(u) P̃(ũ(u))
19A sought-for extension of the Ikeda–Izawa topological open st(ring geometry)– namely, its lift from the
Poisson manifolds Nn, { , }P in [3, 4] to the variational set-up J∞(π), { , }P of jet spaces in [2] – is a
mechanism to quantize Poisson field models. This will be the object of another paper.
221
Acknowledgements. A.V.K. thanks the organizers of international workshop SQS’15 (August
3–8, 2015 at JINR Dubna, Russia) for stimulating discussions and partial financial support.
References
[1] Kontsevich M. Deformation quantization of Poisson manifolds. I // Lett. Math. Phys. 2003. V. 66, n. 3.
P. 157–216. arXiv:q-alg/9709040
[2] Kiselev A.V. Deformation approach to quantisation of field models. Preprint IHÉS/M/15/13. Bures-sur-
Yvette: IHÉS, 2015. P. 1–37.
[3] Cattaneo A. S., Felder G. A path integral approach to the Kontsevich quantization formula // Comm.
Math. Phys. 2000. V. 212, n. 3. P. 591–611. arXiv:q-alg/9902090
[4] Ikeda N. Two-dimensional gravity and nonlinear gauge theory // Ann. Phys. 1994. V. 235, n. 2. P. 435–
464. arXiv:hep-th/9312059
222
Chapter 11
The expansion ⋆ mod ō(h̄4) and
computer-assisted proof schemes in
the Kontsevich deformation
quantization
This chapter is based on the peer-reviewed journal publication R. Buring and
A.V. Kiselev, Experimental Math. 31(3) or (4), 54 p., 2022 (in press).
(doi:10.1080/10586458.2019.168046) (Preprint arXiv:1702.00681 [math.CO] – 77
p.) Appendix A.1 in that paper follows the talk given by the dissertant at the work-
shop Symmetries of Discrete Systems and Processes III (3–7 August 2015, Děčín, Czech
Republic); the entire work has been presented by the dissertant at seminars and confer-
ences multiple times (Oxford, Utrecht, Larnaca, Be￿dlewo, etc.).
Commentary. The theory in this chapter is used in Chapters 1, 3, and 9 of Part I. The
theory of this chapter is explained in more detail in the next chapter. The graph encoding
of Kontsevich’s ⋆ mod ō(h̄4) is contained in Appendix B.1.
223
THE EXPANSION ⋆ MOD ō(ℏ4) AND
COMPUTER-ASSISTED PROOF SCHEMES IN
THE KONTSEVICH DEFORMATION QUANTIZATION
R. BURING∗ AND A.V.KISELEV§
Abstract. The Kontsevich deformation quantization combines Poisson dynamics,
noncommutative geometry, number theory, and calculus of oriented graphs. To ma-
nage the algebra and differential calculus of series of weighted graphs, we present
software modules: these allow generating the Kontsevich graphs, expanding the non-
commutative ⋆-product by using a priori undetermined coefficients, and deriving linear
relations between the weights of graphs. Throughout this text we illustrate the as-
sembly of the Kontsevich ⋆-product up to order 4 in the deformation parameter ℏ.
Already at this stage, the ⋆-product involves hundreds of graphs; expressing all their
coefficients via 149 weights of basic graphs (of which 67 weights are now known ex-
actly), we express the remaining 82 weights in terms of only 10 parameters (more
specifically, in terms of only 6 parameters modulo gauge-equivalence). Finally, we
outline a scheme for computer-assisted proof of the associativity, modulo ō(ℏ4), for
the newly built ⋆-product expansion.
Contents
Introduction 225
1. Weighted graphs 229
2. The Kontsevich ⋆-product 237
3. Associativity of the Kontsevich ⋆-product 246
4. Discussion 259
Conclusion 267
References 273
Appendix A. Approximations and conjectured values of weight integrals 276
Appendix B. C++ classes and methods i
Appendix C. Encoding of the entire ⋆-product modulo ō(ℏ4) v
Appendix D. Encoding of the associator of the ⋆-product modulo ō(ℏ4) x
Appendix E. Gauge transformation that removes 4 master-parameters out of 10xvii
Date: 14 March 2018; in original form 20 December 2017, in final form 5 October 2019.
2010 Mathematics Subject Classification. 05C22, 53D55, 68R10, also 05C31, 16Z05, 53D17, 81R60,
81Q30.
Key words and phrases. Associative algebra, noncommutative geometry, deformation quantization,
Kontsevich graph complex, computer-assisted proof scheme, software module, template library.
∗Address: Institut für Mathematik, Johannes Gutenberg–Universität, Staudingerweg 9, D-55128
Mainz, Germany. ∗E-mail: rburing@uni-mainz.de.
§Address: Johann Bernoulli Institute for Mathematics and Computer Science, University of Gronin-
gen, P.O. Box 407, 9700 AK Groningen, The Netherlands. §E-mail: A.V.Kiselev@rug.nl.
224
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 225
Introduction. On every finite-dimensional affine (i.e. piecewise-linear) manifold Nn,
the Kontsevich star-product ⋆ [32] is an associative but not necessarily commutative
deformation of the usual product × in the algebra of functions C∞(Nn) towards a
given Poisson bracket {·, ·}P on Nn (see a∑lso [19, 4, 5]). Specifically, whenever ⋆ =× + ℏ {·, ·}P + ō(ℏ) is an infinitesimal deformation, it can always be completed to an
associative star-product ⋆ = ×+ℏ {·, ·}P + k⩾2 ℏkBk(·, ·) in the space of formal power
series C∞(Nn)[[ℏ]]; this was proven in [32]. An explicit calculation of the bi-linear bi-
differential terms Bk(·, ·) at high orders ℏk is a computationally hard problem. In this
paper we reach the order k = 4 in expansion of ⋆ by using software modules for the
Kontsevich graph calculus, which we presently discuss.
Convenient in practice, the idea from [32] (see also [28, 29, 31]) is to draw every de-
rivation ∂i ≡ ∂/∂xi (with respect to a local coordinate xi on a chart in the Poisson ma-
nifold Nn at hand) as decorated edge i-, so that large different∑ial expressio∣ns become
ori∣∣ented graphs. For example, the Poisson bracket
←−
{f, g} nP(x) = i,j=1(f)∂ ∣ iji ·P (x) ·−→ x
∂j (g) of two functions ∈ ∞ n is depicted by the graph ←−
i P ij −→jf, g C (N ) (f) (g); here
x
P ij is the skew-symmetric matrix of Poisson bracket coefficients and the summation
over i, j running from 1 to the dimensiorn n of N
n is implicit. In these terms, the known
– from [12] – expansion of trhe Kontsevich star-p(rodruct1 looks asrB r r r
fo)llows:2r r r r r r r r r r	 rℏ1 
A ℏ2  B ℏ2 @R r r 	r ℏ2⋆ = + 
 UA + 
/SwBN + ?	@R + 	@Rr ? +g g r

? r?+
f f 1! f g 2! f g 3 f g f g 6 g
( rC r r r r r r
f
 C
	

JB Ĵ r r	 r r	 @Rr rA
ℏ3 r  C rB  r @R?r r?	 @RAUr 
r

r

 )	
+ 
/SwBNCCWr + r? r?+ 	?r @R r?+ r?	@Rr + r	@Rr?+Rr?	@Rr + r	@Rr?
 +
6 f g( f
g
r ) f
g g g g g
r rr rr (
r f fr r f fQ  H rQs 	+ Hj r r )
ℏ3 r@R r r 	 r	 r r rℏ3   r	 @Rr+ ?	 3H@jR + 	@HRjr? + U?rL R r?+ ?rL R r?
 + ?r	@Rr?+ r?	@Rr? +o(ℏ ). (1)
3 f g f g 6 f g f g f g f g
By construction, every oriented edge carries its own index and every internal vertex
(not containing the arguments f or g) is inhabited by a copy of the coefficient matrix
P = (P ij) of the Poisson bracket {·, ·}P . This means that expansion (1) encodes the
analytic formula (
f ⋆ g = f × g + ℏP ij∂ f∂ g + ℏ2 1i j P ijPkℓ∂ ∂ f∂)∂ g +( 1∂ P ijPkℓk i ℓ j ℓ ∂k∂if∂ g2 3 j
− 1∂ P ijPkℓℓ ∂if∂k∂jg − 1∂ P ijℓ ∂jPkℓ∂if∂kg + ℏ3 1P ijPkℓPmn∂3 6 6 m∂k∂if∂n∂ℓ∂jg
1We recall that the expansion ⋆ mod ō(ℏ2) in [32] was gauge-equivalent to the genuine one so that
the two-cycle graph at ℏ2/6 in the first line of above formula (1) was gauged out: see Example 25 on
p. 246 where we explain how this is done.
2The indication L and R for Left ≺ Right, respectively, matches the indices –which the pairs of
edges carry – with the ordering of indices in the coefficients of the Poisson structure contained in the
arrowtail vertex. Note that exactly two edges are issued from every internal vertex in every graph in
formula (1); not everywhere displayed in (1), the ordering L ≺ R in each term is determined from
same object’s expansion (2).
226 R. BURING AND A. V. KISELEV
− 1∂ ∂ P ij∂ ∂ PkℓPmnm ℓ n j ∂ 1 ij kℓif∂kg − P ∂nP ∂ mnℓP ∂k∂6 6 if∂m∂jg
− 1∂ ∂ P ij∂ Pkℓm ℓ n Pmn∂k∂if∂jg − 1∂m∂ℓP ij∂ kℓ mnnP P ∂if∂k∂6 6 jg
+ 1∂n∂ P ijPkℓPmnℓ ∂m∂k∂if∂jg + 1∂ ∂ P ijPkℓPmn∂ f∂ ∂ ∂ g6 6 n ℓ i m k j
+ 1∂ P ijPkℓPmnn ∂m∂k∂ 1 ij kℓ mnif∂ℓ∂jg − ∂nP P P ∂3 3 k∂if∂m∂ℓ∂jg
− 1∂ P ij∂ ∂ PkℓPmn∂ ∂ f∂ g + 1∂ ∂ P ij∂ Pkℓ mnℓ n j m i k n ℓ j P ∂if∂m∂)kg6 6
− 1∂ P ijPkℓ∂ Pmn∂ ∂ f∂ ∂ g − 1∂ P ij∂ PkℓPmnn ℓ k i m j ℓ n ∂k∂if∂m∂ 36 6 jg + ō(ℏ ). (2)
We now see that the language of Kontsevich graphs is more intuitive and easier to
percept than writing formulae. The calculation of the associator Assoc⋆(f, g, h) =
(f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) can also be done in a pictorial way (see section 2.4 on p. 243).
The coefficients of graphs at ℏk in a star-product expansion are given by the Kontsevich
integrals over the configuration spaces of k distinct points in the Lobachevsky plane H,
see [32] and [15]. Although proven to exist, such weights of graphs are very hard to
obtain in practice.3 Much research has been done on deriving helpful relations between
the weights in order to facilitate their calculation [17, 36, 21, 18, 6]. In Example 26 on
p. 250 we explain how expansion (1) modulo ō(ℏ3) was obtained in [12]. The techniques
which were then sufficient are no longer enough to build the Kontsevich ⋆-product
beyond the order ℏ3; clearly, extra mathematical concepts and computational tools
must be developed. In this paper we present the software in which several known
relations between the Kontsevich graph weights are taken into account; we express the
weights of all graphs at ℏ4 in terms of 10 master-parameters. (To be more precise,
the ten master-parameters are reduced to just 6 by taking the quotient over certain
four degrees of gauge freedom in the associative star-product expansions mod ō(ℏ4).)
This paper is aimed to provide much more than a reference to computer programs: it
also contains a synopsis of the proofs for the ideas in the construction, as well as an
explanation of the parts which require computer implementation. Now, the values of
Kontsevich graph weights and, with more input from the work in progress [2, 34], all
the values which specify ⋆ mod ō(ℏ4) are the main result of this paper. These weights
(as well as the ones of higher-order expansion terms) are subject to conjectures and
open problems (see [17, 2]).
This paper contains four chapters. In chapter 1 we introduce the software to encode
and generate the Kontsevich graphs and operate with series of such graphs. In partic-
ular, the coefficients of graphs in series can be undetermined variables. The series are
then reduced modulo the skew-symmetry of graphs (under the swapping of Left⇄ Right
in their construction). Thirdly, a series can be evaluated at a given Poisson structure:
that is, a copy of the bracket is placed at every internal vertex.
Chapter 2 is devoted to the construction of Kontsevich’s ⋆-product: containing a
given Poisson structure in its leading deformation term, this bi-linear operation is not
necessarily commutative but it is required to be associative; hence the coefficients of a
power series for ⋆ must be specified. For example, at order k = 4 of the deformation
parameter ℏ there are 149 parameters to be found. (The actual number of graphs at ℏ4 is
much greater; we here count the “basic” graphs only.) We review a number of methods
3In fact, there are many other admissible graphs, not shown in (1), in which every internal vertex
is a tail for two oriented edges, but the weights of those graphs are found to be zero.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 227
to obtain the weights of Kontsevich graphs; the spectrum of techniques employed ranges
from complex analysis and direct numeric integration [14] to finding linear relations
between such weights by using abstract geometric reasonings. The associativity of
Kontsevich’s ⋆-product is a major source of relations between the graph weights; at ℏ4
such relations are linear because everything is known about the weights up to order
three. We obtain these relations at order four in chapter 3 and we solve that system
of linear algebraic equations for 149 unknowns. The solution is expressed in terms of
only 10 master-parameters, see formula (11) on pp. 267–272.4 It is readily seen that the
final formula (13) on pp. 280–284, in which the ten parameters are assigned specific real
values so that all the coefficients in ⋆ mod ō(ℏ4) are the values of Kontsevich’s integrals,
is the genuine formula of the Kontsevich ⋆-product. Indeed, our formula ⋆ mod ō(ℏ4) is
obtained according to the Proof scheme for Theorem 9 on p. 251 below. We compute
a big system of equations which is satisfied by Kontsevich’s formula: it is constrained
by Lemmas 1–5 (basic identities), Proposition 7 (cyclic weight relations), Methods 1–3
(associativity), and vanishing of some integrands (cf. Appendix A.1). Having solved
this system, we incorporate external input [34, 2] in the form of direct calculation of
Kontsevich’s integrals.
The algebraic system constructed in section 3.1 was obtained by restricting the asso-
ciativity for ⋆ to (a class of) specific Poisson structures. We want however to prove that
for the newly found collection of graph weights, the ⋆-product is associative for every
Poisson structure on all finite-dimensional affine manifolds. For that, in section 3.2 we
design a computer-assisted proof scheme that is independent of the bracket (and of a
manifold at hand). Specifically, in Theorem 12 on p. 256 we reveal how the associator
for Kontsevich’s ⋆-product, taken modulo ō(ℏ4), is factorised via the Jacobiator Jac(P)
or via its differential consequences that all vanish identically for Poisson structures P
on the manifolds Nn. We discover in( particular that su)ch factorisation,
Assoc⋆(f, g, h) = ♢ P , Jac(P), Jac(P) mod ō(ℏ4),
is quadratic and has differential order two with respect to the Jacobiator. For all Poisson
brackets {·, ·} on finite-dimensional affine manifolds NnP our ten-parameter expression
of the ⋆-product does agree up to ō(ℏ4) with previously known results about the values of
Kontsevich graph weights at some fixed values of the ten master-parameters and about
the linear relations between those weights at all values of the master-parameters.5 In an
extensive Discussion on pp. 259–266, we compare (and, again, verify) our result with
other work, namely by Gutt et al [1], Ben Amar [6], Kathotia [21], Willwacher [38],
and Penkava–Vanhaecke [35]. Further discussion of our result is contained in section
4The values of all these ten master-parameters have recently been claimed by Panzer and Pym [34]
as a result of implementation of another technique to calculate the Kontsevich weights: see Table 4 on
p. 280 in Appendix A.2. In particular, the values which we conjecture in Table 3 fully agree with the
exact values suggested in [34]. Based on this external input, the expansion of the Kontsevich ⋆-product
becomes (13) on pp. 280–284.
5From Theorem 12 we also assert that the associativity of Kontsevich’s ⋆-product does not carry on
but it can leak at orders ℏ⩾4 of the deformation parameter, should one enlarge the construction of ⋆ to
an affine bundle set-up of Nn-valued fields over a given affine manifold Mm and of variational Poisson
brackets {·, ·}P for local functionals F,G,H : C∞(Mm → Nn)→ k, see [23, 24, 25, 26] and [27].
228 R. BURING AND A. V. KISELEV
6.2 on p. 61 in the most recent preprint [2]. The following list of insights is gained as a
byproduct of our approach:
• Relations between the Kontsevich graph weights can be obtained by viewing the
⋆-product associator Assoc⋆(P)(f, g, h) = 0 for a Poisson structure P = P(ψ) as
a polydifferential operator on f, g, h,ψ (see §3.1, Method 3). This new technique
(effective by virtue of computer implementation) yields many new relations. In
particular:
• All the weights of graphs at order 3 in the ⋆-product are uniquely determined
(see Example 26 on p. 250) by the associativity equation up to order 4 for Poisson
structure (10) on R3, the elementary Lemmas 1–5, and the cyclic weight relations
up to order 4. This is one instance of:
• Linear relations between only weights of graphs at order n can be obtained (in
an effective, predictable way) from the associativity equation at orders greater
than n (see Remark 11 on p. 247). This is explained using the decomposition
of a polydifferential operator into homogeneous components and the notion of
“prime” Kontsevich graphs.
• The proof of associativity of the ⋆-product at order 4 must involve a second-
order differential consequence of the Jacobi identity (see the second part of
Theorem 12 on p. 256). In particular, a naive jet space extension of Kontsevich’s
star product, where derivatives are replaced by variations, is in general not
associative at order 4 (see Corollary 13 on p. 257).
• The mechanism of vanishing via differential consequences of the Jacobi identity
may start working for the ⋆-product expansion itself (see Theorem 15 on p. 259).
In fact, the order 4 is the first where this may happen. (It could have happened
at order 3, if the weights of graphs were different.)
• So far, from the work of Willwacher (see [38]) it was known that graphs with
two-cycles, or loops, cannot be eliminated all at once from the star-product
by using gauge transformations. At ℏ2, the only such graph can be removed
indeed (see Example 24); at ℏ3 there are four loopful graphs out of 13 graphs
with nonzero coefficients. We discovered a totally unexpected fact (see p. 284
below): at ℏ4, the graphs with loops are dominant: 138 out of 247.
The software implementation [9] consists of a C++ library and a set of command-line
programs. The latter are specified in what follows; a full list of new C++ subroutines
and their call syntaxis is contained in Appendix B. Whenever a command-line program
refers to just one particular function in C++, we indicate that in the text. The current
text refers to version 0.66 of the software. This and future versions are available from
https://github.com/rburing/kontsevich_graph_series-cpp
All data files constructed and referred to in this article (in plain text format, which can
be appreciated independently of the software) are available in the data subdirectory:
https://github.com/rburing/kontsevich_graph_series-cpp/tree/master/data
© The copyright for all newly designed software modules which are presented in
this paper is retained by R.Buring; provisions of the MIT free software license apply.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 229
1. Weighted graphs
In this section we introduce the software to operate with series of oriented graphs.
1.1. Normal forms of graphs and their machine-readable format. As it was
explained in the introduction, we consider graphs whose vertices contain Poisson struc-
tures and whose edges represent derivatives. To be precise, the class of graphs to deal
with is as follows.
Definition 1. Let us consider a class of oriented graphs on m + n vertices labelled 0,
. . ., m+ n− 1 such that the consecutively ordered vertices 0, . . ., m− 1 are sinks, and
each of the internal vertices m, . . ., m + n − 1 is a source for two edges. For every
internal vertex, the two outgoing edges are ordered using L ≺ R: the preceding edge
is labeled L (Left) and the other is R (Right). An oriented graph on m sinks and n
internal vertices is a Kontsevich graph of type (m,n). We denote by Gm,n the set of
all Kontsevich graphs of type (m,n), and by G̃m,n the subset of Gm,n consisting of all
those graphs having neither double edges nor tadpoles.
Example 1. The star-product expansion (1) contains graphs in G̃2,k for 0 ⩽ k ⩽ 3.
Remark 1. The class of graphs which we consider is not the most general type considered
by Kontsevich in [32]. In the construction of the Formality morphism there also appear
graphs with sources for more or fewer (than two) arrows. However, in our approach to
the problem at hand, which is the construction of a ⋆-product expansion that would
be associative modulo ℏk for some k  0, we shall only meet graphs from the class
of Definition 1. Actually, to be more accurate, the Leibniz graphs in Definition 6 on
p. 254 are Kontsevich graphs where some vertices have three outgoing edges; these are
expanded into ordinary Kontsevich graphs (built of wedges) by inserting the Jacobiator
at the tri-valent vertex; see [11] for more details.
Remark 2. There can be tadpoles or cycles in a graph Γ ∈ Gm,n, see Fig. 1.

?r - r RrI
Figure 1. A tadpole and an “eye”.
A Kontsevich graph Γ ∈ Gm,n is uniquely determined by the numbers n and m
together with the list of ordered pairs of targets for the internal vertices. For reasons
which will become clear immediately below, we now consider a Kontsevich graph Γ
together with a sign s ∈ {0,±1}, denoted by concatenation of the symbols: sΓ.
Implementation 1 (encoding). The format to store a signed graph sΓ with Γ ∈ Gm,n
is the integer number m > 0, the integer n ⩾ 0, the sign s, followed by the (possibly
empty, when n = 0) list of n ordered pairs of targets for edges issued from the internal
vertices m, . . ., m + n − 1, respectively. The full format is then (m, n, s; list of
ordered pairs); in plain text we also write m n s <list of ordered pairs>. In the
software, the class KontsevichGraph represents these signed Kontsevich graphs.
230 R. BURING AND A. V. KISELEV
3rR
L@Rr2R
Example 2. The graph ?r	L@Rr has encoding 2 2 1 0 1 0 2.
We recall that to every Kontsevich graph one associates a polydifferential operator by
placing a copy of the Poisson bracket at each vertex. To a signed graph one associates
the polydifferential operator of the graph multiplied by the sign. The skew-symmetry of
the Poisson bracket implies that the same polydifferential operator may be represented
by several different signed graphs, all having different encodings.
Example 3. Taken with the signs in the first row, the graphs in the second row all
represent the same polydifferential operator:
r+1 -1 -1 +1 +1 -1 -1 +13 3 3 3 2Rr rL rR rL r 2 2 2R rR rL rLr@R 2L r R r@Rr2 r r@Rr2 r r@Rr2 r 3 3 3 3R R L L R L L r@Rr Rr L r@Rr @R @RL R r R R r L?	L@R ?	L@R ?	R@R ?	R@R ?	L@R ?	R@Rr ?r	L@Rr r?	R@Rr
0 1 0 2 0 1 2 0 1 0 0 2 1 0 2 0 0 3 0 1 0 3 1 0 3 0 0 1 3 0 1 0
In the third row the target list (for internal vertices 2 and 3, respectively) is written.
We would like to know whether two (encodings of) signed graphs specify the same
topological portrait — up to a permutation of internal vertices and/or a possible swap
L ⇄ R for some pair(s) of outgoing edges. To compare two given encodings of a
signed graph, let us define its normal form. Such normal form is a way to pick the
representative modulo the action of group S nn × (Z2) on the space Gm,n.
Definition 2 (normal form). The list of targets of a graph Γ ∈ Gm,n can be considered
as a 2n-digit integer written in base-(n+m) notation. By running over the entire group
S nn× (Z2) , and by this over all the different re-labelings of Γ, we obtain many different
integers written in base-(n + m). The absolute value |Γ| of Γ is the re-labeling of Γ
such that its list of targets is minimal as a nonnegative base-(n + m) integer. For a
signed graph sΓ, the normal form is the signed graph t|Γ| which represents the same
polydifferential operator as sΓ. Here we let t = 0 if the graph is zero (see Remark 3
below).
Example 4. The minimal base-4 number in the third column of Example 3 is 0 1 0 2.
Hence the absolute value of each of the graphs in Example 3 is the first graph. The
normal form of each of the signed graphs in Example 3 is the first graph taken with the
appropriate sign ±1; the encodings of the normal forms are then 2 2 ±1 0 1 0 2.
This normal form is implemented in software as the method normalize() of the class
KontsevichGraph. By running over the entire symmetry group, it will be inefficient
when the number of vertices is large. In the future this method could be replaced by a
more efficient one, without requiring changes to the rest of the code.
Remark 3. The graphs Γ ∈ Gm,n for which the associated polydifferential operator
vanishes, by being equal to minus itself, are called zero. This property can be de-
tected during the calculation of the normal form of a signed graph. One starts with
the encoding of a signed graph. Obtain a “sorted” encoding (representing the same
polydifferential operator) by sorting the outgoing edges in every pair in nondecreasing
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 231
order: each swap L ⇄ R entails a reversion of the sign. Now run over the group Sn
of permutations of the internal vertices in the graph at hand, relabeling those vertices.
Should the list of targets in the sorted encoding of a relabeling be equal to the list of
targets in the original sorted encoding, but the sign be opposite, then the graph is zero.
We will see in Chapter 2 (specifically, in Lemma 2 on p. 238) that the weights of these
graphs also vanish, this time by the anticommutativity of certain differentials under
the wedge product.
Example 5. Consider the graph
4 r R-r3
@ B
L@  B
r Rr B .
	2@RBBN r
0 1
with the encoding 2 3 1 0 1 0 1 2 3. For the identity permutation we obtain the
initial sorted encoding 2 3 1 0 1 0 1 2 3 (it was already sorted). For the permu-
tation 2 ⇆ 3 we obtain the encoding 2 3 1 0 1 0 1 3 2; upon sorting the pairs
it becomes 2 3 -1 0 1 0 1 2 3. The list of pairs coincides with the initial sorted
encoding but the sign is opposite; hence the graph is zero.
The notion of normal form of graphs allows one to generate lists of graphs with
different topological portraits (e.g., Kontsevich graph series, see section 1.2 below) by
using the following algorithm. Initially, the list of generated graphs is empty. For every
possible encoding (according to Implementation 1) in a run-through, its normal form
with sign +1 or 0 is added to the list if it is not contained there (otherwise, the offered
encoding is skipped).
Implementation 2. To generate all the Kontsevich graphs with m sinks and n internal
vertices in G̃m,n (without tadpoles or double edges), the command is
> generate_graphs n m
The procedure lists all such graphs (one per line) in the standard output. The second
argument m may be omitted: the default value is m = 2.
Similarly, to generate only normal forms (with sign +1 or 0), the call is
> generate_graphs n m --normal-forms=yes
The optional argument --with-coefficients=yes indicates that (numbered) unde-
termined coefficients should be listed alongside the graphs (the default is no); see §1.2.
(Accordingly, see KontsevichGraph::graphs in Appendix B.)
Example 6. The Kontsevich graphs in G̃m,n with one internal vertex
> generate_graphs 1
2 1 1 0 1
2 1 1 1 0
consist of the wedge with its two different labellings. We can verify that the number of
Kontsevich graphs on n internal vertices and two sinks is (n(n+ 1))n:
232 R. BURING AND A. V. KISELEV
> generate_graphs 2 | wc -l
36
> generate_graphs 3 | wc -l
1728
> generate_graphs 4 | wc -l
160000
> generate_graphs 5 | wc -l
24300000
Here, “| wc -l” counts the number of lines in the output (wc is from GNU coreutils).
Let us remember that while a list of graphs is generated, more options can be chosen
to restrict the graphs: e.g., only prime graphs can be taken into account, graphs of
which the mirror-reflection is already on the list can be skipped, and/or only those
graphs in which each sink receives at least one arrow can be taken. The purpose and
implementation of these options will be explained in the next chapter (see p. 239 below).
1.2. Series of graphs: file format. We now specify how formal power series expan-
sions of graphs are implemented in software. Denote by ℏ the formal parameter; in
machine-readable format, a power series expansion in ℏ is a list of coefficients of ℏk,
k ⩾ 0. The coefficients are formal sums of signed graphs (see KontsevichGraphSum in
Appendix B) in which the coefficients can be of any type, e.g.,
• integer or floating point numbers (e.g., 0.333),
• rational numbers (e.g., 1/3),
• undetermined variables (resp., OneThird).
To be precise, the library [9] contains the class KontsevichGraphSeries which depends
on a template parameter T; it specifies the type of all the coefficients of graphs in
the series. In the command-line programs, the external type GiNaC::ex, which is the
expression type of the GiNaC library [3], allows all of the above values (and combinations
of them). Hence a series under study can contain coefficients of all types at once; the
coefficient of a graph itself can be a sum of different types of objects (e.g., p16+ 0.25).
Implementation 3 (series encoding). In the file format for formal power series expan-
sions, two kinds of lines are possible: either
h^k:
or (separated by whitespace)
<encoding of a graph> <coefficient>
The precision of the formal power series expansion is indicated by the highest k
occurring in lines of the form “h^k:”. Hence one can control this bound by adding such
a line with a high k at the end of the file.
Example 7. The Kontsevich ⋆-product (see §2) is a graph series given up to the second
order in the deformation parameter ℏ in the file star2w.txt which reads
h^0:
2 0 1 1
h^1:
2 1 1 0 1 1
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 233
h^2:
2 2 1 0 1 0 1 1/2
2 2 1 0 1 0 2 w_2_1
2 2 1 0 1 1 2 w_2_2
2 2 1 0 3 1 2 w_2_3
Implementation 4. The substitution of undetermined coefficients by their actual val-
ues, as well as re-expression of indeterminates via other such objects, is done by using
the program
> substitute_relations <graph-series-file> <subsitutions-file>
Its command line arguments are two file names: the first file contains the series and the
second file consists of a list of substitutions (one per line), each substitution written in
the form
<variable>==<what it is set equal to>
The command line program sends the series with all those substitutions to the standard
output.
Example 8. The values of the unknowns in Example 7 are written in weights2.txt:
w_2_1==1/3
w_2_2==-1/3
w_2_3==-1/6
Whence the star-product is given modulo ō(ℏ2) as follows:
$ substitute_relations star2w.txt weights2.txt > star2.txt
$ cat star2.txt
h^0:
2 0 1 1
h^1:
2 1 1 0 1 1
h^2:
2 2 1 0 1 0 1 1/2
2 2 1 0 1 0 2 1/3
2 2 1 0 1 1 2 -1/3
2 2 1 0 3 1 2 -1/6
Here cat from GNU coreutils is used to display the file.
In practice one may encounter graph series containing many graphs and undetermined
coefficients. To split a graph series into parts, the following command is helpful.
Implementation 5. To extract the part of a graph series proportional to a given
expression, use the call
> extract_coefficient <graph-series-file> <expression>
In the standard output one obtains a modification of the original graph series: each
graph coefficient c is now replaced by the coefficient of <expression> in c. If the
coefficient of <expression> in c is identically zero, then the graph is skipped. The
special value <expression> = 1 yields the constant part of the graph series (all the
undetermined variables in the input are set to zero).
234 R. BURING AND A. V. KISELEV
Example 9. From the file in Example 8, we extract the part proportional to w_2_1:
> extract_coefficient star2w.txt w_2_1
h^0:
h^1:
h^2:
2 2 1 0 1 0 2 1
It is just one graph.
1.3. Reduction modulo skew-symmetry. Let us recall that for every internal vertex
in a Kontsevich graph, the pair of out-going edges is ordered by the relation Left≺ Right
and by a mark-up of those two edges using L and R. By construction, the coefficients of
a graph series are sums of signed graphs; each signed graph is specified by its encoding,
see Implementation 1 on p. 229 above. Starting from the vector space of formal sums
of signed graphs with real coefficients, we pass to its quotient. Namely, we postulate
that graphs which differ only by their internal vertex labeling are equal. Further, we
proclaim that every reversal of the edge order in any pair (from the same internal
vertex) entails the reversion of the graph sign. Lastly, we introduce the relations
<coeff> · (sign)Γraph = <sign · coeff> · (+1)Γraph,
for each signed graph (sign)Γraph with any coefficient <coeff>.
The combined effect of these relations is that each sum of signed graphs may be
reduced to a sum of normal forms (see Definition 2) with sign +1. Recall that the
ordering mechanism Left ≺ Right creates graphs that equal zero because they are equal
to minus themselves (see Remark 3 and Example 5).
Remark 4. To avoid such comparison of graphs with zero all the time and so, to increase
efficiency, every signed graph is brought to its normal form as soon as it is constructed.
It is this moment when zero graphs acquire zero signs.
The algorithm to reduce a sum of graphs modulo skew-symmetry runs as follows.
For the starting graph or every next graph in the list, its sign (if nonzero) is set equal
to +1 and its coefficient is modified, if necessary, by using the rule
<coeff> · <sign> = <sign · coeff> · (+1). (3)
Every graph with sign 0 is removed. Then the graph at hand (in its normal form, times
a coefficient) is compared, disregarding signs, with all the graphs which follow in the
list. A match found, its coefficient is added – using relation (3) – to the coefficient of
the graph we started with; the match itself is removed. By this reduction procedure for
graph sums, all vanishing graphs with zero signs are excluded from the list.
Implementation 6. To reduce a graph series expansion modulo skew-symmetry, call
> reduce_mod_skew <graph-series-file> [--print-differential-orders]
The resulting graph series is sent to the standard output. The optional argument
--print-differential-orders controls whether the differential orders of the graphs
(as operators acting on the sinks) are included in the output, with lines such as
# 2 1
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 235
indicating subsequent graphs have differential order (2, 1). (The corresponding methods
are KontsevichGraphSeries<T>::reduce_mod_skew() and KontsevichGraphSum<T>
::reduce_mod_skew() in Appendix B.)
Example 10. We put the zero graph from Example 5 with the coefficient +1 into a
file zerograph3.txt:
h^3:
2 3 1 0 1 0 1 2 3 1
We confirm that reduce_mod_skew kills it:
> reduce_mod_skew zerograph3.txt
h^3:
The output is an empty list of graphs.
Remark 5. An alternative for the implementation of reduce_mod_skew is to make use
of the plain text file format, in three passes. In the first pass, put all graphs in normal
form with sign +1 (updating the coefficients). Recall that graphs are listed first in the
file format for graph series, so the problem of collecting terms is the same as sorting the
file. In the second pass, use sort from GNU coreutils to sort the file (this uses the
very efficient “external R-way merge” algorithm). In the third pass: for each normal
form, add up the coefficients of every copy in the list (since the list is sorted, one need
not look far). The implementation of this algorithm is left as an exercise to the reader.
Remark 6. Sums of graphs may also be reduced modulo the (graphical) Jacobi identity
and its (pictorial) differential consequences; this is the subject of section 3.2.
1.4. Evaluate a given graph series at a given Poisson structure. Let us recall
that every Kontsevich graph contains at least one sink. Every edge (decorated with
an index, say i, over which the summation runs from 1 to n = dimNn) denotes the
derivation with respect to a local coordinate xi at a given point x of the affine man-
ifold Nn (hence the edge denotes ∂/∂xi|x). Every internal vertex (if any) encodes a
copy of a given Poisson structure P . Should the labellings of two outgoing edges be i-
and j- so that the edge with i precedes that with j, the Poisson structure in that
vertex is P ij(x) (that is, the ordering i ≺ j is preserved; moreover, the reference to a
point x is common to all vertices). Now, every Kontsevich graph (with a coefficient
after it) represents a (poly)differential operator with respect to the content of sink(s); to
build that operator, we apply the derivations (at x ∈ Nn) to objects in the arrowhead
vertices, multiply the content of all vertices at a fixed set of index values, and then sum
over all the indices.
Example 11 (Jacobi identity). For all Poisson structures P and all triples of arguments
from the algebra C∞(Nn) of functirons on the Poissron manifolrd at hand, we have that
• • r	
i j

 ?BBN := 	r@Rr@ r r
L
	
R @Rr
@Rk − ri j H j
r
 Hjk r − 	ri 	r@Rkr = 0. (4)
1 2 3 1 2 3 1 2 3 1 2 3
In formulae, by ascribing the index ℓ to the unlabeled edge, the identity reads
(∂ P ijPℓkℓ + ∂ PjkPℓiℓ + ∂ Pkiℓ Pℓj)∂i(1 )∂j(2 )∂k(3 ) = 0.
236 R. BURING AND A. V. KISELEV
Indeed, the coefficient of ∂i ⊗ ∂j ⊗ ∂k is the familiar form of the Jacobi identity.
In fact, the graph itself is the most convenient way to transcribe the formulae which
one constructs from it, see [25, §2.1] for more details.6 The computer implementation is
straightforward. We acknowledge however that it is one of the most needed instruments.
Implementation 7. The call is
> poisson_evaluate <graph-series-filename> <poisson-structure>
and options for <poisson-structure> are7
• 2d-polar,
• 3d-generic,
• 3d-polynomial,
• 4d-determinant,
• 4d-rank2,
• 9d-rank6.
The output is a list of coefficients of the differential operator that the graph series
represents, filtered by (a) powers of ℏ, (b) the differential order as an operator acting
on the sinks, and (c) the actual derivatives falling on the sinks.
Example 12. Put the graph sum for the Jacobiator Jac(P) in jacobiator.txt:
3 2 1 0 1 2 3 -1
3 2 1 0 2 1 3 1
3 2 1 0 4 1 2 -1
We evaluate it at a Poisson structure:
> poisson_evaluate jacobiator.txt 2d-polar
Coordinates: r t
Poisson structure matrix:
[[0, r^(-1)]
[-r^(-1), 0]]
h^0:
6In the variational set-up of Poisson field models, the affine manifold Nn is realised as fibre in an
affine bundle π over another affine manifold Mm equipped with a volume element. The variational
Poisson brackets {·, ·}P are then defined for integral functionals that take sections of such bundle π
to numbers. The encoding of variational polydifferential operators by the Kontsevich graphs now
reads as follows. Decorated by an index i, every edge denotes the variation with respect to the
ith coordinate along the fibre. By construction, the variations act by first differentiating their argument
with respect to the fibre variables (or their derivatives along the base Mm); secondly, the integrations
by parts over the underlying space Mm are performed. Whenever two or more arrows arrive at a graph
vertex, its content is first differentiated the corresponding number of times with respect to the jet fibre
variables in J∞(π) and only then it can be differentiated with respect to local coordinates on the base
manifold Mm. The assumption that both the manifolds Mm and Nn be affine makes the construction
coordinate-free, see [23, 27] and [24, 26].
7The current version of the software does not allow specification of an arbitrary Pois-
son structure at runtime (e.g. input as a matrix of functions); however, in the source file
util/poison_structure_examples.hpp the list of Poisson structures (as matrices) can be extended
to one’s heart’s desire.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 237
# 1 1 1
# [ r ] [ r ] [ r ]
0
# [ r ] [ r ] [ t ]
0
# [ r ] [ t ] [ r ]
0
# [ r ] [ t ] [ t ]
0
# [ t ] [ r ] [ r ]
0
# [ t ] [ r ] [ t ]
0
# [ t ] [ t ] [ r ]
0
# [ t ] [ t ] [ t ]
0
For example, the pair of lines
# [ r ] [ t ] [ r ]
0
indicates that the coefficient of ∂r ⊗ ∂t ⊗ ∂r is zero in the polydifferential operator.
Restriction of graph series to Poisson structures will be essential in section 3.1 below
where systems of linear algebraic equations between the Kontsevich graph weights in ⋆
will be obtained by restricting the associativity equation Assoc⋆(f, g, h) = 0 to a given
Poisson bracket.
2. The Kontsevich ⋆-product
The star-product ⋆ = ×+ℏ{·, ·}P+ō(ℏ) in C∞(Nn)[[ℏ]] is an associative unital noncom-
mutative deformation of the associative unital commutative product × in the algebra
of functions C∞(Nn) on a given affine manifold Nn of dimension n <∞. The bi-linear
bi-differential ⋆-product is realized as a formal power series in ℏ by using the weighted
Kontsevich graphs. In fact, the bi-differential operator at ℏk is a sum of all Kontsevich
graphs Γ ∈ G̃2,k without tadpoles, with k internal vertices (and two sinks) taken with
some weights w(Γ). Let us recall their original definition [32].
Definition 3. Every Kontsevich graph Γ ∈ G̃2,k can be embedded in the closed upper
half-plane H ∪ R ⊂ C by placing the internal vertices at pairwise distinct points in H
and the external vertices at 0 and 1; the edges are drawn as geodesics with respect to
the hyperbolic metric, i.e. as vertical lines and circular segments. The angle φ(p, q)
between two distinct points p, q ∈ H is the angle between the geodesic from p to q and
the geodesic from p to ∞ (measured counterc(lockwis)e from the latter):
q − p
φ(p, q) = Arg ,
q − p̄
238 R. BURING AND A. V. KISELEV
and it can be extended toH∪R by continuity. The weight of a Kontsevich graph Γ ∈ G̃2,k
is given by the integral8 ∫ ∧k1
w(Γ) = dφ(pj, p2k Left(j)) ∧ dφ(pj, pRight(j)), (5)(2π) Ck(H) j=1
over the configuration space of k points in the upper half-plane H ⊂ C,
Ck(H) = {(p1, . . . , pk) ∈ Hk : pi pairwise distinct};
the integrand is defined pointwise at (p1, . . . , pk) by considering the embedding of Γ in
H that sends the jth internal vertex to pj; the numbers Left(j) and Right(j) are the
left and right targets of jth vertex, respectively. (If Left(j) is the first or the second
sink, put pLeft(j) = 0 or 1 respectively; the same goes for pRight(j) if Right(j) is a sink.)
Theorem (Kontsevich [32]). For every Poisson bi-vector P on Nn and an infinitesimal
deformation × 7→ ×+ℏ{·, ·}P+ō(ℏ) towards the respective Poisson bracket, the ℏ-linear
star-produ∑ct ℏk ∑
⋆ = ×+ w(Γ) Γ(P)(·, ·) : C∞(Nn)[[ℏ]]× C∞(Nn)[[ℏ]]→ C∞(Nn)[[ℏ]] (6)
k!
k⩾1 Γ∈G̃2,k
is associative.
Lemma 1. Permuting the internal vertex labels of a Kontsevich graph leaves the weight
unchanged.
Proof. Such a permutation re-orders the factors in a wedge product of two-forms. □
Lemma 2. Swapping L ⇄ R at an internal vertex of a Kontsevich graph Γ ∈ G̃2,k
implies the reversal of the sign of its weight.
Proof. Anticommutativity of wedge product of two differentials in formula (5). □
Lemma 3. The weight of a graph Γ ∈ G̃2,k and its mirror-reflection Γ̄ are related by
w(Γ̄) = (−)kw(Γ).
Proof. Taking the reflection of a graph (with respect to the vertical line <(z) = 1/2) is
an orientation-reversing coordinate change on each of the k “factors” H in Ck(H). □
Lemma 4 ([16]). For a Kontsevich graph such that at least one sink receives no edge(s),
its weight is zero.9
Lemma 5. The map w : tk G̃2,k → R that assigns weights to graphs is multiplicative,
w(Γi×̄Γj) = w(Γi)× w(Γj), (7)
with respect to the product ×̄/ of graphs,
Γ ×̄Γ = (Γ t Γ ) {ath sink in Γ thi j i j i = a sink in Γj, 0 ⩽ a ⩽ 1},
which identifies the respective sinks.
8We omit the factor 1/k! that was written in [32], to make the weight multiplicative (see Lemma 5).
9The fact that the differential order of ⋆ is positive with respect to either of its arguments should
be expected, in view of the required property of the ⋆-product to be unital: f ⋆ 1 = f = 1 ⋆ f .
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 239
Proof. The integrals converge absolutely [32]; apply Fubini’s theorem and linearity. □
Exam(plre 1)3. Som(e wr ei)ght rela(tionrs o)btained (fromr th)e lemm(ars abo)ve: ( r)
r B
2
w  rB r = w r 
A r ; w Lr 
A Rr = −w Rr 
A Lr ; w @rRr@ r = w r	 .
/SwBN 
 UA 
 AU 
 UA ?	R 	r@Rr?
Lemma 5 motivates the following definition.
Definition 4. A Kontsevich graph Γ ∈ G̃2,k is called composite if Γ is equal to the
×̄-product of some Kontsevich graphs on two sinks and positive number of internal
vertices in both of the co-factors. Otherwise (if such a realization is not possible), the
graph is called prime.
Using Lemma 5 and induction, we obtain that the weight of a composite graph Γ =
Γ1×̄ · · · ×̄Γt is the product of the weights of its factors: w(Γ) = w(Γ1)× · · · × w(Γt).
2.1. Basic set of graphs. We identify a set of graphs such that the weights of those
graphs would suffice to determine all the other weights.
Definition 5. A basic set of graphs on k internal vertices is a set of pairwise distinct
normal forms (the signs of which are discarded) of only those Kontsevich graphs Γ ∈ G̃2,k
which are prime, and in which every sink receives at least one edge. By definition, the
basic set contains the normal form of a graph but not its mirror reflection if it differs
from the graph at hand. To decide whether a graph or its mirror-reflection Γ̄ 6= Γ
is included into a basic set, we take the graph whose absolute value is minimal as a
base-(k + 2) number. Note that a basic set on k ⩾ 3 vertices does contain zero graphs.
Corollary 6. To build ⋆-product (6) up to ō(ℏk) for some power k ⩾ 1, knowing the
Kontsevich weights w(Γi) only for a basic set of graphs Γi ∈ G̃2,ℓ at all ℓ ⩽ k is enough.
Indeed, the weights of all other graphs with ℓ internal vertices are calculated from
Lemmas 1, 2, 3, 4, and 5. r
@Rr r r	
Example 14. Consider the prime graph r?	@Rr and its mirror-reflection r	@Rr?. The en-
codings of their normal formrs are 2 2 1 0 1 0 2 and 2 2 1 0 1 1 2 respectively.Since 0 1 0 2 < 0 1 1 2 as base-4 numbers, only the first graph is included in the
basic set. The fork graph r 
A r is mirror-symmetric hence it is included anyway.
 UA
The basic set at order 3 is displayed in Figure 2.
2.2. “All” graphs in ⋆ mod ō(ℏ4). In Table 1 we list the number of basic graphs
at every order k ⩽ 6 in the Kontsevich ⋆-product. The actual number of graphs with
respect to which the sums in formula (6) expand is of course much greater.
Implementation 8. To obtain the list of normal forms for graphs from a basic set at
order k, the following command is available:
> generate_graphs k --basic=yes
240 R. BURING AND A. V. KISELEV
r -r
@ B r
@  B rA r- r r r rr
rR r B r r@RAUr r A @rRr r r@Rr	 @Rr
	@RBBN R?	@R AAU?	@R 	?@Rr? r?	@Rr?
0 w_3_1 w_3_2 w_3_3 w_3_4
r
	r
@rR?r r r
r r	 r r r
@Rr
	?@R @R r	@Rr rJĴr

	@Rr r	JĴr r+r Hr

 ? ?r

j6rJ
	 JĴ r
w_3_5 wr_3_6 w_3r_7 w_3_8 w_3_9
r r	 r rQ	Qsr r	

JĴ r r r r r r rr    
-@R? r? Ur? r? r? r? r
 BBNr r? r?
w_3_10 w_3_11 w_3_12 w_3_13 w_3_14
Figure 2. Basic set at order 3, with undetermined weights for nonzero
graphs. (The weights are determined in Example 26 on p. 250 below.)
Table 1. How many basic graphs there are at low orders k.
Order = k 0 1 2 3 4 5 6
#(Basic set) 0 1 2 15 156 2307 43231
#(Nonzero in basic set) 0 1 2 14 149 2218 42050
The list of normal forms is then sent to the standard output. This command is equiv-
alent to
> generate_graphs k --prime=yes --normal-forms=yes \
--postive-differential-order=yes --modulo-mirror-images=yes
Example 15. The list of basic graphs with ⩽ 3 internal vertices – with undetermined
coefficients at orders 1, 2, 3 – is constructed using the following commands:
$ cat > basic3w.txt
h^0:
2 0 1 1
h^1:
^D (press Ctrl+D)
$ generate_graphs 1 --basic=yes --with-coefficients=yes \
>> basic3w.txt
$ echo 'h^2:' >> basic3w.txt
$ generate_graphs 2 --basic=yes --with-coefficients=yes \
>> basic3w.txt
$ echo 'h^3:' >> basic3w.txt
$ generate_graphs 3 --basic=yes --with-coefficients=yes \
>> basic3w.txt
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 241
The file basic3w.txt now contains the basic set.
Starting from a basic set, the ⋆-product is built up to a certain order k ⩾ 0 in ℏ.
Implementation 9. The program
> star_product <basic-set-filename>
takes as its input a graph series with a basic set of graphs at each order; the graphs go
with coefficients of any nature (i.e. number or indeterminate). The program’s output is
an expansion of the ⋆-product up to the order that was specified by the input. In other
words, all the graphs which are produced from the ones contained in a given basic set
are generated and their coefficients are (re)calculated from the ones in the input (using
Lemmas 2, 3, and 5).
Example 16. To generate the star-product up to order 3 with all weights of nonzero
basic graphs undetermined (from Example 15), one proceeds as follows:
$ star_product basic3w.txt > star3w_unreduced.txt
$ reduce_mod_skew --print-differential-orders star3w_unreduced.txt \
> star3w.txt
The file star3w.txt now contains the desired star-product.
2.3. Methods to obtain the weights of basic graphs. We deduce that to build
the ⋆-product modulo ō(ℏ4) as many as 149 weights of nonzero basic graphs Γi ∈ G̃2,4
at k = 4 must be found (or at least expressed in terms of as few master-parameters as
possible). In fact, direct calculation of all of the 149 Kontsevich integrals is not needed
to solve the problem in full because there exist more algebraic relations between the
weights of basic graphs. In the following proposition we recall a class of such relations.10
Proposition 7 (cyclic weight relations [17]). Let Γ be a Kontsevich graph on m = 2
ground vertices. Let E ⊂ Edge(Γ) be a subset of edges in Γ such that for every e ∈ E,
target(e) 6= 0. (That is, every edge from the subset E lands on the sink 1 or an internal
vertex.) For every such subset E, define the graph ΓE as follows: let its vertices be the
same as in Γ and for every edge e ∈ Edge(Γ), preserve it in ΓE if e 6= E, but if e ∈ E
replace that edge by a new edge in ΓE going from source(e) to the sink 0. By definition,
the ordering L ≺ R of outgoing edges is inherited in ΓE from E even if the targets of
any of those edges are new. Thirdly, denote by N0(ΓE) the number of edges in ΓE such
that their target is the sink 0. Then the Kontsevich weight of a graph Γ is related to
the weights of all such graphs ΓE obtain∑ed from Γ by the formula
w(Γ) = (−)n (−)N0(ΓE)w(ΓE). (8)
E⊂Edge(Γ)
∀e∈E,target(e)̸=0
Note that this relation is linear in the weights of all graphs.
10A convenient approach to calculation of Kontsevich weights (5) at order 3 by using direct inte-
gration (and for that, using methods of complex analysis such as the Cauchy residue theorem) was
developed in [14], see Appendix A.1 on p. 276 below. However, we note that most successful at k = 3,
this method is no longer effective for all graphs at k ⩾ 4. More progress is badly needed to allow k ⩾ 5.
242 R. BURING AND A. V. KISELEV
If the graph Γ or, in practice, some of the new graphs ΓE in (8) is composite, Lemma 5
provides a further, nonlinear reduction of w(Γ) by using graphs with fewer internal
vertices.
Example 17. Consider the graph Γ3,8 in Figure 2 with weight w(Γ3,8) = w_3_8. For
every non-empty subset E (with target(e) =6 0 for every e ∈ E) the graph (Γ3,8)E is
a zero-weight graph by virtue of one of the Lemmas at the beginning of this chapter.
Hence the only term in the sum on the right-hand side in (8) is the weight of the graph
corresponding to the empty set: w((Γ3,8)∅) = w(Γ3,8). Since n = 3 and N0(Γ3,8) = 2
we get the cyclic relation w(Γ3,8) = −w(Γ3,8); whence w(Γ3,8) = 0.
Remark 7. It is readily seen that only prime, that is, non-composite graphs Γ need be
used to generate all relations (8). Indeed, every subset E of edges for a composite graph
Γ = Γ1×̄Γ2 splits to a disjoint union E1 t E2 of such subsets for the graphs Γ1 and Γ2
separately. Therefore the re-direction of edges in a composite graph would inevitably
yield the composite graph Γ1 2E1×̄ΓE2 . Now, the multiplicativity of Kontsevich weights
and the additivity of the count N0(ΓE) = N0(Γ1E1) +N 20(ΓE2) can be used to conclude
that the relations obtained from composite graphs are redundant.
Implementation 10. The command
> cyclic_weight_relations <star-product-file>
treats the input ⋆-product as a clothesline for graphs and their weights. For each graph
Γ in the ⋆-product, it outputs the relation (8) between the weights of the respective
graphs in the form LHS - RHS == 0.
Example 18. At the order three with the ⋆-product from Example 16:
> cyclic_weight_relations star3w.txt
...
1/3*w_3_6+1/6*w_2_3==0
1/3*w_3_8==0
...
Remark 8. For some (basic) graphs it happens that the weight integrand in (5), as
a differential 2k-form, vanishes identically, even if the graph is not zero due to skew-
symmetry. This is the case for 21 out of 149 nonzero basic graphs at k = 4; see also
Appendix A.1.
For calculations of particular weight integrals we refer to the literature in section 4.
Remark 9 (rationality). Willwacher and Felder [17] express the weight of a graph in
G̃2,7 as p · ζ(3)2/π6 + q where p and q are rational numbers and ζ is the Riemann ζ-
function. Whether ζ(3)2/π6 is rational or not is an open problem. The software which
we presently discuss supports – through GiNaC [3] – the input of ζ-values as coefficients,
e.g. the expression ζ(3)2/π6 can be input as zeta(3)^2/Pi^6. This can be used e.g. to
express other weights in terms of such values. According to Banks–Panzer–Pym in [2],
Q[(2π)−1]-linear combinations of multiple zeta values actually start to appear in the
harmonic weight coefficients in the ⋆-product, but at order ℏ6. They do not yet appear
at order 4 (or rather, they are all seen to be rational at order 4).
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 243
Is any of the weights transcendental? This has not been proved, so that it remains
unknown whether any of them is or none of them are.
All the above being said about methods to obtain the values w(Γ) for Kontsevich
graph weights and about the schemes to generate linear relations between these num-
bers, we observe that the requirement of associativity for the ⋆-product modulo ō(ℏk),
whenever that structure is completely known at all orders up to ℏk, is an ample source
of relations of that kind. This will be used intensively in chapter 3 from p. 246 on-
wards. In particular, we mention here that the values of weights of graphs at order ℓ
may be restricted by the associativity requirement at orders > ℓ, by restriction to fixed
differential orders (i, j, k) (see Lemma 10 on p. 253).
2.4. How graphs act on graphs. Let us have a closer look at the equation of asso-
ciativity for the sought-for ⋆-product:
Assoc⋆(f, g, h) = (f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) = 0.
We see that the graph series f ⋆ g and g ⋆h serve as the left- and right co-multiples of h
and f , respectively, in yet another copy of the star-product. To realize the associator
by using the Kontsevich graphs, we now explain how graphs act on graphs (here, in
every composition ⋆ ◦ ⋆ the graph series acts on a graph series by linearity).
We postulate that the action of graph series on graph series is k[[ℏ]]-linear and k[G∗,∗]-
linear with respect to both the graphs that act and that become the arguments.
Recall that every Poisson bracket is a derivation in each of its arguments. In conse-
quence, every derivation falling on a sink – in a graph Γ1 that acts on a given graph Γ2
taken as the new content of that sink – acts on the sink’s content via the Leibniz rule;
all the Leibniz rules for the derivations in-coming to that sink work independently from
each other. Recall that the vertices of a graph represent factors in an expression.
Example 19. Consider the action of a wedge graph Λ on the two-sinks graph (••) ∈
G2,0, taken as its second argumenr t. We havre that r
r 
A = 
A +
 AUr r r
 UA r r r 
Ar r .
 AU
The result is a sum of Kontsevich graphs of type (3, 1). Let us remember that the
sinks are distinguished by their ordering; in particular the two Kontsevich graphs on
the right-hand side are not equal.
Example 20. Now let the wedge graph act on a wedge graph (again, as the former’s
second argument): r r r rr 
AU
 
r
rAUr	= r 
AU
 
r r r + r 
A r r + 
 r .AU 
 UA
AUr r
 
rAUr
Example 21. Finally, consider a graph in which two arrows fall on the first sink and
let its content be (••r) ∈ G2,0: r r r rr@R?	r
r r = @Rr @Rr @Rr @Rr@R ?r	r@ +Rr r r	?@ r +R r r +	@Rr r=r .?@Rr
244 R. BURING AND A. V. KISELEV
These three examples basically cover all the situations; we shall refer to them again,
namely, from the next chapter where the restrictions by using the total differential
orders are discussed.
So far, we have focused on graphs; under the action of a graph on a graph, their
coefficients are multiplied. (This is why the associativity of the ⋆-product is an infinite
system of quadratic equations for the coefficients of all the graphs).
Implementation 11. In the class KontsevichGraphSeries<T>, the method
KontsevichGraphSeries<T>::operator()
allows function-call syntax for the insertions described above. As its argument it takes a
std::vector (that is, a list) of the Kontsevich graph series in ℏ; these are the m respec-
tive arguments for a Kontsevich graph series. It returns a KontsevichGraphSeries<T>.
The method is called for the object of the class, that is, for the graph series which is to
be evaluated at the m specified arguments.
For example, this allows the realization of Examples 19 and 20 in C++ expressions as
wedge({ dot, twodots }) and wedge({ dot, wedge }) respectively.
Implementation 12. To calculate the associator Assoc⋆(f, g, h) for a given ⋆-product
and ordered objects f, g, h, the call is
> star_product_associator <star-product-filename>
where the input file <star-product-filename> contains the (truncated) power series
for the ⋆-product. In the standard output one obtains a (truncated at the same order
in ℏ as in the input) power series containing, at each power ℏk, the sums of graphs from
G3,k with coefficients (their admissible types were introduced in §1.2 above).
Example 22. The associator for the ⋆-product up to order 2 (from Example 8):
$ star_product_associator star2.txt
h^0:
h^1:
h^2:
# 1 1 1
3 2 1 0 1 2 3 -2/3
3 2 1 0 2 1 3 2/3
3 2 1 0 4 1 2 -2/3
It is 2ℏ2 times the Jacobiator (4), whose encoding we saw before in Example 12.
3
2.5. Gauge transformations. At first glance, the concept of gauge transformations
for (graphs in the) ⋆-products is an extreme opposite of plugging a list of graph series as
arguments of a given graph series. Namely, the idea of a gauge transformation is that
a graph series (possibly of finite length) is towered over a single vertex • ∈ G1,0. By
definition, a gauge transformation of a vertex • is a map of the form • 7→ [•] = •+ℏ·(...)
taking G1,0 → k[G1,∗][[ℏ]].
Example 23. The map • 7→ • ℏ2
q+r q+ AU3
 is a gauge transformation of • ∈ G12 1,0. This
graph series is encoded in the following file gaugeloop.txt:
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 245
h^0:
1 0 1 1
h^2:
1 2 1 0 2 1 0 1/12
The construction of gauge transformations is extended from G1,0 by k[[ℏ]]- and
k[G∗,∗]-linearity. This effectively means that in the course of action by a gauge trans-
formation t on a graph series f ∈ k[G∗,∗][[ℏ]], all the arrows work over the vertices in
every graph in f via the Leibniz rule (as it has been explained in the previous section).
This is how one expands [f ] ⋆ [g], that is, the Kontsevich ⋆-product (6) of two gauged
arguments [f ] and [g]. Let us recall further that the shape [•] = •+ ℏ · (. . .), where the
gauge tail of • is given by some graphs from k[G1,∗][[ℏ]], guarantees the existence of a
formal left inverse t−1 to the original transformation t, so that (t−1 ◦ t)(•) = •.
Lemma 8. If • 7→ = t(•) = •+ℏΓ (•)+. . .+ℏℓ1 Γℓ(·)+ ō(ℏℓ) is a gauge transformation,
let
t−1( ) = + ℏγ1( ) + . . .+ ℏℓγℓ( ) + ō(ℏℓ)
by setting
m∑−1
γ0 = id, γm( ) := − γk(Γm−k( )).
k=0
Then t−1(t(•)) = •, that is, the transformation t−1 : k[G1,∗][[ℏ]]→ k[G1,∗][[ℏ]] is the left
inverse of t up to ō(ℏ).
It is readily seen that the assembly of the entire t−1 can require infinitely many
operations even if the direct transformation t took only finitely many of them, e.g., as
in Example 23.
In these terms, for the Kontsevich ⋆-product (6) we obtain, by operating with gauge
transformations and their formal inverses, a class of star products ⋆′ which are defined
by the relation
t(f ⋆′ g) = t(f) ⋆ t(g), f, g ∈ C∞(Nn)[[ℏ]]. (9)
Clearly, all these gauged star-products ⋆′ remain associative (because ⋆ was) but the
coefficients of graphs at an order k ⩾ 2 in ℏ are no longer necessarily equal to the
respective values in (6). The use of gauge transformations for products allows to gauge
out some graphs, often at aq ceqrtain order ℏ
k in the star-product expansion.
q	Example 24. The graph ? q?with a loop is gauged oqutq from the Kontsevich ⋆-pro-+
duct (6) by using the gauge transformation t : • 7→ •+ ℏ2 AU3r
 , see Example 23. Note that12
taking the formal inverse t−1 does create loop-containing graphs at higher orders ℏ⩾3
in the gauged star-product ⋆′ which is specified by (9).
Remark 10. Not every graph taken in the Kontsevich star-product ⋆ at a particular
order ℏk can be gauged out. For example, such are the graphs Γ ∈ G̃2,∗ containing an
internal vertex v with edges running from it to both the ground vertices.
Implementation 13. The command for gauge transformation is
> gauge <star-product-filename> <gauge-transformation-filename>
246 R. BURING AND A. V. KISELEV
where
• the file <star-product-filename> contains a machine-format graph encoding
of star-product ⋆ truncated modulo ō(ℏk) for some k ⩾ 0;
• the content of <gauge-transformation-filename> is a gauge transformation t(•),
that is, a truncated modulo ō(ℏℓ⩾0) series in ℏ consisting of the Kontsevich
graphs built over one sink vertex •.
In the standard output one obtains the truncation, modulo ō(ℏmin(k,ℓ)), of the graph
series for the gauged star-product ⋆′ defined by f ⋆′ g = t−1(t(f) ⋆ t(g)).
(The corresponding method is KontsevichGraphSeries<T>::gauge_transform() in
Appendix B.)
Example 25. Let the gauge transformation from Example 24 be stored in the file
gaugeloop.txt, and recall the ⋆-product up to order two from Example 8 in the file
star2.txt. The gauge transformation kills the loop graph:
$ gauge star2.txt gaugeloop.txt > star2gauged_unreduced.txt
$ reduce_mod_skew star2gauged_unreduced.txt > star2gauged.txt
$ cat star2gauged.txt
h^0:
2 0 1 1
h^1:
2 1 1 0 1 1
h^2:
2 2 1 0 1 0 1 1/2
2 2 1 0 1 0 2 1/3
2 2 1 0 1 1 2 -1/3
Indeed, we see that the line
2 2 1 0 3 1 2 -1/6
containing the loop graph has disappeared.
Let us note at once that every gauge transformation t given by a Kontsevich graph
polynomial in ℏ of degree ℓ can clearly be viewed formally as a polynomial transfor-
mation of any degree greater or equal than ℓ. This is why by using the same software
we can actually obtain the gauged star-product ⋆′ modulo ō(ℏ4) starting with the Kon-
tsevich star-product ⋆ modulo ō(ℏ4) and applying the gauge transformation of nominal
degree ℓ = 2 from Example 23. In other words, the precision in ⋆′ with respect to ℏ is
the same as in ⋆ even though the degree of the polynomial gauge transformation t is
smaller. In practice, this is achieved by adding an empty list of graphs at the power ℏk
to a given gauge transformation of degree ℓ < k.
3. Associativity of the Kontsevich ⋆-product
In the final section of this paper we explore two complementary matters. On the
one hand, we analyse how the associativity postulate for the Kontsevich ⋆-product
contributes to finding the values of weights w(Γ) for graphs Γ in ⋆. On the other
hand, a point is soon reached when no new information can be obtained about the
values of w(Γ): specifically, neither from the fact of associativity of the ⋆-product nor
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 247
from any proven properties of the Kontsevich weights. We outline a computer-assisted
scheme of reasoning that, working uniformly over the set of all Poisson structures under
study, reveals the associativity of ⋆-product on the basis of our actual knowledge about
the weights w(Γ) of graphs Γ in it.
In [12] we reported an exhaustive description of the Kontsevich ⋆-product up to ō(ℏ3).
At the next expansion order ō(ℏ4) in ⋆, we now express the weights of all the 160 000 =
(5 · 4)4 graphs Γ ∈ G̃ 42,4 (of which up to 10 000 = (5 · 4/2) are different modulo signs)
in terms of only 10 parameters; those ten master-parameters themselves are the (still
unknown) Kontsevich weights of the four internal vertex graphs portrayed in Fig. 3. By
following the second strategy we prove that for any values of those ten parameters the
⋆-product exr pansion modulo ōr(ℏ
4) is associrative, also up to ō(ℏ
4).
r	r@R r r rr r r@R r r r6 P rq r r r r r@rR	r rHj6 r rI@	r XXz	 ?	 ? ?	 ? ?	 ? CCWr	 r? @Rr? r?
p1 =rw_4_100 p2 = w_4_101 p3 = w_4_102 p4 = w_4_103 p5 = w_4_104
r r Ur
r r rr r r r r rr r
Ĵ r Kr rjr r

r U*rKr r
-r?
 BBN? 
) BBN ?+ ? Ur r? r
K rBBNr
p6 = w_4_107 p7 = w_4_108 p8 = w_4_109 p9 = w_4_119 p10 = w_4_125
Figure 3. The ten graphs whose unknown weights11 are taken as the
master-parameters pi; in fact, the four graphs whose weights are under-
lined can be gauged out from ⋆ so that there remain only 6 parameters
that determine it modulo ō(ℏ4).
3.1. Restriction of the ⋆-product associativity equation Assoc⋆(f, g, h) = 0 to a
Poisson structure P. We now view the postulate of associativity for the Kontsevich
⋆-product as an equation for coefficients in the graph expansion of ⋆. Whenever an
expansion modulo ō(ℏℓ) is known for the ⋆-product, one passes to the next order ō(ℏℓ+1)
by taking all the graphs Γ ∈ G̃2,ℓ+1 with undetermined coefficients, and then expands
(with respect to graphs) the associator Assoc⋆(f, g, h) up to the order ō(ℏℓ+1). This
expansion now runs over all the graphs with at most ℓ+1 internal vertices. It is readily
seen that by construction this associativity equation Assoc⋆(f, g, h) = ō(ℏℓ+1) is always
linear12 with respect to the coefficients of graphs from G̃2,ℓ+1.
Remark 11. One can still get linear relations between the weights w(Γ) of graphs Γ ∈
G̃2,ℓ+1 at order ℏℓ+1 in ⋆ by inspecting the associativity of ⋆ at higher orders – ranging
from ℓ+2 till 2ℓ+1 – in ℏ. Indeed, a linear relation containing the unknown weights (and
the already known lower-order part of ⋆ as coefficients) but not the weights of graphs
with ⩾ ℓ + 2 internal vertices can appear whenever a properly chosen homogeneous
11Numerical approximations of two of these weights are listed in Table 3 in Appendix A.1.
12Should a graph Γ ∈ G̃2,ℓ+1 be composite so that its Kontsevich weight is factorized using for-
mula (7), the resulting nonlinearity with respect to the weights would actually involve only the graphs
with at most ℓ internal vertices.
248 R. BURING AND A. V. KISELEV
component of the tri-differential operator Assoc⋆(f, g, h) does not contain any weights
from higher orders. For instance, this is the component at homogeneity orders (i, j, k)
such that prime graphs Γ ∈ G̃2,⩾ℓ+2 of homogeneity orders (i+ j, k) and (i, j+k) (when
viewed as bi-differential operators) do not exist or if the weights of all such graphs are
known in advance.
3.1.1. Let us also note that in the graph equation Assoc⋆(f, g, h) = 0 that holds by
virtue of the Jacobi identity Jac(P) = 0, not every coefficient of every graph in the
expansion should be expected to vanish. Indeed, the Jacobiator is a vanishing sum of
three graphs that evaluates to zero at every Poisson structure P which we put into every
internal vertex. This is why the restriction of associativity equation to a given Poisson
structure (or to a class of Poisson structures) is a practical way to proceed in solution
of the problem of finding the coefficients of graphs in ⋆. More specifically, after the
restriction of associator Assoc⋆(f, g, h) to a structure P which is known to be Poisson
so that all the instances and all derivatives of the Jacobiator Jac(P) are aut∣omati-
cally trivialized, the left-hand side of the associativity equation Assoc⋆(f, g, h)∣P = 0
mod ō(ℏℓ+1) becomes an analytic expression (linear with respect to the unknowns w(Γ)
for Γ ∈ G̃2,ℓ+1). At this point one can proceed in several ways.
We now outline three methods to obtain systems of linear equations upon the un-
known weights w(Γ) of basic graphs Γ ∈ G̃2,ℓ+1. Working in local coordinates, we ensure
that the unknowns’ coefficients in the equations which we derive are real numbers.13
Method 1. Let the associator’s arguments∣ be given functions f, g, h ∈ C∞(Nn). Re-
strict the analytic expression A∣ ssoc⋆(f, g, h)∣P to a point x of the manifold N
n equipped
with a Poisson structure P . For every choice of f, g, h ∈ C∞(Nn) and of a point x ∈ Nn,
the restriction Assoc⋆(f, g, h)∣P(x) = 0 mod ō(ℏℓ+1) yields one linear relation between
the weights of graphs at order ℏℓ+1. Taking the restriction at several points x1, . . .,
xk ∈ Nn, one obtains a system of such equations, the rank of which does not exceed
the number k of such points in Nn. Bounded by the number of unknowns w(Γ), the
rank would always stabilize as k →∞.
Examples of Poisson structures P – for instance, on the manifolds Rn – are available
from [20] (here n ⩾ 3) and [37]; from Proposition 2.1 on p. 74 in the latter one obtains
a class of Poisson (in fact, symplectic) structures with polynomial coefficients on even-
dimensional affine spaces R2k. Besides, there is a regular construction (by using the
R-matrix formalism, see [33, p. 287]) of Poisson brackets on the vector space of square
matrices 2Mat(R, k × k) ∼= Rk (e.g., in this way one has a rank-six Poisson structure
on R9).
Method 1 is the least computationally expensive, so it can be used effectively at the
initial stage, e.g., to detect the zero values of certain graph weights: once found, such
trivial values allow to decrease the number of unknowns in the further reasoning.
13From the factorization of associator for ⋆ via differential consequences of the Jacobi identity for
a Poisson structure P, which will be revealed in section 3.2 below, it will be seen in hindsight that
the construction of linear relations between the graph weights is overall insensitive to a choice of
local coordinates in a chart within a given Poisson manifold. Indeed, the factorization will have been
achieved simultaneously for all Poisson structures on all the manifolds at once, irrespective of any local
coordinates.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 249
Method 2. Now let f, g, h ∈ k[x1, . . . , xn] be polynomials referred to local coordinates
x1, . . ., xn on Nn. On that coordin∣ate chart U ⊂ Nnα , take a Poisson structure thecoefficients P ij(x) of which would a∣ lso be polynomial. In consequence, the left-handside of the equation Assoc⋆(f, g, h) P = 0 mod ō(ℏℓ+1) then becomes polynomial as
well. Linear in the unknowns w(Γ), all the coefficients of this polynomial equation
vanish (independently from each other). Again, this yields a system of linear algebraic
equations for the unknown weights w(Γ) of the Kontsevich graphs Γ ∈ G̃2,ℓ+1 in the
⋆-product.
We observe that the linear equations obtained by using Method 2 better constrain
the set of unknowns w(Γ), that is, the rank of this system is typically higher than in
Method 1. Intuitively, this is because the polynomials at hand are not collapsed to
their values at points x ∈ N .
Method 3. Keep the associator’s arguments f, g, h unspecified and consider a class of
Poisson structures P [ψ1, . . . , ψm] depending in a differential polynomial way on func-
tional parameters ψα, that is, on arbitrary functions, whenever P is referred to local
coordinates. (For example, let n = 3 and on R3 with Cartesian coordinates x, y, z
introduce the class of Poisson br(ackets using the Jac)obian determinants,
{u, v}P = p · det ∂(q, u, v)/∂(x, y, z) , q ∈ C∞(R3), (10)
supposing that∣∣ the density p(x, y, z) is also smooth on R3.) Now view the associatorAssoc⋆(f, g, h) P as a polydifferential operator in the parameters f, g, h (with[ψ1,...,ψm]
respect to which it is linear) and in ψ1, . . ., ψm from P . By splitting the associator,
which is postulated to vanish modulo ō(ℏℓ+1), into homogeneous differential-polynomial
components, we obtain a system of linear algebraic equations upon the graph weights.
It is readily seen that, whenever the parameters ψ1, . . ., ψm are chosen to be poly-
nomials (here let us suppose for definition that the resulting Poisson structure P(x)
itself is polynomial), the rank of ∣the algebraic system obtained by Method 3 can begreater than the rank of an analo∣gous system from Method 2. This is because theanalytic expression Assoc⋆(f, g, h) P keeps track of all the parameters, whereas[ψ1,...,ψm]
in Method 2 they are merged to a single polynomial.
We finally note that the linear algebraic systems which are produced by each method
should be merged. Indeed, the goal is to maximize the rank and by this, reduce the
number of free parameters in the solution.14
It has been seen in §2.4, Implementation 12 how the associator is calculated in terms
of graphs. The next step – namely, restriction of the associator to a given Poisson struc-
ture – can be performed by using a call poisson_evaluate as it has been explained
in §1.4. However, the further restriction as described in the Methods has been imple-
mented in a separate program (similar to poisson_evaluate) which directly outputs
the desired relations, as follows.
Implementation 14. The command
14If the rank of the resulting linear algebraic system is equal to the number of unknowns – and if
all the coefficients coming from lower orders ⩽ ℓ within the ⋆-product expansion with respect to ℏ are
also rational – then all the solution components are rational numbers as well, cf. [17].
250 R. BURING AND A. V. KISELEV
> poisson_make_vanish <graph-series-file> <poisson-structure>
sends to the standard output relations such as
-1/24+w_3_1+4*w_3_2==0
between the undetermined coefficients in the input, which must hold if the input graph
series is to vanish as a consequence of the Jacobi identity for the specified Poisson
structure. The implementation is described in the Methods above. The choice of
Poisson structure is made in the same way as in Implementation 7. If the optional
extra argument --linear-solve is specified, the program will assume that the relations
which will be obtained are linear, and attempt to solve the linear system.
Example 26. To obtain all the weights of basic graphs Γ ∈ G̃2,3 at ℏ3 in the Kontsevich
star-product ⋆, it was enough to build the linear system of algebraic equations that
combined (i) cyclic relations (8), (ii) the relations which Method 3 produces for generic
Poisson structure (10), and (iii) those linear relations between the weights of Γ ∈ G̃2,3
which – in view of Remark 11 on p. 247 – still do appear at the next power ℏ4 in
Assoc⋆(f, g, h) = 0, by using the same generic Poisson structure (10). The resulting
expansion of ⋆-product modulo ō(ℏ3) is shown in formula (1) on p. 225. This result is
achieved by using the software as follows. Starting from the sets of basic graphs up to
the order 2 (with known weights) in the file basic2.txt, generate lists of basic graphs
(with undetermined weights) up to the order four:
$ cp basic2.txt basic3+4w.txt
$ echo 'h^3:' >> basic3+4w.txt
$ generate_graphs 3 --basic=yes --with-coefficients=yes \
>> basic3+4w.txt
$ echo 'h^4:' >> basic3+4w.txt
$ generate_graphs 4 --basic=yes --with-coefficients=yes \
>> basic3+4w.txt
Build the ⋆-product expansion up to the order 4 from these basic sets:
$ star_product basic3+4w.txt > star3+4w_unreduced.txt
$ reduce_mod_skew star3+4w_unreduced.txt > star3+4w.txt
Generate cyclic weight relations:
$ cyclic_weight_relations star3+4w_unreduced.txt \
> weight_relations_3+4w-cyclic.txt
Build the associator expansion up to the order 4 from the ⋆-product expansion:
$ star_product_associator star3+4w.txt > assoc3+4w.txt
Obtain relations from the requirement of associativity for the Poisson structure (10):
$ poisson_make_vanish assoc3+4w.txt 3d-generic \
> weight_relations_3+4w-3d.txt
Merge the systems of linear relations:
$ cat weight_relations_3+4w-* > weight_relations_3+4w_all.txt
Solving the linear system in weight_relations_3+4w_all.txt yields the solution
w_3_1=1/24, w_3_2=0, w_3_3=0, w_3_4=-1/48, w_3_5=-1/48
w_3_6=0, w_3_7=0, w_3_8=0, w_3_9=0, w_3_10=0
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 251
w_3_11=-1/48, w_3_12=-1/48, w_3_13=0, w_3_14=0.
Store the set of basic graphs at ℏ3 with their true weights in the file basic3.txt (not
removing graphs with zero weights); store the true Kontsevich ⋆-product up to ℏ3 in
the file star3.txt and its associator in the file assoc3.txt.
Instead of evaluating the associator in full, we could also have selected (e.g. by
reading the file assoc3+4w.txt, which also contains lines of the form “# i j k”) those
differential orders (i, j, k) at ℏ4 at which only weights from order 3 appear, in view of
Remark 11: such are (1, 3, 2), (2, 3, 1), (2, 1, 3), (3, 2, 1), (3, 1, 2), (1, 2, 3) and (2, 2, 2).
Remark 12. A substitution of the values of certain graph weights expressed via other
weights is tempting but not always effective. Namely, we do not advise repeated running
of any of the three methods with such expressions taken into account in the input.
Usually, the gain is disproportional to the time consumed; for instead of a coefficient to-
express the program now has to handle what typically is a linear combination of several
coefficients. This shows that the only types of substitutions which are effective are either
setting the coefficients to fixed numeric values (e.g., to zero) or the shortest possible
assignments of a weight value via a single other weight value (like w(Γ1) = −w(Γ2) for
some graphs Γ1 and Γ2).
3.1.2. The ⋆-product expansion at order four. At order four in the expansion of the Kon-
tsevich ⋆-product with respect to ℏ, there are 149 basic graphs Γ ∈ G̃2,4. The knowledge
of their coefficients would completely determine the ⋆-product modulo ō(ℏ4). By using
Methods 1–3 from §3.1, we found the exact values of 67 basic graphs and we expressed
the remaining 82 weights in terms of the 10 master-parameters (themselves the weights
of certain graphs from G̃2,4; the other 72 weights are linear functions of these ten).
Theorem 9. The weights of basic Kontsevich graphs at order 4 are subdivided as follows.
The weights of 27 basic graphs are equal to zero. Of these 27, the integrands of 21 weights
are identically zero, and the other 6 weight values were found to be equal to zero. The
remaining 122 weights of basic graphs Γ ∈ G̃2,4 are arranged as follows:
· 40 nonzero weights are known explicitly;
· the values of the remaining 82 weights are expressed linearly in terms of the
weights of those ten graphs which are shown in Fig. 3.
• The encoding of entire ⋆-product modulo ō(ℏ4), that is, its part up to ō(ℏ3) known
from formula (1) plus ℏ4 times the sum of all the prime and composite weighted graphs
with four internal vertices, is given in Appendix C. (In that table the weights of com-
posite graphs are numbers; for they are expressed via the known coefficients of graphs
from G̃2,⩽3.) The weights of basic graphs at ℏ4 are expressed in Table 7 in terms of the
ten master-parameters, see p. viii in Appendix C.
Moreover (as stated in Theorem 12 on p. 256 below), the associativity Assoc⋆(f, g, h) =
0 mod ō(ℏ4) is established (up to order four) for the star product ⋆ mod ō(ℏ4) at all
values of the ten master-parameters.
Proof scheme (for Theorem 9). We run the software as follows. First one generates the
sets of basic graphs up to order 4, with undetermined weights at order 4 (the weights
at order 2 and 3 are known from e.g. Example 8 and Example 26):
252 R. BURING AND A. V. KISELEV
$ cp basic3.txt basic4w.txt
$ echo 'h^4:' >> basic4w.txt
$ generate_graphs 4 --basic=yes --with-coefficients=yes \
>> basic4w.txt
(The output is listed in Table 5 of Appendix C.)
Build the ⋆-product expansion up to order 4:
$ star_product basic4w.txt > star4w_unreduced.txt
$ reduce_mod_skew --print-differential-orders star4w_unreduced.txt \
> star4w.txt
(The output is listed in Table 6 of Appendix C.)
Generate the linear cyclic weight relations at order 4:
$ cyclic_weight_relations star4w_unreduced.txt \
> weight_relations_4w-cyclic.txt
Find 21 relations of the form w_4_xxx==0 which hold by virtue of the weight integrand
vanishing in formula (5), by using Implementation 17 in Appendix A.1, and place these
relations in the file weight_relations_4w-integrandvanishes.txt.
Build the expansion of the associator for the ⋆-product up to the order 4:
$ star_product_associator star4w.txt > assoc4w.txt
(The output is listed in Table 8 of Appendix C.)
Obtain relations from the requirement of associativity for the Poisson structure (10):
$ poisson_make_vanish assoc4w.txt 3d-generic \
> weight_relations_4w-3d.txt
Merge the systems of linear equations:
$ cat weight_relations_4w-* > weight_relations_4w_total.txt
Solve the resulting system (contained in weight_relations_4w_total.txt) by using
any relevant software. One obtains the relations listed in Table 7 in Appendix C,
e.g. in the file weight_relations_4w_intermsof10.txt. To express the star-product
(respectively, the associator for the ⋆-product) in terms of the 10 parameters, run
$ substitute_relations star4w.txt \
weight_relations_4w_intermsof10.txt \
> star4_intermsof10_unreduced.txt
$ reduce_mod_skew star4_intermsof10_unreduced.txt \
> star4_intermsof10.txt
(respectively, substitute into assoc4w.txt to obtain assoc4_intermsof10.txt); see
Implementation 4. □
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 253
Remark 13. Numerical approximations of weights are listed in Tables 2 and 3 in Ap-
pendix A.1. In particular, we have the approximate values of the master-parameters
p4 = w_4_103 ≈ −1/11520 and p5 = w_4_104 ≈ 1/2880.15
Remark 14. Out of the 149 weights of basic graphs in the Kontsevich ⋆-product, as many
as 28 weights do not appear in the equation Assoc⋆(f, g, h) = 0 at ℏ4. A mechanism
which works towards such disappearance is that some graphs Γ ∈ G̃2,4 which do not
show up are bi-derivations with respect to the sinks. Combined at order four in the
associator with only the original undeformed product ×, every such graph is cancelled
out from (f ⋆[g) ⋆ h− f ⋆ ](g ⋆ h) according to the mechanism which we illustrate here:
r 
A r , • • =
 AU r 
r
A r + r 
A r r− r 
AAUr r− r r 
A r = 0.AU 
 UA 
 
 AU
In this way the ten master-parameters are split into the six which do show up in the
associativity equation and the four weights which do not show up in Assoc⋆(f, g, h) = 0
at ℏ4 but which do appear through the cyclic weight relations (see formula (8) on
p. 241).
3.2. Computer-assisted proof scheme for associativity of ⋆ for all {·, ·}P . In
practice, the methods from §3.1 stop producing linear relations that would be new
with respect to the already known constraints for the graph weights. As soon as such
“saturation” is achieved, the number of master-parameters in ⋆-product expansion may
in effect be minimal. That is, the ⋆-product, known so far up to a certain order ō(ℏk),
may in fact be always associative –modulo ō(ℏk) – irrespective of a choice of the Poisson
structure(s) P .
In this section we outline a scheme of computer-assisted reasoning that allows to re-
veal the factorization Assoc⋆(f, g, h) = ♢(P , Jac(P))(f, g, h) of associator for ⋆ via the
Jacobiator Jac(P) that vanishes by definition for every Poisson structure P . At order
k = 2 the factorization ♢(Jac(P)) is readily seen; the factorizing operator ♢(Jac(P)) =
2ℏ2 Jac(P)+ ō(ℏ2) is a differential operator of order zero, acting on its argument Jac(P)
3
by multiplication. Involving the Jacobi identity and only seven differential conse-
quences from it at the next expansion order k = 3, the factorization Assoc⋆(f, g, h) =
♢(P , Jac(P))(f, g, h) was established by hand in [12]. For higher orders k ⩾ 4 the use
of software allows to extend this line of reasoning; the scheme which we now provide
works uniformly at all orders ⩾ 2.
Let us first inspect how sums of graphs can vanish by virtue of differential conse-
quences of the Jacobi identity Jac(P) = 0 for Poisson structures P on finite-dimensional
affine real manifolds Nn. ∑
Lemma 10 ([12]). ∑A tri-differential operator C =
IJK
|I|,|J |,|K|⩾0 c ∂I ⊗ ∂J ⊗ ∂K with
coefficients cIJK ∈ C∞(Nn) vanishes identically if and only if all its homogeneous
components C IJKijk = |I|=i,|J |=j,|K|=k c ∂I ⊗ ∂J ⊗ ∂K vanish for all differential orders
(i, j, k) of the respective multi-indices (I, J,K); here ∂ = ∂α1◦· · ·◦∂αnL 1 n for a multi-index
L = (α1, . . . , αn).
15The values of ten master-parameters have been suggested by Pym and Panzer [34], see Table 4
on p. 280 in Appendix A.2 below. Their prediction completely agrees with our numeric data.
254 R. BURING AND A. V. KISELEV
Lemma 10 states in practice that for every arrow falling on the Jacobiator (for which,
in turn, a triple of arguments is specified), the expansion of the Leibniz rule yields
four fragments which vanish separately. Namely, there is the fragment such that the
derivation acts on the content P of the Jacobiator’s two internal vertices, and there
are three fragments such that the arrow falls on the first, second, or third argument of
the Jacobiator. It is readily seen that the action of a derivative on an argument of the
Jacobiator effectively amounts to an appropriate redefinition of its respective argument
(cf. Examples 19–21 on p. 243). Therefore, a restriction to the order (1, 1, 1) is enough
in the run-through over all the graphs which contain Jacobiator (4) and which stand
on the three arguments f, g, h of the operator ♢(P , Jac(P)) at hand.
Definition 6. A Leibniz graph is a graph whose vertices are either sinks, or the sources
for two arrows, or the Jacobiator (which is a source for three arrows). There must be
at least one Jacobiator vertex. The three arrows originating from a Jacobiator vertex
must land on three distinct vertices (and not on the Jacobiator itself). Each edge falling
on a Jacobiator works by the Leibniz rule on the two internal vertices in it.
An example of a Leibniz graph is given in Fig. 4. Every Leibniz graph can be
expanded to a sum of Kontsevich graphs, by expanding both the Leibniz rule(s) and all
copies of the Jacobiator. In this way (sums of) Leibniz graphs also encode (poly)differe-
ntial operators ♢(P , Jac(P)), depending on the bi-vector P and the tri-vector Jac(P).
? • There is a cycle,
• • • there is a loop,r • there are no tadpoles in this
6
r@@R
r 
 B graph,
 B • an arrow falls back on Jac(P),B
 B • and Jac(P) does not stand on? ? BN
( ) ( ) ( ) all of the three sinks.
Figure 4. A nontrivial example of Leibniz graph.
By design, we have
Proposition 11. For every Poisson bi-vector P the value – at the Jacobiator Jac(P) –
of every (poly)differential operator encoded by the Leibniz graph(s) is zero.
Proof. By induction on the number of arrows falling on the Jacobiator. In case of
zero arrows, the operator is a multiple of a Jacobiator and hence zero. In general, the
operator associated to a Leibniz graph is of the form
(∂L Jac(P))(A,B,C) ·D,
where ∂L are the incoming arrows on the Jacobiator. Now, Jac(P)(A,B,C) = 0 implies
0 = ∂L(Jac(P)(A,B,C)) ·D ∑
= (∂L Jac(P))(A,B,C) ·D + (∂H Jac(P))(∂IA, ∂JB, ∂KC) ·D
H+I+J+K=L
H≠ L
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 255
where the terms in the sum on the right are Leibniz graphs with fewer arrows falling on
the Jacobiator, hence they are zero by induction. The same proof works for a Leibniz
graph with more than one Jacobiator (the extraneous ones – in D – are irrelevant). □
Hence, to show that a sum of Kontsevich graphs vanishes at every Poisson structure,
it suffices to write it as a sum of Leibniz graphs.
In particular, the mechanism of factorization of the associator for the Kontsevich
⋆-product is known from [32]; it has been discussed in [11]. Namely, by [32] the Ja-
cobi identity is the only obstruction to the Kontsevich ⋆-product associativity. This
is because an expression of the ⋆-product associator as a (possibly, non-unique) sum
of Leibniz graphs can be predicted in advance, based on the graphs in the Formality
morphism (see [11] for more details).
Example 27. Consider the associator Assoc⋆(f, g, h) mod ō(ℏ3) for the ⋆-product
which is fully known up to order 3. The assembly of factorizing operator ♢(P , ·) acting
on Jac(P) is explained in [12]; linear in its argument, the operator ♢ has differential
order one with respect to the Jacobiator.
Remark 15. The same technique, showing the vanishing of a sum of Kontsevich graphs
by writing it as a sum of Leibniz graphs, has been used in [7]. The underlying mechanism
from [31] is analyzed in detail in [13].
Implementation 15 (Encoding of Leibniz graphs). For a Leibniz graph with ℓ Jaco-
biators and n− 2ℓ remaining bi-vector vertices, an encoding is defined in terms of the
encoding of a Kontsevich graph in its expansion, plus the data which tells where the
Jacobiators are. The full encoding is the integer ℓ, followed by the Kontsevich graph
encoding with n internal vertices, followed by the ℓ pairs of Jacobiator vertices (j1, j2),
where the internal Jacobiator edge is j1 ← j2. Each target in the Kontsevich graph
encoding which is a Jacobiator vertex ji from a Jacobiator (j1, j2) (except for the target
of the internal Jacobiator edge j1 ← j2) should be interpreted as a placeholder for a
Leibniz rule acting on both j1 and j2.
Example 28. The Leibniz graph from Fig. 4 (with n = 5 and ℓ = 1) has the encoding
1 3 5 1 0 5 3 6 3 4 3 1 6 2 6 7
Here the first 6 should be interpreted as a placeholder for the Jacobiator containing the
last two vertices 6 and 7; the three arguments of the Jacobiator are 3, 1, 2. To expand
this encoding into Kontsevich graph encodings, cyclically permute the arguments of the
Jacobiator and replace the placeholder by 6 or 7 (in all possible ways):
3 5 1 0 5 3 6 3 4 3 1 6 2
3 5 1 0 5 3 7 3 4 3 1 6 2
3 5 1 0 5 3 6 3 4 1 2 6 3
3 5 1 0 5 3 7 3 4 1 2 6 3
3 5 1 0 5 3 6 3 4 2 3 6 1
3 5 1 0 5 3 7 3 4 2 3 6 1
One obtains six terms.
Implementation 16. Let the input file <graph-series-filename> contain a graph
series S with constant (e. g., rational, real or complex) coefficients; here S is supposed
256 R. BURING AND A. V. KISELEV
to vanish by virtue of the Jacobi identity and its differential consequences. Now run
the command
> reduce_mod_jacobi <graph-series-filename>
The program finds a particular solution ♢ of the factorization problem
S(f, g, h) = ♢(P , Jac(P), . . . , Jac(P))(f, g, h).
In the standard output one obtains the list of encodings of Leibniz graphs in ♢ that
specify differential consequences of the Jacobi identity; every such graph encoding is
followed in the output by its sought-for nonzero coefficient.16 Two extra options can
be set equal to nonnegative integer values, by passing these two numbers as extra
command-line arguments. Namely,
• the pa(rameter ma)x-jacobiators restricts the number of Jacobiators in eachLeibniz graph, so that by(the assignment max-jacobiators = 1 the right-handside ♢ P , Jac(P) is linear in the Jacobiat)or, whereas if max-jacobiators =
2, the right-hand side ♢ P , Jac(P), Jac(P) can be quadratic in Jac(P), and
so on;
• independently, the parameter max-jac-indegree restricts (from above) the
number of arrows falling on the Jacobiator(s) in each of the Leibniz graphs
that constitute the factorizing operator ♢.
Furthermore, if --solve is specified as the third extra argument, the input graph series
is allowed to contain undetermined coefficients; these are then added as variables to-
solve-for in the linear system.
Theorem 12. For every component S(i) of the associator (for ⋆ from Theorem 9)
Assoc (f, g, h) mod ō(ℏ4⋆ ) =: S(0) + p S(1)1 + . . .+ p S(10)10 ,
there exists a factorizing operator (♢(i) such th)at
S(i)(f, g, h) = ♢(i) P , Jac(P) (f, g, h), 0 ⩽ i ⩽ 10∑.
• At no values of the master-parameters pi would the solution ♢ = i ♢(i) of factor-
ization problem be a first-order differential operator acting on the Jacobiator.
Proof scheme. Take the associator Assoc⋆(f, g, h) mod ō(ℏ4) for the ⋆-product expan-
sion modulo ō(ℏ4), in the file assoc4_intermsof10.txt which was obtained in The-
orem 9. The associator is linear in the ten master-parameters. Let us split it into
the constant term (e.g., at the zero value of every parameter) plus the ten respective
components S(i):
$ extract_coefficient assoc4_intermsof10.txt 1 \
> assoc4_intermsof10_constantpart.txt
$ extract_coefficient assoc4_intermsof10.txt w_4_100 \
> assoc4_intermsof10_part100.txt
$ extract_coefficient assoc4_intermsof10.txt w_4_101 \
> assoc4_intermsof10_part101.txt
16Sample outputs of specified type are contained in Table 9 in Appendix D.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 257
(and so on, for each parameter pi). In fact, four of the parameters do not show up
in the associator (see Remark 14): the corresponding files do not contain any graphs.
Now run the command reduce_mod_jacobi for each input file with S(i), e.g., for S(1):
$ reduce_mod_jacobi assoc4_intermsof10_part100.txt
For each S(i) a solution is found: the series vanishes modulo the Jacobi identity. The
output for S(1) is written in Table 9 in Appendix D. For the second part of the theorem,
we run reduce_mod_jacobi with the options max-jac-indegree = 1 and --solve:
$ reduce_mod_jacobi assoc4_intermsof10.txt 1 1 --solve
(Our setting of max-jacobiators = 1 here makes no difference.) No solution is found.
Inspecting the output, we find that the following term in the associator cannot be
produced by a first-order differential consequrence of the Jacobi identity:rA
@RUArrA
− 2 ℏ4R	?r@Rr A15 Ur
Indeed one can show this graph arises only in a differential consequence of order two. □
Corollary 13 (⋆-product non-extendability from {·, ·}P to {·, ·} at order ℏ4P ). Because
there are at least two arrows falling on the object Jac(P) in ♢ at every value of the ten
master-parameters pi, the associativity can be broken at order ℏ4 for extensions of the ⋆-
product to infinite-dimensional set-up6 on p. 236 of Nn-valued fields ϕ ∈ C∞(Mm → Nn)
over a given affine manifold Mm, of local functionals F,G,H taking such fields to
numbers, and of variational Poisson brackets {·, ·}P on the algebra of local functionals.
Indeed, the Jacobiator Jac(P) ∼= 0 for a variational Poisson bi-vector P is a coho-
mologically trivial variational tri-vector on the jet space J∞(Mm → Nn), whence the
first variation of Jac(P) brought on it by a unique arrow would of course be vanishing
identically. Nevertheless, that variational tri-vector’s density is not necessarily equal
to zero on J∞(Mm → Nn) over Mm for those variational Poisson structures whose
coefficients P ij explicitly depend on the fields ϕ or their derivatives along Mm. This
is why the second and higher variations of the Jacobiator Jac(P) would not always
vanish. (Such higher-order variations of functionals are calculated by using the tech-
niques from [23, 27].) We know from [12] that Assoc⋆(F,G,H) ∼= 0 mod ō(ℏ3), i.e.
the associator is trivial up to order ℏ3 for all variational Poisson brackets {·, ·}P but
we now see that it can contain cohomologically nontrivial terms proportional to ℏ4.
Consequently, it is the order four at which the associativity of ⋆-products can start to
leak in the course of deformation quantization of Poisson field models.
We now claim that four master-parameters can simultaneously be gauged out of the
star-product. (That is, either some of the four or all of them at once can be set equal
to zero, although this may not necessarily be their true value given by formula (5).)17
17Let us recall that the property of a parameter in a family of star-products to be removable by
some gauge transformation is not the same as setting such parameter to zero (or any other value).
Indeed, other graph coefficients, not depending on the parameter at hand, might get modified by that
gauge transformation. However – and similarly to the removal of the loop graph at ℏ2 in the Kontsevich
258 R. BURING AND A. V. KISELEV
Theorem 14. For each j ∈ {2, 3, 9, 10} there exists a gauge transformation id+ℏ4pjZj
(listed in Table 10 in Appendix E) such that the master-parameter pj is reset to zero in
the deformed star-product ⋆′. This is achieved in such a way that no graph coefficients
which initially did not contain the parameter to(g∑auge ou)t would change at all.
• Moreover, the gauge transformation id+ ℏ4 · j pjZj removes at once all the four
master-parameters, still preserving those coefficients of graphs in ⋆ which did not depend
on any of them.
Proof scheme. Let the ⋆-product expansion in terms of 10 parameters (obtained in The-
orem 9) be contained in star4_intermsof10.txt. Construct a gauge transformation
of the form id + ℏ4G, where G is the sum over all possible graphs with four internal
vertices over one sink which are nonzero, without double edges, without tadpoles, and
with positive differential order, taken with undetermined coefficients gi:
$ cat > gauge4.txt
1 0 1 1
h^4:
^D (press Ctrl+D)
$ generate_graphs 4 1 --normal-forms=yes --zero=no \
--positive-differential-order=yes \
--with-coefficients=yes >> gauge4.txt
$ sed -i 's/w/g/' gauge4.txt # replace coefficient prefix 'w' by 'g'
Obtain gauged star-product expansion ⋆′ by applying the gauge transformation to ⋆:
$ gauge star4_intermsof10.txt gauge4.txt \
> star4_intermsof10_gauged_unreduced.txt
Reduce the graph series for ⋆′ modulo skew-symmetry:
$ reduce_mod_skew star4_intermsof10_gauged_unreduced.txt \
> star4_intermsof10_gauged.txt
Inspect which of the 10 parameters pj cannot be gauged out, by checking for the exis-
tence of graph coefficients containing pj but not any gi. For example, for p1 = w_4_100:
$ grep w_4_100 star4_intermsof10_gauged.txt \
| grep -v g | wc -l
17
There are 17 graphs with such coefficients, so p1 = w_4_100 cannot be gauged out.
Following this procedure for all the 10 parameters, we find that the only candidates to be
gauged out are p2 = w_4_101, p3 = w_4_102, p9 = w_4_119, and p10 = w_4_125. Now
inspect the file star4_intermsof10_gauged.txt for the lines containing these pj and
(necessarily, some) gi. For each pj, find a choice of gi so that pj is completely removed
from the file. (The gi will be of the form gi = αijpj for αij ∈ R.) It turns out that this
is always possible. Hence this choice of gi defines the sought-for gauge transformation
id + ℏ4pjZj which gauges out the parameter pj. The gauge-transformations which
⋆-product (see Examples 24 and 25) – the trivialization of four parameters at no extra cost is the case
which Theorem 14 states.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 259
kill the∑(four) parameters separately may be combined into the gauge-transformation
id + ℏ4( j pjZj) that kills all (four) of them simultaneously. □
Remark 16. The master-parameters which we can gauge out are exactly the ones which
do not show up in the associativity equation (see Remark 14).
Let us finally address a possible origin of so ample a freedom in the ten-parameter
family of star-products (now known up to ō(ℏ4)). We claim that the mechanism of
vanishing via differential consequences of the Jacobi identity, which was recalled in
Lemma 10 and used in Theorem 12, starts working not only for the associator built
over ⋆, but it may even start working for the ⋆-product expansion itself.
∑ (Theorem)15. The ten-parameter family of star-product expansions ⋆ = ...+ ℏ4 ⋆(0) +10 p (i) 4i=1 i⋆ + ō(ℏ ) does contain, in the ten-dimensional affine subspace parametrized by
p1, . . . , p10 in k[G2,4], a unique one-dimensional (null or ‘improper’) subspace such that
every point α · (⋆(9)−2⋆(6)) = α ·⋆(9|6) in it admits a Leibniz graph factorization (via the
Jacobiator) ⋆(9|6) = ∇(P , Jac(P)) ∈ k[G2,4]. This null space is the span of the direction
w_4_119 : w_4_107 : . . . = 1 : (−2) : 0 : ... : 0 ∈ RP9, that is, the master-parameters p9
and p6 occur in proportion 1 : (−2) and all the other pi’s are zero.
In effect, the respective part of the star-product always cancels out for every given
Poisson structure P . This factorization and uniqueness of the direction ⋆(9|6) is estab-
lished by using the same computer-assisted scheme of reasoning which worked in the
proof of Theorem 12.
4. Discussion
The coefficients of (sometimes different, sometimes gauge-inequivalent) star-product
expansions up to low orders were previously obtained in the papers [21, 35, 1, 6, 38].
Let us compare the result in this paper with those publications, and let us use the
software described in this paper to verify some results about other star-products.
4.1. Previously known weights. The values of some (families of) Kontsevich graph
weights are given in the literature. The graphs in the Bernoulli family have scaled
Bernoulli numbers as weights (see [6, Corollary 6.3] or [21, Proposition 4.4.1]), e.g.
w_3_2 = B3/3! = 0 and w_4_12 = B4/4! = −1/720. The weights of a family of graphs
containing cycles are obtained in [6, Corollary 6.3], e.g. w_3_9 = ±B3/(2 · 3!) = 0
and w_4_72 = −B4/(2 · 4!) = 1/1440. Willwacher states in [38] the vanishing of three
graph weights at the order 3 (they are w_3_7, w_3_13, w_3_14 in Figure 2) and the
non-vanishing one other (it is w_3_12 in Figure 2); this agrees with our calculation in
Example 26.
4.2. Numerical approximation. In Tables 2 and 3 in Appendix A.1 we list numerical
approximations of several weights. These approximations are consistent with the exact
weights (and relations) obtained in this paper.
4.3. Independent symbolic calculation. The values for the weights found in this
paper agree with a symbolic calculation of the graph weights reported by Pym and
Panzer [34] and reproduced in Table 4 on p. 280 in Appendix A.2.
260 R. BURING AND A. V. KISELEV
4.4. The obstruction to the existence of a loopless star product. In [38],
Willwacher establishes that any universal star-product (defined by the Kontsevich
graphs, possibly with different coefficients) which is gauge-equivalent to Kontsevich’s
⋆-product must contain graphs with 2-cycles. To obtain the same result using our
software, we proceed as follows.
Example 29. We start with Kontsevich’s ⋆-product up to ℏ3 in star3.txt. The unique
graph with a loop at order 2 can be removed by extending the gauge transformation
from Example 23 which was stored in gaugeloop.txt:
$ cp gaugeloop.txt gaugeloop3.txt
$ echo "h^3:" >> gaugeloop3.txt
$ gauge star3.txt gaugeloop3.txt > star3_gauge2_unreduced.txt
$ reduce_mod_skew star3_gauge2_unreduced.txt > star3_gauge2.txt
The gauged ⋆-product is obtained in star3_gauge2.txt:
h^0:
2 0 1 1
h^1:
2 1 1 0 1 1
h^2:
2 2 1 0 1 1 2 -1/3
2 2 1 0 1 0 2 1/3
2 2 1 0 1 0 1 1/2
h^3:
2 3 1 0 1 1 2 1 2 1/6
2 3 1 0 1 0 1 1 2 -1/3
2 3 1 0 1 0 2 0 2 1/6
2 3 1 0 1 0 1 0 2 1/3
2 3 1 0 1 0 1 0 1 1/6
2 3 1 0 3 1 2 2 3 -1/6
2 3 1 0 1 2 4 2 3 1/12
2 3 1 0 1 0 2 1 3 -1/6
2 3 1 0 1 0 4 1 2 -1/6
2 3 1 0 3 1 4 1 3 -1/6
2 3 1 0 1 1 2 2 3 -1/6
2 3 1 0 3 1 2 1 2 1/6
2 3 1 0 1 1 4 2 3 1/6
2 3 1 0 3 0 2 1 2 1/6
2 3 1 0 1 0 2 2 3 -1/6
2 3 1 0 3 1 2 0 3 -1/6
2 3 1 0 1 0 4 2 3 1/6
Willwacher denotes this ⋆-product by a = a0 + a1 + a2 + a3 + . . ., and supposes that
our (desirably loopless) ⋆-product reads b = a0 + a1 + a2 + (a3 + b3) + (a4 + b4) + . . .
The Maurer-Cartan associativity equation [b, b] = 0 implies in particular [a0, b3] = 0
and [a1, b3] + [a0, b4] = 0 (here [−,−] is the Gerstenhaber bracket).
Claim. For any solution of [b, b] = 0, the sum a3 + b3 contains graphs with cycles.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 261
We carry out Willwacher’s proof, with some minor corrections. Since b3 is a Hochschild
cocycle it suffices to assume b3 is in the image of the (graphical) Hochschild-Kostant-
Rosenberg map, so it is a skew-symmetric bi-derivation.18 There are two terms −αA
and −βB in a3 (α, β =6 0) which are skew-symmetric bi-derivations and graphs with
cycles. So for a3+b3 to have no cycles, b3 must be the linear combination b3 = αA+βB.
The proof proceeds by showing that [a1, b3] + [a0, b4] = 0 cannot hold: indeed, simplify-
ing the 3-cochain [a1, b3] modulo the image of [a0,−] and the Jacobi identity results in
a sum of graphs (called Shoikhet’s obstruction) which does not vanish. Let us illustrate
all of this in detail.
Example 30. The skew bi-derivation terms in a3 are:
2 3 1 0 1 2 4 2 3 1/12
2 3 1 0 3 1 2 2 3 -1/6
(We have α = −1/12 and β = 1/6.) We store the correction term b3 in the file b3.txt:
2 3 1 0 1 2 4 2 3 -1/12
2 3 1 0 3 1 2 2 3 1/6
Calculate the Gerstenhaber bracket [a1, b3] (using a1 in wedge.txt)19:
$ echo '2 1 1 0 1 1' > wedge.txt
$ gerstenhaber_bracket wedge.txt b3.txt > \[wedge,b3\]_unreduced.txt
$ reduce_mod_skew \[wedge,b3\]_unreduced.txt > \[wedge,b3\].txt
Generate an ansatz for b4:
$ generate_graphs 4 --normal-forms=yes --with-coefficients=yes \
--positive-differential-order=yes --zero=no > b4.txt
Calculate [a0, b4] (using a0 in dotdot.txt):
$ echo '2 0 1 1' > dotdot.txt
$ gerstenhaber_bracket dotdot.txt b4.txt > \[dotdot,b4\]_unreduced.txt
$ reduce_mod_skew \[dotdot,b4\]_unreduced.txt > \[dotdot,b4\].txt
Store Shoikhet’s obstruction with undetermined coefficients A,B in shoikhet_obs.txt:
3 4 1 0 1 2 3 3 4 3 4 A
3 4 1 2 1 0 3 3 4 3 4 -A
3 4 1 0 1 2 3 3 4 4 5 B
3 4 1 2 1 0 3 3 4 4 5 -B
Add [a0, b4] and Shoikhet’s obstruction to [a1, b3]:
$ cat \[wedge,b3\].txt \[dotdot,b4\].txt shoikhet_obs.txt \
> \[wedge,b3\]+\[dotdot,b4\]+shoikhet_obs_unreduced.txt
$ reduce_mod_skew \
\[wedge,b3\]+\[dotdot,b4\]+shoikhet_obs_unreduced.txt \
> \[wedge,b3\]+\[dotdot,b4\]+shoikhet_obs.txt
18However, this does not imply that each individual graph in it is a skew-symmetric bi-derivation.
Rather, each graph which is a bi-derivation can be skew-symmetrized, which yields either the original
graph or the sum of two graphs which are mirror-reflections of each other. It is clear that this is what
Willwacher intended, e.g. because the mirror-reflection of his graph D is not drawn.
19The graphical calculation of [a1, b3] in [38] contains errors, e.g. the first graph has a vertex with
three outgoing edges and the term with coefficient β has arrows in the wrong direction.
262 R. BURING AND A. V. KISELEV
Reduce modulo the Jacobi identity and solve (for the expression to be equal to zero):
$ reduce_mod_jacobi \[wedge,b3\]+\[dotdot,b4\]+shoikhet_obs.txt \
1 10 --solve
Indeed, there is a solution A = β = 1/6 and B = −4(α + β) = −1/3. So, modulo
the image of [a0,−] and the Jacobi identity, [a1, b3] is equal to Shoikhet’s obstruction
with A = −β = −1/6 and B = 4(α + β) = 1/3 (the sign changed because we added
Shoikhet’s obstruction instead of subtracting it).20
Example 31. An example of a Poisson structure for which Shoikhet’s obstruction
doesn’t vanish is given by 3d-polynomial:
$ poisson_evaluate shoikhet_obs.txt 3d-polynomial
...
# [ x ] [ x ] [ y ]
-4*A*y^3*z^2*x^2+2*y^3*B*z*x^3-2*y*B*z^3*x^3+y*B*z^4*x^2+...
...
For example, the coefficient of the differential monomial x2y3z2∂x⊗∂x⊗∂y is −4A 6= 0.
In this section we traced Willwacher’s steps. There is a much simpler proof of the
claim when all the coefficients of graphs in a3 are known (which was not the case in [38]):
there are loopful graphs with nonzero coefficients in a3 which cannot be gauged out.
4.5. Penkava–Vanhaecke deformations. In [35] M. Penkava and P. Vanhaecke give
(among other things) deformations π⋆ = π + hπ1 + h2π2 + . . . + hnπn + ō(hn) where
π is the pointwise product, π1 is the Poisson bracket, h is the formal parameter, and
associativity holds modulo ō(hn) for arbitrary polynomial Poisson algebras. Note that
every ⋆-product (which is associative as a formal power series in h) induces such an
expansion modulo ō(hn) for every n, but not every deformation modulo ō(hn) can be
extended to higher orders.21 Indeed, Penkava–Vanhaecke exhibit deformations which
can be extended and some which cannot be extended. (Namely, already at order 3
there exist formulas which do not extend to higher orders — although such formulas
are clearly not the genuine Kontsevich star-product.) In the following sequence of
examples, we verify some of their results and compare them with ours.
Example 32. Proposition 5.1 in [35] gives a deformation π+hπ 21+h π2+ ō(h2), and in
fact it coincides with Kontsevich’s ⋆-product modulo ō(h2) with the loop graph gauged
out; see Example 25 in this text.
Example 33. Theorem 5.6 in [35] provides a deformation π⋆ = π+hπ +h21 π2+h3π3+
ō(h3). The differential polynomials in it can be viewed as Kontsevich graphs; we store
their encodings with their numerical coefficients in star3pv5.6.txt:
h^0:
2 0 1 1
20The solution reported here differs from Willwacher’s, not in sign (which is left ambiguous in [38])
but in proportion: he claims A = ±2β and B = ±2(α+ β).
21Note that this problem is different from the computation of obstructions to Kontsevich’s Formality
[32, 31]. Specifically, in “Formality Conjecture” [31], Kontsevich reports the absence of obstructions
to Formality up to n ⩽ 6. Formality is now a theorem: Kontsevich’s ⋆-product exists at all orders.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 263
h^1:
2 1 1 0 1 1
h^2:
2 2 1 0 1 1 2 -1/3
2 2 1 0 1 0 2 1/3
2 2 1 0 1 0 1 1/2
h^3:
2 3 1 0 1 1 2 2 3 -1/3
2 3 1 0 1 0 2 2 3 -1/3
2 3 1 0 1 0 1 0 1 1/6
2 3 1 0 1 0 2 1 3 -1/6
2 3 1 0 1 0 4 1 2 -1/6
2 3 1 0 1 1 2 1 2 1/6
2 3 1 0 1 0 2 0 2 1/6
2 3 1 0 1 0 1 0 2 1/3
2 3 1 0 1 0 1 1 2 -1/3
Calculate the associator in terms of graphs (see Implementation 12):
$ star_product_associator star3pv5.6.txt > assoc3pv5.6.txt
It vanishes as a consequence of the Jacobi identity (see Implementation 16):
$ reduce_mod_jacobi assoc3pv5.6.txt
Hence π⋆ is associative modulo ō(h3) (for arbitrary Poisson structures on Rd). This
deformation is not equal to Kontsevich’s ⋆-product modulo ō(h3), nor is it Kontsevich’s
⋆-product with the loop graph gauged out, but the next example gives the explicit
relation between this product and Kontsevich’s.
Example 34. Theorem 5.6 in [35] further relates arbitrary deformations to π⋆ = π +
hπ1 + h
2π2 + h
3π3 + ō(h
3) from Example 33. Namely, every deformation modulo ō(h3)
is gauge-equivalent to π̃⋆ = π + hπ1 + h2(π + φ ) + h32 2 (π3 + φ3 + ψ3) for some choice
of (φ2, φ3, ψ3), where φ2 and φ3 are antisymmetric biderivations and ψ3 is a symmetric
2-cochain satisfying ∂ψ3 = [π1, φ2]. Let us show that this holds for Kontsevich’s ⋆-
product expansion πK 3⋆ mod ō(h ). In Section 4.4 we obtained Kontsevich’s ⋆-product
with the loop graph gauged out; let us denote it by π̃K⋆ mod ō(h3). Up to ō(h2) the
deformations π̃K⋆ and π⋆ are equal (as we observed in Example 32), so we choose φ2 = 0.
Subtracting π⋆ from π̃K⋆ yields the file star3_gauge2_minus_pv5.6.txt:
h^3:
# 1 1
2 3 1 0 3 1 2 2 3 -1/6
2 3 1 0 1 2 4 2 3 1/12
# 1 2
2 3 1 0 3 1 4 1 3 -1/6
2 3 1 0 1 1 2 2 3 1/6
2 3 1 0 3 1 2 1 2 1/6
2 3 1 0 1 1 4 2 3 1/6
# 2 1
2 3 1 0 3 0 2 1 2 1/6
264 R. BURING AND A. V. KISELEV
2 3 1 0 1 0 2 2 3 1/6
2 3 1 0 3 1 2 0 3 -1/6
2 3 1 0 1 0 4 2 3 1/6
and it can be seen that it is antisymmetric, so this must be h3φ3 and hence ψ3 = 0. But
the terms are not all of differential order (1, 1), so how can φ3 be a bi-derivation? The
answer is that all other terms vanish due to two first-order differential consequences of
the Jacobi identity for the Poisson structure. In other words, there is a Leibniz graph
– with one arrow incoming on the Jacobiator – which expands to the homogeneous
component of order (1, 2), and naturally the mirror-reflection of that Leibniz graph
expands to the order (2, 1) component. This can be verified using reduce_mod_jacobi.
Example 35. Theorem 6.1 in [35] gives the obstruction to extending π⋆ from Ex-
ample 33 to the fourth order. The proof shows that this obstruction is the skew-
symmetrization of the degree-(1, 1, 1) homogeneous component of the associator at h4.
We reproduce this as follows. First create a file representing π⋆ mod ō(ℏ4):
$ cp star3pv5.6.txt star4pv6.1.txt
$ echo "h^4:" >> star4pv6.1.txt
Calculate the associator:
$ star_product_associator star4pv6.1.txt > assoc4pv6.1.txt
Skew-symmetrize (see §4.7 below):
$ skew_symmetrize assoc4pv6.1.txt > obs4pv6.1_unreduced.txt
Reduce modulo skew-symmetry:
$ reduce_mod_skew obs4pv6.1_unreduced.txt \
--print-differential-orders > obs4pv6.1.txt
Finally, we see that the degree-(1, 1, 1) homogeneous component at ℏ4 is
3 4 1 0 1 2 3 3 4 4 5 4/3
3 4 1 0 2 1 3 3 4 4 5 -4/3
3 4 1 0 4 1 2 3 4 3 5 -4/3
3 4 1 0 1 2 3 3 4 3 4 -2/3
3 4 1 0 2 1 3 3 4 3 4 2/3
3 4 1 0 4 1 2 3 4 3 4 -2/3
which is (up to an irrelevant constant factor) the sum of six terms written in Theo-
rem 6.1. We store this component in the file obs4pv6.1_111.txt and the others in
obs4pv6.1_rest.txt. The latter vanish as a consequence of the Jacobi identity:
$ reduce_mod_jacobi obs4pv6.1_rest.txt
as claimed in Theorem 6.1. The (1, 1, 1)-component does not vanish in general: indeed,
$ reduce_mod_jacobi obs4pv6.1_111.txt
does not find any solution. An explicit Poisson structure for which the obstruction does
not vanish is given at the end of [35, §9]. We can do the same computation in our
software: the respective Poisson structure was added under the name 4d-pv, so that
$ poisson_evaluate obs4pv6.1_111.txt 4d-pv
shows a multi-vector field which is not identically zero.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 265
Example 36. Finally Lemma 8.2 in [35] gives the correction term φ3 to π3 for the
deformation to extend to the fourth order. In terms of graph encodings with coefficients,
φ3 is:
2 3 1 0 3 1 2 2 3 -1/6
2 3 1 0 1 2 4 2 3 1/12
which is exactly the correction term we found in Example 34 to make the deformation
equal to Kontsevich’s ⋆-product modulo ō(h3) with the loop graph gauged out. This
proves that the deformation extends to the fourth order. Alternatively, we can store
the full expansion in star3pv8.2.txt and confirm that it extends to the order 4 as
follows. First, add graphs with undetermined coefficients at ℏ4:
$ cp star3pv8.2.txt star4pv8.2.txt
$ echo 'h^4:' >> star4pv8.2.txt
$ generate_graphs 4 --normal-forms=yes --with-coefficients=yes \
>> star4pv8.2.txt
Calculate the graphical associator:
$ star_product_associator star4pv8.2.txt > assoc4pv8.2.txt
Finally, run
$ reduce_mod_jacobi assoc4pv8.2.txt 1 2 --solve
and observe that there is a solution.
4.6. Universal star-products. We do work on affine Poisson manifolds, so that for-
mulae are coordinate-independent because of the contraction of upper versus lower
indices in all tensor objects and because all the Jacobians are constant in the course
of affine coordinate reparametrizations. S. Gutt et al in [1] provide star-products mod-
ulo ō(ℏ3) which are universal with respect to all Poisson structures P on all smooth
finite-dimensional manifoldsMd equipped with a torsion-free, not necessarily flat, lin-
ear connection ∇. Then the formula of ⋆ mod ō(ℏ3) is expressed in terms of differential
polynomials in not only the bi-vector P – clearly, our ∂i replaced by∇i in every instance
– but also in the curvature R of ∇.
Example 37. To compare with Kontsevich’s formula up to ō(ℏ3) which is given in the
present paper (also in [12]), we can put R = 0 in the formula by Gutt et al. The terms
up to ō(ℏ2) clearly match. A-priori there are (5−1)×(5−1) = 16 terms with coefficient
−1/6 at ℏ3. Two pairs of terms double, and they become two terms with coefficients
±1/3. One term vanishes identically, because it is the zero graph from Example 5.
The resulting 13 terms are exactly those in Kontsevich’s formula (1) at ℏ3. Hence
the formula obtained by Gutt et al. restricted to R = 0 coincides with Kontsevich’s
⋆-product up to ō(ℏ3).
It would be interesting to recover such a univeral formula ⋆(P , R) – depending also
on the curvature R – modulo ō(ℏ4) and beyond.
4.7. Universal flows on spaces of Poisson structures. The software presented in
this paper has been extended to operate on first-order differential operators which rep-
resent (skew-symmetric) multi-vector fields. In particular skew-symmetrization was im-
plemented in skew_symmetrize and the graphical Schouten bracket was implemented
266 R. BURING AND A. V. KISELEV
in schouten_bracket. This has been applied by the authors jointly with A. Bouis-
aghouane in [7] to confirm the existence of a universal flow on the spaces of Poisson
structures, which was suggested by Kontsevich. The explicit mechanism that explains
why these universal flows exist, based on work by Kontsevich, Willwacher, and Jost, is
given by the authors in [13].
4.8. Open problems. The following two questions, posed by M. Kontsevich (private
communication) can be approached up to finite orders in ℏ by using the software mod-
ules which we have presented:
• Which quadratic weight relations are determined by the associativity alone?
(We refer to the preprint [2, p. 61] for discussion.)
• How many degrees of freedom in the graph weights (at a fixed order in ℏ) are
due to gauge transformations?
Independently (Kevin Morand, private communication), an open problem is to de-
scribe the action of the graph complex (with suitable cocycles γ ∈ ker[•−•,−]) on the
⋆-product under the infinitesimal symmetries ℏP 7→ ℏP+ε O⃗r(γ)(ℏP)+ ō(ε) of Poisson
structures (see [7, 8, 13] and [29, 30, 31]).
For a long time, the third and fourth order expansion of Kontsevich’s ⋆-product was
unknown to the physics community, which may have delayed the implementation of
deformation quantization in the study of Nature. No model of that theory could be
tested approximately because it could not be known what any such model actually was.
This is why we present the formula ⋆ mod ō(ℏ4) in Eq. (13) on pp. 280–284 in this
paper.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 267
Conclusion
The expansion of Kontsevich’s star-product modulo ō(ℏ4) is (here f, g ∈ C∞(Nn))
(
f ⋆ g = f × g + ℏP ij∂if∂jg + ℏ2 1P ijPkℓ∂ ∂ 1 ij kℓ2 k if∂ℓ)∂jg +(3∂ℓP P ∂k∂if∂jg
− 1∂ P ijPkℓ3 ℓ ∂if∂ ∂
1
k jg − 6∂ℓP
ij∂ Pkℓj ∂if∂kg + ℏ3 1P ijPkℓPmn6 ∂m∂k∂if∂n∂ℓ∂jg
+ 13∂nP
ijPkℓPmn∂ ∂ ∂ f∂ ∂ g − 1∂ P ijPkℓm k i ℓ j 3 n P
mn∂k∂if∂m∂ℓ∂jg
− 1P ij∂ Pkℓ∂ Pmn6 n ℓ ∂k∂if∂m∂jg +
1
6∂ ∂ P
ijPkℓPmnn ℓ ∂m∂k∂if∂jg
+ 1∂ ∂ P ijPkℓPmn∂ f∂ ∂ ∂ g − 1∂ ∂ P ij∂ PkℓPmn6 n ℓ i m k j 6 m ℓ n ∂k∂if∂jg
− 1∂ ∂ P ij∂ PkℓPmn6 m ℓ n ∂if∂k∂jg −
1∂ P ijPkℓ∂ Pmn6 n ℓ ∂k∂if∂m∂jg
− 1∂ P ij∂ PkℓPmn6 ℓ n ∂k∂
1
if∂m∂jg + 6∂ ∂ P
ij
n ℓ ∂jPkℓPmn∂if∂m∂kg
( )− 1 ij6∂ℓP ∂n∂ PkℓPmnj ∂m∂if∂kg − 1∂ ij kℓ mn6 m∂ℓP ∂n∂jP P ∂if∂kg
+ ℏ4 16∂qP
ijPkℓPmnPpq∂p∂m∂k∂ f∂ ∂ ∂ g − 1∂ P ijPkℓPmnPpqi n ℓ j 6 q ∂m∂k∂if∂p∂n∂ℓ∂jg
+ 16∂q∂nP
ijPkℓPmnPpq∂p∂m∂k∂if∂ℓ∂jg + 16∂q∂nP
ijPkℓPmnPpq∂k∂if∂p∂m∂ℓ∂jg
− 16∂p∂nP
ijPkℓ∂qPmnPpq∂m∂k∂if∂ℓ∂jg − 16∂p∂nP
ijPkℓ∂qPmnPpq∂k∂if∂m∂ℓ∂jg
− 1∂ P ijPkℓ6 q P
mn∂ Ppqn ∂m∂k∂if∂p∂ℓ∂jg − 16∂
ij
nP Pkℓ∂ PmnPpqq ∂m∂k∂if∂p∂ℓ∂jg
+ 1P ij∂ ∂ Pkℓ6 q n ∂ P
mn
ℓ Ppq∂k∂if∂p∂ ∂ g − 1P ijm j 6 ∂
kℓ
nP ∂q∂ Pmnℓ Ppq∂p∂k∂if∂m∂jg
− 1 ij6P ∂p∂nP
kℓ∂ mn pq 1 ij kℓ mn pqq∂ℓP P ∂k∂if∂m∂jg − 9∂nP ∂qP P P ∂m∂k∂if∂p∂ℓ∂jg
− 19∂ ∂ P
ij
p n ∂qPkℓPmnPpq∂ ∂ ∂ f∂ 1 ij kℓ mn pqm k i ℓ∂jg − 9∂p∂nP ∂qP P P ∂k∂if∂m∂ℓ∂jg
+ 1 P ijPkℓPmn24 P
pq∂p∂m∂k∂if∂q∂n∂ℓ∂jg − 1 ij12P P
kℓ∂ mnqP ∂ Ppqn ∂m∂k∂if∂p∂ℓ∂jg
+ 118∂nP
ij∂ kℓqP PmnPpq∂p∂ ∂ ∂ f∂ ∂ g − 1 ∂ P ijPkℓ∂ Pmn∂ Ppqm k i ℓ j 18 ℓ q n ∂m∂k∂if∂p∂jg
+ 1 ∂ P ij∂ kℓ mn pq18 n qP P P ∂ ∂ f∂ ∂ ∂ ∂ g +
1 ∂ P ij∂ Pkℓ∂ Pmn pqk i p m ℓ j 18 q n ℓ P ∂k∂if∂p∂m∂jg
+ 1 ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq72 ℓ j q n ∂m∂if∂p∂kg −
1
18∂ P
ij
n ∂ Pkℓp ∂ mn pqqP P ∂m∂k∂if∂ℓ∂jg
− 118∂
ij
nP ∂ Pkℓp ∂qPmnPpq∂ ∂ f∂ ∂ ∂ g + 1k i m ℓ j 30∂q∂n∂ P
ijPkℓℓ PmnPpq∂p∂m∂k∂if∂jg
− 130∂q∂n∂
ij
ℓP PkℓPmnPpq∂if∂p∂m∂k∂jg + 2 ij kℓ mn pq45∂n∂ℓP ∂qP P P ∂p∂m∂k∂if∂jg
− 2 ∂ ∂ P ij∂ kℓ mn pq 1 ij kℓ mn pq45 n ℓ qP P P ∂if∂p∂m∂k∂jg − 30∂q∂n∂ℓP P P P ∂m∂k∂if∂p∂jg
+ 1 ∂ ∂ ∂ P ijPkℓPmnPpq 730 q n ℓ ∂k∂if∂p∂m∂jg − 90∂p∂n∂ℓP
ij∂ kℓqP PmnPpq∂m∂k∂if∂jg
+ 7 ∂ ∂ ∂ P ij∂ Pkℓ90 p n ℓ q P
mnPpq∂if∂m∂k∂ 1jg + 30∂ℓP
ij∂ ∂ PkℓPmnq n Ppq∂p∂m∂k∂if∂jg
− 1 ∂ P ij∂ ∂ PkℓPmnPpq30 ℓ q n ∂if∂
1 ij kℓ mn pq
p∂m∂k∂jg − 45∂ℓP ∂nP ∂qP P ∂p∂m∂k∂if∂jg
+ 145∂ℓP
ij∂ Pkℓ∂ PmnPpq∂ f∂ ∂ ∂ ∂ g + 1 ∂ ∂ P ij∂ Pkℓn q i p m k j 45 q ℓ n P
mnPpq∂m∂k∂if∂p∂jg
− 1 ∂ ij kℓ mn pq 1 ij kℓ mn pq45 n∂ℓP P ∂qP P ∂k∂if∂p∂m∂jg + 30∂ℓP ∂q∂nP P P ∂m∂k∂if∂p∂jg
− 130∂ P
ijPkℓ∂ ∂ PmnPpq∂ ∂ f∂ ∂ ∂ g − 1 ∂ P ij kℓ mn pqn q ℓ k i p m j 90 ℓ ∂nP ∂qP P ∂m∂k∂if∂p∂jg
+ 190∂qP
ijPkℓ∂ Pmn∂ Ppq∂ ∂ f∂ ∂ ∂ g + 1 ∂ ∂ P ij∂ ∂ Pkℓ mn pqℓ n k i p m j 90 p ℓ q n P P ∂m∂k∂if∂jg
− 1 ij kℓ mn pq 1 ij kℓ mn pq90∂p∂ℓP ∂q∂nP P P ∂if∂m∂k∂jg − 30∂p∂ℓP ∂nP ∂qP P ∂m∂k∂if∂jg
+ 1 ∂ ∂ P ij30 p ℓ ∂ P
kℓ
n ∂ Pmnq Ppq∂if∂ ∂ ∂ g − 2 ij kℓ mn pqm k j 45∂ℓP ∂p∂nP ∂qP P ∂m∂k∂if∂jg
268 R. BURING AND A. V. KISELEV
+ 245∂
ij
ℓP ∂ ∂ Pkℓp n ∂qPmnPpq∂if∂ ∂ ∂ g − 1 ij kℓ mn pqm k j 30∂q∂ℓP P P ∂nP ∂m∂k∂if∂p∂jg
+ 1 ∂ ∂ P ij∂ PkℓPmnPpq∂ ∂ f∂ ∂ ∂ g − 1 ∂ P ij30 n ℓ q k i p m j 15 ℓ ∂qP
kℓPmn∂nPpq∂m∂k∂if∂p∂jg
+ 1 ∂ P ij∂ Pkℓ∂ PmnPpq∂ ∂ f∂ ∂ ∂ g − 1 ij kℓ mn pq15 n q ℓ k i p m j 30∂p∂ℓP ∂qP P ∂nP ∂m∂k∂if∂jg
+ 1 ∂ ∂ P ij∂ PkℓPmn∂ Ppq∂ f∂ ∂ ∂ g − 1 ∂ ∂ P ijPkℓ∂ Pmn∂ Ppq30 p ℓ q n i m k j 45 p ℓ q n ∂m∂k∂if∂jg
+ 1 ∂ ∂ P ijPkℓ∂ Pmn∂ Ppq∂ f∂ ∂ ∂ g + 1 ∂ P ij kℓ mn pq45 p ℓ q n i m k j 90 ℓ ∂pP ∂qP ∂nP ∂m∂k∂if∂jg
− 1 ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq∂ f∂ ∂ ∂ g + 2 ∂ ∂ ∂ P ij∂ PkℓPmnPpq90 ℓ p q n i m k j 45 p n ℓ q ∂k∂if∂m∂jg
− 2 ∂ ∂ ∂ P ijPkℓ∂ PmnPpq45 p n ℓ q ∂k∂if∂m∂jg +
1
15∂
ij
ℓP ∂ ∂ kℓ mn pqq nP P P ∂k∂if∂p∂m∂jg
− 1 ∂ P ijPkℓ15 n ∂
mn
q∂ℓP Ppq∂p∂k∂if∂m∂ g + 1 ∂ P ij∂ Pkℓ∂ Pmnj 90 ℓ n q P
pq∂k∂if∂p∂m∂jg
− 1 ∂ P ijPkℓ∂ Pmn∂ Ppq90 q ℓ n ∂m∂k∂
1 ij kℓ mn pq
if∂p∂jg + 90∂p∂ℓP P ∂qP ∂nP ∂k∂if∂m∂jg
− 1 ∂ ∂ P ij∂ Pkℓ∂ PmnPpq∂ ∂ f∂ ∂ g + 1 ∂ P ij∂ Pkℓ∂ Pmn pq90 q m n ℓ k i p j 90 p q ℓ ∂nP ∂m∂k∂if∂jg
− 1 ∂ P ij∂ kℓ mn pq 1 ij kℓ mn pq90 p qP ∂ℓP ∂nP ∂if∂m∂k∂jg + 90∂pP P ∂q∂ℓP ∂nP ∂m∂k∂if∂jg
− 1 ∂ P ijPkℓ90 p ∂
mn
q∂ℓP ∂nPpq∂ f∂ ∂ ∂ g − 1 ∂ P ij∂ Pkℓ∂ ∂ Pmn pqi m k j 30 m n q ℓ P ∂p∂k∂if∂jg
+ 1 ∂ P ij∂ Pkℓ∂ ∂ PmnPpq∂ f∂ ∂ ∂ g + 1 ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq30 m n q ℓ i p k j 90 q j ℓ n ∂m∂k∂if∂pg
+ 1 ∂ P ij90 q ∂ P
kℓ
j ∂ Pmnℓ ∂nPpq∂if∂ ∂ ∂ g + 1 P ij∂ ∂ Pkℓ mnp m k 90 q j ∂ℓP ∂ P
pq
n ∂m∂k∂if∂pg
+ 1 ∂ P ij90 n ∂
kℓ
q∂jP ∂ℓPmnPpq∂ f∂ 1 ij kℓ mn pqi p∂m∂kg + 90P ∂jP ∂q∂ℓP ∂nP ∂m∂k∂if∂pg
+ 190∂ P
ij
ℓ ∂n∂jPkℓ∂ mnqP Ppq∂if∂p∂ 1m∂kg − 30∂ P
ij
n ∂q∂jPkℓ∂ mn pqℓP P ∂p∂k∂if∂mg
− 1 ∂ ij kℓ mn pq 1 ij kℓ mn pq30 nP ∂jP ∂q∂ℓP P ∂if∂p∂m∂kg − 45∂nP ∂jP ∂q∂ℓP P ∂p∂k∂if∂mg
− 1 ∂ ∂ ij kℓ mn pq 1 ij45 q nP ∂jP ∂ℓP P ∂if∂p∂m∂kg − 90∂nP ∂ P
kℓ
j ∂q∂ℓPmnPpq∂k∂if∂p∂mg
− 1 ij90P ∂q∂jP
kℓ∂ Pmnℓ ∂ pqnP ∂k∂if∂p∂mg − 160P
ij∂q∂n∂jPkℓ∂ mnℓP Ppq∂p∂k∂if∂mg
− 1 ∂ P ij∂ ∂ ∂ PkℓPmnPpq60 ℓ q n j ∂if∂p∂m∂kg −
1 ij
45P ∂
kℓ mn pq
n∂jP ∂q∂ℓP P ∂p∂k∂if∂mg
− 145∂ ∂ P
ij
n ℓ ∂q∂ PkℓPmnj Ppq∂if∂p∂ 1m∂kg + 30P
ij∂q∂n∂jPkℓ∂ mn pqℓP P ∂k∂if∂p∂mg
+ 1 ∂ P ij∂ ∂ ∂ PkℓPmnPpq∂ ∂ f∂ ∂ g − 1 P ij∂ ∂ Pkℓ∂ ∂ PmnPpq30 ℓ q n j m i p k 90 n j q ℓ ∂k∂if∂p∂mg
− 1 P ij∂ Pkℓ45 j ∂q∂ℓP
mn∂ Ppqn ∂ 1p∂k∂if∂mg − 45∂ ∂ P
ij∂ Pkℓ∂ Pmnn ℓ j q Ppq∂if∂p∂m∂kg
− 120∂
ij
ℓP ∂q∂n∂ kℓjP PmnPpq∂p∂ ∂ f∂ g − 1m i k 20∂q∂n∂ℓP
ij∂ PkℓPmnPpqj ∂if∂p∂m∂kg
− 13∂ ∂ P ij∂ PkℓPmn90 n ℓ q P
pq∂m∂ ∂
13 ij kℓ mn pq
k if∂p∂jg + 90∂q∂nP P ∂ℓP P ∂k∂if∂p∂m∂jg
+ 13∂ ∂ P ij∂ ∂ PkℓPmn90 q ℓ n j P
pq∂m∂if∂ ∂ g − 16p ∂ ∂ ∂ P ij∂ Pkℓ mn pqp k 4 p m ℓ n ∂qP P ∂k∂if∂jg
+ 16p ∂ ∂ ∂ P ij4 p m ℓ ∂ Pkℓn ∂ PmnPpqq ∂if∂k∂jg − 16p5∂p∂ P ij∂ kℓ mn pqm nP ∂q∂ℓP P ∂k∂if∂jg
+ 16p ∂ ij kℓ5 p∂mP ∂nP ∂ ∂ Pmnq ℓ Ppq∂ ij kℓ mn pqif∂k∂jg − 16p4∂p∂mP ∂qP ∂ℓP ∂nP ∂k∂if∂jg
+ 16p4∂ ∂ P ijp m ∂qPkℓ∂ Pmnℓ ∂ pqnP ∂if∂k∂jg + 16p ij kℓ4∂mP ∂pP ∂ ∂ Pmnq ℓ ∂nPpq∂k∂if∂jg
− 16p ∂ P ij∂ Pkℓ4 m p ∂q∂ℓPmn∂ Ppqn ∂if∂ ∂ ij kℓ mn pqk jg + 16p5∂pP ∂mP ∂q∂ℓP ∂nP ∂k∂if∂jg
− 16p ∂ P ij∂ Pkℓ∂ mn pq5 p m q∂ℓP ∂nP ∂if∂k∂jg + 16p ∂ P ij1 m ∂q∂ Pkℓj ∂ Pmn∂ Ppqℓ n ∂k∂if∂pg
− 16p ∂ P ij1 m ∂q∂ Pkℓj ∂ Pmn∂ Ppqℓ n ∂if∂p∂kg + 16p ∂ P ij∂ kℓ mn pq2 m jP ∂q∂ℓP ∂nP ∂k∂if∂pg
− 16p ∂ P ij∂ kℓ mn pq2 k q∂jP ∂ℓP ∂nP ∂if∂p∂mg + 16p ∂ P ij∂ kℓ mn pq3 q m∂jP ∂ℓP ∂nP ∂k∂if∂pg
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 269
− 16p ∂ P ij3 p ∂ kℓjP ∂q∂ℓPmn∂ Ppqn ∂if∂m∂kg + 16p4P ij∂ ∂ ∂ Pkℓq m j ∂ Pmnℓ ∂ Ppqn ∂k∂if∂pg
− 16p ∂ P ij4 m ∂q∂n∂jPkℓ∂ PmnPpqℓ ∂if∂ ijp∂kg + 16p5P ∂m∂ Pkℓ∂ ∂ Pmnj q ℓ ∂ pqnP ∂k∂if∂pg
− 16p ij kℓ mn pq5∂kP ∂n∂jP ∂q∂ℓP P ∂if∂p∂mg + 16p6∂ P ijℓ ∂n∂ Pkℓj ∂qPmnPpq∂m∂if∂p∂kg
+ 16p ∂ ij kℓ mn6 q∂ℓP ∂jP P ∂ Ppqn ∂m∂if∂p∂kg + 16p7∂p∂ P ijℓ ∂q∂n∂jPkℓPmnPpq∂m∂if∂kg
− 16p7∂p∂ ∂ P ij∂ ∂ PkℓPmnPpqn ℓ q j ∂if∂m∂kg + 16p8∂p∂ℓP ij∂n∂jPkℓ∂ PmnPpqq ∂m∂if∂kg
+ 16p ∂ ∂ P ij8 n ℓ ∂p∂jPkℓ∂ mnqP Ppq∂if∂m∂kg + 16p9∂pP ij∂ kℓ mn pqq∂jP ∂ℓP ∂nP ∂m∂if∂kg
− 16p ∂ ∂ P ij∂ Pkℓ9 q m j ∂ℓPmn∂ pqnP ∂if∂p∂kg − 32p ∂ P ij∂ ∂ ∂ Pkℓ∂ Pmn4 m q n j ℓ Ppq∂p∂if∂kg
+ 32p ij kℓ mn4∂q∂n∂kP ∂jP ∂ℓP Ppq∂ f∂ ∂ g + 16p ∂ ∂ P ij∂ ∂ ∂ Pkℓ∂ Pmni p m 10 p m q n j ℓ Ppq∂if∂kg
+ 16p ∂ ∂ ij kℓ mn pq ij kℓ mn pq10 p n∂kP ∂jP ∂q∂ℓP P ∂if∂mg + 32p5P ∂m∂jP ∂q∂ℓP ∂nP ∂p∂if∂kg
+ 32p5∂q∂kP ij∂ ∂ Pkℓn j ∂ mnℓP Ppq∂if∂p∂mg + ( 1 ij kℓ mn pq12 + 8p7)∂p∂m∂ℓP ∂q∂nP P P ∂k∂if∂jg
+ (− 1 − 8p )∂ ∂ ∂ P ij∂ ∂ PkℓPmnPpq12 7 p m ℓ q n ∂if∂k∂jg
+ (− 112 − 8p7)∂
ij kℓ mn pq
mP ∂p∂nP ∂q∂ℓP P ∂k∂if∂jg
+ ( 1 + 8p )∂ P ij∂ ∂ Pkℓ∂ ∂ PmnPpq12 7 m p n q ℓ ∂if∂k∂jg
+ (− 112 − 8p7)P
ij∂p∂n∂jPkℓ∂ ∂ PmnPpqq ℓ ∂k∂if∂mg
+ ( 112 + 8p7)∂ ∂ P
ij
p ℓ ∂
kℓ
q∂n∂jP PmnPpq∂if∂m∂kg
+ ( 1 + 8p )∂ ∂ ∂ P ij∂ ∂ ∂ PkℓPmnPpq45 7 p m ℓ q n j ∂if∂kg
+ (− 160 − 24p7)∂
ij kℓ mn pq
p∂ℓP ∂q∂nP P P ∂k∂if∂m∂jg
+ ( 160 + 24p )∂ ∂ P
ijPkℓ∂ mn pq7 p n q∂ℓP P ∂k∂if∂m∂jg
+ ( 4 − 16p )∂ P ij∂ ∂ Pkℓ45 6 q n j ∂ P
mnPpqℓ ∂k∂if∂p∂mg
+ ( 445 − 16p )∂ P
ij∂ ∂ PkℓPmn6 ℓ q j ∂ Ppqn ∂m∂if∂p∂kg
+ (1790 + 24p7)∂p∂mP
ij∂q∂nPkℓPmnPpq∂k∂if∂ℓ∂jg
+ (− 112 + 16p
ij kℓ mn pq
6 + 48p5)∂mP ∂pP ∂qP ∂nP ∂k∂if∂ℓ∂jg
+ ( 7 − 16p − 48p )∂ P ijPkℓ90 6 5 p ∂
mn pq
q∂ℓP ∂nP ∂k∂if∂m∂jg
+ (− 7 + 16p + 48p )∂ P ij∂ ∂ Pkℓ∂ Pmn90 6 5 m q n ℓ P
pq∂k∂if∂p∂jg
+ (− 1 ij kℓ mn pq36 + 16p4 − 16p5 − 8p7)∂pP ∂q∂mP ∂ℓP ∂nP ∂k∂if∂jg
+ ( 136 − 16p4 + 16p5 + 8p )∂ P
ij∂ ∂ Pkℓ∂ Pmn7 p q m ℓ ∂ Ppqn ∂if∂k∂jg
+ (−1 + 16p − 48p + 48p )∂ P ij9 6 4 5 n ∂
kℓ mn pq
q∂jP ∂ℓP P ∂k∂if∂p∂mg
+ (−19 + 16p6 − 48p4 + 48p
ij
5)P ∂q∂ Pkℓ∂ Pmnj ℓ ∂nPpq∂m∂if∂p∂kg
+ (− 1 ij kℓ36 + 16p4 − 16p5 − 8p7)P ∂p∂jP ∂q∂ P
mn
ℓ ∂nPpq∂k∂if∂mg
+ ( 136 − 16p4 + 16p5 + 8p7)∂ ∂ P
ij
n k ∂
kℓ
jP ∂ ∂ Pmnq ℓ Ppq∂if∂p∂mg
+ (− 13 + 8p + 24p − 8p )∂ ∂ P ij∂ ∂ Pkℓ∂ PmnPpq180 6 5 1 m ℓ p n q ∂k∂if∂jg
+ ( 13 − 8p − 24p + 8p )∂ ∂ P ij∂ ∂ Pkℓ∂ PmnPpq180 6 5 1 m ℓ p n q ∂if∂k∂jg
+ ( 445 − 16p6 + 48p4 − 48p5)∂
ij kℓ
q∂nP ∂jP ∂ Pmnℓ Ppq∂k∂if∂p∂mg
270 R. BURING AND A. V. KISELEV
+ ( 4 − 16p + 48p − 48p )∂ P ij∂ ∂ Pkℓ45 6 4 5 n q j ∂ P
mnPpqℓ ∂p∂if∂m∂kg
+ (− 118 + 32p − 32p − 16p )P
ij∂ ∂ Pkℓ4 5 7 p j ∂q∂ℓPmn∂ Ppqn ∂m∂if∂kg
+ (− 118 + 32p4 − 32p5 − 16p7)∂q∂ P
ij
m ∂n∂ Pkℓj ∂ℓPmnPpq∂if∂p∂kg
+ (1790 − 32p6 + 96p4 − 96p5)∂ P
ij∂ Pkℓq j ∂ mnℓP ∂ Ppqn ∂k∂if∂p∂mg
+ ( 23360 + 16p4 − 8p1 + 12p )∂ ∂ ∂ P
ij∂ kℓ mn pq7 p m ℓ n∂jP ∂qP P ∂if∂kg
+ (− 23 ij kℓ mn pq360 − 16p4 + 8p1 − 12p7)∂m∂ℓP ∂p∂n∂jP ∂qP P ∂if∂kg
+ ( 49180 − 16p − 48p + 24p )∂ ∂ P
ij∂ Pkℓ∂ PmnPpq6 5 7 p m n q ∂k∂if∂ℓ∂jg
+ (− 19180 − 32p4 + 16p1 − 16p7)∂m∂
ij
ℓP ∂ ∂ Pkℓ∂ mn pqp j qP ∂nP ∂if∂kg
+ (− 31180 + 32p6 − 96p4 + 96p )∂ P
ij
5 q ∂
kℓ
jP ∂ Pmnℓ ∂nPpq∂m∂if∂p∂kg
+ (− 190 − 8p6 − 24p5 + 8p1 − 8p7)∂p∂
ij
mP Pkℓ∂q∂ℓPmn∂ Ppqn ∂k∂if∂jg
+ ( 190 + 8p6 + 24p5 − 8p1 + 8p7)∂p∂mP
ijPkℓ∂ ∂ Pmn∂ Ppqq ℓ n ∂if∂k∂jg
+ (2 ij9 − 48p8 + 96p4 − 96p5 + 48p7)∂ℓP ∂p∂ P
kℓ∂ PmnPpqn q ∂k∂if∂m∂jg
+ (29 − 48p8 + 96p4 − 96p5 + 48p )∂ P
ij
7 n ∂
kℓ
pP ∂ ∂ PmnPpqq ℓ ∂k∂if∂m∂jg
+ (16 − 32p8 + 64p − 64p + 32p )∂ P
ij
4 5 7 p ∂q∂n∂ Pkℓj ∂ mn pqℓP P ∂k∂if∂mg
+ (−16 + 32p8 − 64p4 + 64p5 − 32p
ij
7)∂ℓP ∂ ∂ ∂ Pkℓp n j ∂ mn pqqP P ∂if∂m∂kg
+ (−19 + 16p8 − 32p4 + 32p5 − 16p
ij
7)∂kP ∂ ∂ ∂ Pkℓq n j ∂ Pmnℓ Ppq∂p∂if∂mg
+ (19 − 16p8 + 32p4 − 32p5 + 16p7)∂kP
ij∂ kℓ mn pqq∂n∂jP ∂ℓP P ∂if∂p∂mg
+ ( 1 − 8p + 16p − 24p + 4p )∂ ∂ P ij∂ ∂ ∂ Pkℓ∂ PmnPpq120 8 4 5 7 p k q n j ℓ ∂if∂mg
+ ( 1120 − 8p8 + 16p
ij kℓ mn pq
4 − 24p5 + 4p7)∂kP ∂p∂n∂jP ∂q∂ℓP P ∂if∂mg
+ ( 518 − 48p8 + 96p4 − 96p5 + 48p7)∂
ij
ℓP ∂ Pkℓp ∂qPmn∂ Ppqn ∂k∂if∂m∂jg
+ ( 518 − 48p8 + 96p4 − 96p5 + 48p7)∂ P
ij∂ Pkℓ∂ Pmn∂ Ppqq m ℓ n ∂k∂if∂p∂jg
+ ( 415 − 48p8 + 96p4 − 96p5 + 48p7)∂ P
ij∂ Pkℓ∂ ∂ PmnPpqm n q ℓ ∂k∂if∂p∂jg
+ (− 415 + 48p8 − 96p4 + 96p5 − 48p7)∂
ij kℓ mn pq
mP P ∂q∂ℓP ∂nP ∂k∂if∂p∂jg
+ (− 118 + 16p8 − 32p4 + 32p5 − 16p7)∂
ij kℓ mn pq
ℓP ∂p∂n∂jP ∂qP P ∂m∂if∂kg
+ (− 118 + 16p8 − 32p4 + 32p5 − 16p7)∂
ij kℓ mn pq
p∂n∂ℓP ∂jP ∂qP P ∂if∂m∂kg
+ (− 736 + 32p8 − 64p4 + 64p5 − 32p7)∂
ij kℓ mn pq
p∂ℓP ∂jP ∂qP ∂nP ∂m∂if∂kg
+ (− 736 + 32p8 − 64p4 + 64p5 − 32p7)∂ℓP
ij∂ ∂ Pkℓ∂ Pmnp j q ∂ pqnP ∂if∂m∂kg
+ (− 112 + 16p8 − 32p4 + 32p5 − 16p7)∂ℓP
ij∂ ∂ Pkℓp j ∂ mnqP ∂ Ppqn ∂m∂if∂kg
+ (− 112 + 16p8 − 32p
ij kℓ mn pq
4 + 32p5 − 16p7)∂p∂ℓP ∂jP ∂qP ∂nP ∂if∂m∂kg
+ (− 190 − 16p8 + 32p
ij kℓ mn pq
4 − 80p5 + 16p1)∂p∂kP ∂n∂jP ∂q∂ℓP P ∂if∂mg
+ ( 1360 − 16p8 − 16p3 + 32p4 − 48p5)∂p∂kP
ij∂ Pkℓ∂ ∂ Pmnj q ℓ ∂ pqnP ∂if∂mg
+ ( 11 − 16p + 32p − 16p + 16p )∂ P ij∂ kℓ mn pq120 8 4 5 7 k m∂jP ∂q∂ℓP ∂nP ∂if∂pg
+ ( 190 + 8p6 + 16p4 + 24p5 − 8p1 + 8p7)∂m∂ P
ij∂ Pkℓ∂ Pmnℓ p q ∂ pqnP ∂k∂if∂jg
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 271
+ (− 190 − 8p6 − 16p4 − 24p5 + 8p1 − 8p7)∂ ∂ P
ij
m ℓ ∂pPkℓ∂ mnqP ∂ Ppqn ∂if∂k∂jg
+ (− 1 ij kℓ mn pq15 + 8p8 + 8p4 + 8p2 + 16p10 − 16p7)∂mP ∂p∂jP ∂q∂ℓP ∂nP ∂if∂kg
+ (− 115 + 8p
ij kℓ mn pq
8 + 8p4 + 8p2 + 16p10 − 16p7)∂p∂nP ∂q∂jP ∂kP ∂ℓP ∂if∂mg
+ ( 120 − 8p8 + 24p4 − 16p5 − 8p2 + 8p7)∂ P
ij
k ∂q∂m∂
kℓ
jP ∂ Pmn∂ Ppqℓ n ∂if∂pg
+ (− 120 + 8p8 − 24p4 + 16p5 + 8p2 − 8p7)∂
ij kℓ mn pq
pP ∂q∂n∂jP ∂kP ∂ℓP ∂if∂mg
+ (− 140 + 8p8 + 16p4 + 8p5 + 16p10 − 12p
ij
7)∂mP ∂ ∂ ∂ Pkℓ∂ ∂ PmnPpqp n j q ℓ ∂if∂kg
+ ( 140 − 8p8 − 16p4 − 8p5 − 16p10 + 12p7)∂ ∂ ∂ P
ij∂ kℓ mn pqp n k q∂jP ∂ℓP P ∂if∂mg
+ (1190 + 8p6 − 16p4 + 40p5 − 8p1 + 24p7)∂p∂mP
ij∂q∂nPkℓ∂ PmnPpqℓ ∂k∂if∂jg
+ (−11 − 8p + 16p − 40p + 8p − 24p )∂ ∂ P ij∂ ∂ Pkℓ∂ Pmn pq90 6 4 5 1 7 p m q n ℓ P ∂if∂k∂jg
+ (15 − 32p8 − 48p5 − 32p10 + 16p1 + 48p
ij kℓ mn pq
7)∂p∂nP ∂q∂jP ∂ℓP P ∂k∂if∂mg
+ (15 − 32p8 − 48p5 − 32p10 + 16p1 + 48p )∂ P
ij∂ ∂ Pkℓ7 n p j ∂q∂ℓPmnPpq∂if∂m∂kg
+ (−16 + 16p8 − 16p3 + 32p4 − 16p1 − 32p
ij kℓ mn pq
7)∂pP ∂jP ∂q∂ℓP ∂nP ∂m∂if∂kg
+ (1 − 16p + 16p − 32p + 16p + 32p )∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq6 8 3 4 1 7 q m j ℓ n ∂if∂p∂kg
+ (19 − 16p8 + 16p4 − 32p5 + 16p1 + 16p7)∂ P
ij
m ∂ ∂
kℓ mn pq
n jP ∂q∂ℓP P ∂if∂p∂kg
+ (−19 + 16p8 − 16p4 + 32p5 − 16p1 − 16p
ij kℓ
7)∂n∂kP ∂q∂jP ∂ℓPmnPpq∂p∂if∂mg
+ (−1 ij kℓ mn pq9 + 16p8 − 32p4 + 48p5 + 16p2 − 16p7)∂mP ∂jP ∂q∂ℓP ∂nP ∂p∂if∂kg
+ (−19 + 16p8 − 32p4 + 48p + 16p − 16p )∂ P
ij∂ ∂ Pkℓ5 2 7 n q j ∂ Pmn∂ pqk ℓP ∂if∂p∂mg
+ (19 − 16p8 + 32p4 − 48p5 + 16p + 16p )∂ P
ij∂ Pkℓ∂ ∂ Pmn∂ Ppq2 7 m j q ℓ n ∂if∂p∂kg
+ (−19 + 16p8 − 32p4 + 48p5 − 16p2 − 16p7)∂ P
ij
k ∂q∂ Pkℓ∂ Pmnj ℓ ∂ pqnP ∂p∂if∂mg
+ ( 790 − 16p8 + 40p4 − 40p5 + 8p2 + 12p )∂ P
ij∂ ∂ Pkℓ7 k p j ∂q∂ℓPmn∂nPpq∂if∂mg
+ ( 7 − 16p + 40p − 40p + 8p + 12p )∂ ∂ P ij kℓ90 8 4 5 2 7 m k ∂jP ∂ ∂ P
mn
q ℓ ∂
pq
nP ∂if∂pg
+ ( 1180 − 16p8 − 16p4 − 16p5 − 32p10 + 8p )∂ P
ij
7 n ∂p∂ Pkℓj ∂ ∂ PmnPpqq ℓ ∂k∂if∂mg
+ (− 1 + 16p + 16p + 16p + 32p − 8p )∂ ∂ P ij∂ Pkℓ∂ mn pq180 8 4 5 10 7 p n j q∂ℓP P ∂if∂m∂kg
+ (3790 − 48p8 − 16p6 + 96p4 − 96p5 + 48p7)∂
ij
pP ∂ ∂ Pkℓ∂ PmnPpqq n ℓ ∂k∂if∂m∂jg
+ (−37 ij kℓ90 + 48p8 + 16p6 − 96p4 + 96p5 − 48p7)∂pP ∂nP ∂q∂ℓP
mnPpq∂k∂if∂m∂jg
+ ( 29360 − 16p8 − 16p4 − 16p10 + 8p
ij
1 + 20p7)∂p∂mP ∂ ∂ Pkℓ∂ ∂ PmnPpqn j q ℓ ∂if∂kg
+ ( 29 − 16p − 16p − 16p + 8p + 20p )∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmn360 8 4 10 1 7 n k p j q ℓ P
pq∂if∂mg
+ (34 ij45 − 96p8 − 32p6 + 240p4 − 288p5 + 96p7)∂pP ∂ P
kℓ∂ Pmnq ℓ ∂ pqnP ∂k∂if∂m∂jg
+ (−34 + 96p + 32p − 240p + 288p − 96p )∂ P ij∂ Pkℓ∂ Pmn∂ Ppq45 8 6 4 5 7 m q ℓ n ∂k∂if∂p∂jg
+ (− 245 + 8p9 + 4p6 − 8p3 + 4p5 − 4p
ij kℓ mn pq
1 − 4p7)∂pP ∂q∂m∂jP ∂ℓP ∂nP ∂if∂kg
+ ( 2 − 8p − 4p + 8p − 4p + 4p + 4p )∂ ∂ ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq45 9 6 3 5 1 7 q m k j ℓ n ∂if∂pg
+ ( 1 + 8p + 32p + 8p + 32p − 8p + 8p )∂ ∂ P ij∂ Pkℓ∂ ∂ PmnPpq30 6 4 5 10 1 7 p n j q ℓ ∂k∂if∂mg
+ (− 130 − 8p6 − 32p4 − 8p5 − 32p10 + 8p1 − 8p7)∂p∂nP
ij∂ kℓ mn pqq∂jP ∂ℓP P ∂if∂m∂kg
272 R. BURING AND A. V. KISELEV
+ (− 7 + 8p − 16p + 16p − 8p − 8p − 16p )∂ P ij∂ ∂ Pkℓ mn90 8 3 4 5 1 7 p m j ∂q∂ℓP ∂nP
pq∂if∂kg
+ ( 790 − 8p8 + 16p3 − 16p4 + 8p5 + 8p1 + 16p7)∂q∂kP
ij∂m∂ Pkℓj ∂ Pmnℓ ∂ Ppqn ∂if∂pg
+ (− 7180 + 16p8 − 8p6 − 16p4 + 8p5 + 8p1 − 16p7)∂mP
ij∂ ∂ kℓ mn pqn jP ∂q∂ℓP P ∂p∂if∂kg
+ ( 7180 − 16p8 + 8p6 + 16p4 − 8p5 − 8p1 + 16p )∂ ∂ P
ij
7 n k ∂
kℓ
q∂jP ∂ Pmnℓ Ppq∂if∂p∂mg
+ ( 13 − 8p + 24p − 32p − 8p + 8p + 4p )∂ ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq360 8 4 5 2 1 7 m k q j ℓ n ∂if∂pg
+ (− 13360 + 8p8 − 24p4 + 32p5 + 8p2 − 8p1 − 4p7)∂
ij
pP ∂ ∂ Pkℓn j ∂q∂ Pmnk ∂ℓPpq∂if∂mg
+ ( 1 ij kℓ mn pq15 − 32p8 + 8p6 + 48p4 − 72p5 + 24p1 + 24p7)∂pP ∂n∂jP ∂q∂ℓP P ∂k∂if∂mg
+ (− 115 + 32p8 − 8p6 − 48p4 + 72p − 24p − 24p )∂ ∂ P
ij∂ ∂ Pkℓ∂ PmnPpq5 1 7 p ℓ n j q ∂if∂m∂kg
+ (− 11180 − 16p9 + 16p8 − 8p6 + 8p + 8p − 16p )∂ P
ij∂ ∂ Pkℓ∂ Pmn∂ ∂ Ppq5 1 7 p q j k n ℓ ∂if∂mg
+ (− 17180 + 16p8 − 8p6 − 32p4 + 40p5 − 8p1 − 16p7)∂ ∂ P
ij
p ℓ ∂q∂ PkℓPmnj ∂ pqnP ∂m∂if∂kg
+ ( 17180 − 16p8 + 8p6 + 32p4 − 40p5 + 8p1 + 16p )∂ ∂ P
ij
7 p ℓ ∂
kℓ
q∂jP Pmn∂nPpq∂if∂m∂kg
+ ( 61180 − 48p8 − 8p6 + 96p4 − 120p5 + 24p1 + 48p7)∂ ∂ P
ij∂ kℓ mnp ℓ qP P ∂nPpq∂k∂if∂m∂jg
+ (− 61180 + 48p8 + 8p6 − 96p4 + 120p5 − 24p1 − 48p7)∂ ∂ P
ijPkℓ∂ Pmn∂ Ppqq m ℓ n ∂k∂if∂p∂jg
+ (5390 − 96p8 − 16p6 + 192p4 − 240p5 + 48p1 + 96p7)∂n∂ P
ij∂ Pkℓ∂ Pmnℓ p q Ppq∂k∂if∂m∂jg
+ (−4990 + 48p8 + 24p6 − 144p4 + 168p5 − 24p1 − 72p7)∂p∂ P
ij∂ Pkℓ∂ Pmnℓ n q Ppq∂k∂if∂m∂jg
+ (4990 − 48p8 − 24p6 + 144p4 − 168p5 + 24p1 + 72p7)∂p∂nP
ij∂ Pkℓ∂ Pmnq ℓ Ppq∂k∂if∂m∂jg
+ ( 190 − 16p
ij kℓ mn pq
8 + 8p6 − 16p3 + 16p4 − 24p5 − 8p1 + 8p7)∂pP ∂jP ∂q∂ℓP ∂nP ∂k∂if∂mg
+ (− 1 + 16p − 8p + 16p − 16p + 24p + 8p − 8p )∂ ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq90 8 6 3 4 5 1 7 q k j ℓ n ∂if∂p∂mg
+ ( 320 + 16p9 − 32p8 + 8p6 + 16p4 − 40p − 8p
ij kℓ mn pq
5 1 + 32p7)∂pP ∂q∂jP ∂ℓP ∂nP ∂k∂if∂mg
+ (− 320 − 16p9 + 32p8 − 8p6 − 16p4 + 40p5 + 8p
ij kℓ mn pq
1 − 32p7)∂nP ∂jP ∂q∂kP ∂ℓP ∂if∂p∂mg
+ (− 7180 − 16p9 + 16p
ij
8 − 8p6 + 16p4 + 8p5 − 8p1 − 16p7)∂q∂mP ∂ Pkℓ∂ mnj ℓP ∂nPpq∂k∂if∂pg
+ ( 7180 + 16p9 − 16p
ij kℓ mn pq
8 + 8p6 − 16p4 − 8p5 + 8p1 + 16p7)∂pP ∂q∂jP ∂ℓP ∂nP ∂if∂m∂kg
+ ( 7120 + 16p9 − 8p8 + 8p6 + 16p4 + 16p10 − 8p1 + 12p7)∂p∂mP
ij∂ ∂ Pkℓq j ∂ mnℓP ∂ Ppqn ∂if∂kg
+ (− 7 − 16p + 8p − 8p − 16p − 16p + 8p − 12p )∂ ∂ P ij kℓ mn pq120 9 8 6 4 10 1 7 p n ∂jP ∂q∂kP ∂ℓP ∂if∂mg
+ (8p ij9 − 8p8 + 4p6 − 8p3 − 8p4 + 4p5 − 8p2 − 4p1 + 4p7)∂p∂kP ∂ ∂ Pkℓ∂ Pmn∂ Ppqq j ℓ n ∂if∂mg
+ (−8p + 8p − 4p + 8p + 8p − 4p + 8p + 4p − 4p )∂ P ij∂ Pkℓ∂ ∂ Pmn∂ pq9 8 6 3 4 5 2 1 7 p j q k n∂ℓP ∂if∂mg
+ ( 23360 + 8p9 − 16p8 + 4p6 − 8p3 + 8p4 − 20p5 + 8p2 − 16p10 − 4p1 + 16p7)
∂ ∂ P ij∂ Pkℓ∂ mn pqp m j q∂ℓP ∂nP ∂if∂kg
+ (− 23360 − 8p9 + 16p8 − 4p6 + 8p3 − 8p4 + 20p5 − 8p2 + 16p10 + 4p1 −) 16p7)
∂ P ijn ∂p∂jPkℓ∂q∂kPmn∂ Ppqℓ ∂if∂mg + ō(ℏ4). (11)
The ten master-parameters in (11) are the still unknown weights of the prime graphs
which are portrayed in Fig. 3 on p. 247. The four underlined parameters can be gauged
out (without modifying the coefficients of any other Kontsevich graphs with four internal
vertices), see Theorem 14 on p. 258. At all values of the ten master-parameters, that
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 273
is, irrespective of their true values given by formula (5), the ⋆-product is proven in
Theorem 12 to be associative modulo ō(ℏ4).
Acknowledgements. The authors are grateful to the anonymous referees for their critical
comments and suggestions which helped us improve this text, to prof. S. Tabachnikov
(Editor-in-Chief) for persistence and constructive criticism, and B. Pym and E. Panzer
for communicating the values of ten master-parameters obtained via a different tech-
nique [34]. We thank prof. M. Gerstenhaber and M. Kontsevich for their attention to
our work.
This research was supported in part by JBI RUG project 106552 (Groningen, The
Netherlands) and IM JGU project 5020 (Mainz, Germany). The authors also thank the
Center for Information Technology of the University of Groningen for providing access
to Peregrine high performance computing cluster. A part of this research was done
while the authors were visiting at the IHÉS in Bures-sur-Yvette, France and AVK was
visiting at the MPIM Bonn, Germany; warm hospitality and partial financial support
by these institutions are gratefully acknowledged.
References
[1] Ammar M., Chloup V., Gutt S. (2008) Universal Star Products, Lett. Math. Phys.
84:2–3, 199–215.
[2] Banks P., Panzer E., Pym B. (2018) Multiple zeta values in deformation quanti-
zation, Preprint arXiv:1812.11649 [math.QA].
[3] Bauer C., Frink A., Kreckel R. (2002) Introduction to the GiNaC Framework for
Symbolic Computation within the C++ Programming Language, J. Symb. Comp.
33, 1–12. See also http://www.ginac.de.
[4] Bayen F., Flato M., Frønsdal C., Lichnerowicz A., Sternheimer D. (1978) Deforma-
tion theory and quantization. I. Deformations of symplectic structures, Ann. Phys.
(N. Y.) 111:1, 61–110.
[5] Bayen F., Flato M., Frønsdal C., Lichnerowicz A., Sternheimer D. (1978) Deforma-
tion theory and quantization. II. Physical applications, Ann. Phys. (N. Y.) 111:1,
111–151.
[6] Ben Amar N. (2007) A comparison between Rieffel’s and Kontsevich’s deformation
quantizations for linear Poisson tensors, Pac. J. Math. 229:1, 1–24.
[7] Bouisaghouane A., Buring R., Kiselev A.V. (2017) The Kontsevich tetrahedral flow
revisited, J. Geom. Phys. 119, 272–285. (Preprint arXiv:1608.01710 [q-alg]).
[8] Bouisaghouane A., Kiselev A.V. (2017) Do the Kontsevich tetrahedral flows pre-
serve or destroy the space of Poisson bi-vectors ? J. Phys.: Conf. Ser. 804, Pa-
per 012008, 1–10. (Preprint arXiv:1609.06677 [q-alg])
[9] Buring R. Software package kontsevich-graph-series-cpp, see link:
https://github.com/rburing/kontsevich_graph_series-cpp
[10] Buring R. List of integrands for the 149 basic Kontsevich graphs at ℏ4, see link:
http://rburing.nl/kontsevich_graph_weight_integrands4.txt
[11] Buring R., Kiselev A.V. (2019) Formality morphism as the mechanism of ⋆-product
associativity: how it works. Collected Works Inst. Math. Kiev 16:1, 22–43. (Preprint
arXiv:1907.00639 [math.QA])
274 R. BURING AND A. V. KISELEV
[12] Buring R., Kiselev A.V. (2017) On the Kontsevich ⋆-product associativity
mechanism, Physics of Particles and Nuclei Letters 14:2, 403–407. (Preprint
arXiv:1602.09036 [q-alg])
[13] Buring R., Kiselev A.V. (2019) The orientation morphism: from graph cocycles
to deformations of Poisson structures, J. Phys.: Conf. Ser. 1194 Proc. 32nd Int.
colloquium on Group-theoretical methods in Physics: Group32 (9–13 July 2018,
CVUT Prague, Czech Republic), Paper 012017, 10 p. (Preprint arXiv:1811.07878
[math.CO])
[14] Buring R., Kiselev A.V. (2015) The table of weights for graphs with ⩽ 3 inter-
nal vertices in Kontsevich’s deformation quantization formula. (3rd International
workshop on symmetries of discrete systems & processes, 3–7 August 2015, CVUT
Děčín, Czech Republic), see Appendix A.1 in this paper.
[15] Cattaneo A. S., Felder G. (2000) A path integral approach to the Kontsevich
quantization formula, Comm. Math. Phys. 212:3, 591–611.
[16] Dito G. (1999) Kontsevich star product on the dual of a Lie algebra, Lett. Math.
Phys. 4, 291–306.
[17] Felder G., Willwacher T. (2010) On the (ir)rationality of Kontsevich weights, Int.
Math. Res. Not. 2010:4, 701–716.
[18] Felder G., Shoikhet B. (2000) Deformation quantization with traces, Lett. Math.
Phys. 53, 75–86.
[19] Gerstenhaber M. (1964) On the deformation of rings and algebras, Ann. Math. 79,
59–103.
[20] Grabowski J., Marmo G., Perelomov A.M. (1993) Poisson structures: towards a
classification, Mod. Phys. Lett. A8:18, 1719–1733.
[21] Kathotia V. (1998) Kontsevich’s universal formula for deformation quantiza-
tion and the Campbell–Baker–Hausdorff formula, I. Preprint arXiv:9811174 (v2)
[math.QA]
[22] Kiselev A.V. (2012) The twelve lectures in the (non)commutative geometry of
differential equations, Preprint IHÉS/M/12/13 (Bures-sur-Yvette, France), 140 p.
[23] Kiselev A. V. (2013) The geometry of variations in Batalin–Vilkovisky formalism,
J. Phys.: Conf. Ser. 474, Paper 012024, 1–51. (Preprint arXiv:1312.1262 [math-
ph])
[24] Kiselev A.V. (2014) The Jacobi identity for graded-commutative variational
Schouten bracket revisited, Physics of Particles and Nuclei Letters 11:7, 950–953.
(Preprint arXiv:1312.4140 [math-ph])
[25] Kiselev A.V. (2017) The deformation quantization mapping of Poisson to associa-
tive structures in field theory, Banach Center Publ. 113 50th Sophus Lie Seminar
219–242. (Preprint arXiv:1705.01777 [q-alg])
[26] Kiselev A.V. (2016) The right-hand side of the Jacobi identity: to be naught or
not to be ? J. Phys.: Conf. Ser. 670 Proc. XXIII Int. conf. ‘Integrable Systems
and Quantum Symmetries’ (23–27 June 2015, CVUT Prague, Czech Republic),
Paper 012030, 1–17. (Preprint arXiv:1410.0173 [math-ph])
[27] Kiselev A.V. (2017) The calculus of multivectors on noncommutative jet spaces,
J. Geom. Phys. 130, 130–167. (Preprint arXiv:1210.0726 [math.DG])
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 275
[28] Kontsevich M. (1993) Formal (non)commutative symplectic geometry, The
Gel’fand Mathematical Seminars, 1990-1992 (L. Corwin, I. Gelfand, and J. Lepow-
sky, eds), Birkhäuser, Boston MA, 173–187.
[29] Kontsevich M. (1994) Feynman diagrams and low-dimensional topology. First Eu-
rop. Congr. of Math. 2 (Paris, 1992), Progr. Math. 120, Birkhäuser, Basel, 97–121.
[30] Kontsevich M. (1995) Homological algebra of mirror symmetry. Proc. Intern. Congr.
Math. 1 (Zürich, 1994), Birkhäuser, Basel, 120–139.
[31] Kontsevich M. (1997) Formality conjecture. Deformation theory and symplectic
geometry (Ascona 1996, D. Sternheimer, J. Rawnsley and S.Gutt, eds), Math. Phys.
Stud. 20, Kluwer Acad. Publ., Dordrecht, 139–156.
[32] Kontsevich M. (2003) Deformation quantization of Poisson manifolds, Lett. Math.
Phys. 66:3, 157–216. (Preprint q-alg/9709040)
[33] Laurent–Gengoux C., Picherau A., Vanhaecke P. (2013) Poisson structures. Gründ-
lehren der mathematischen Wissenschaften 347, Springer–Verlag, Berlin.
[34] Panzer E., Pym B. (2017) Private communication. See also:
https://www.mathematics.uni-bonn.de/veranstaltungskalender/17251
[35] Penkava M.,Vanhaecke P. (2000) Deformation Quantization of Polynomial Poisson
Algebras, J. Algebra 227:1, 365–393. (Preprint arXiv:math/9804022)
[36] Polyak M. (2003) Quantization of linear Poisson structures and degrees of maps,
Lett. Math. Phys. 66:1, 15–35.
[37] Vanhaecke P. (1996) Integrable systems in the realm of algebraic geometry, Lect.
Notes Math. 1638, Springer–Verlag, Berlin.
[38] Willwacher T. (2014) The obstruction to the existence of a loopless star product,
C. R. Math. Acad. Sci. Paris 352:11, 881–883.
This text was submitted in its original form on 20 December 2017.
276 R. BURING AND A. V. KISELEV
Appendix A. Approximations and conjectured values of weight
integrals
The material presented here is an expanded version of section 3 of the note [14] by the
authors.
A.1. The weight integral in Cartesian coordinates. Recall the integral formula
for the weight of a graph Γ ∈ ∫G̃2,k (s∧ee section 2):k1
w(Γ) = dφ(pj, p2k Left(j)) ∧ dφ(pj, pRight(j)), (5)(2π) Ck(H) j=1
such that the integral is taken over the configuration space of k points in the upper
half-plane H ⊂ C,
Ck(H) = {(p1, . . . , pk) ∈ Hk : pi pairwise di(stinc)t},
and where φ : C2(H)→ [0, 2π) was defined by φ(p, q) = Arg q−p− .q p̄
For nonzero z = x + iy in H we have Arg(x + iy) ∼= arctan(y/x), where ∼= denotes
equality of functions up to a constant. Since d arctan(t) = 1/(1 + t2), the weight
dt
integrand is a rational function of the C(artesian coordinates: for p)= a+ib and q = x+iy,
∼ 2b(a− x)φ(p, q) = arctan . (12)
(a− x)2 + (y + b)(y − b)
In Cartesian coordinates (x1, y1, . . . , xk, yk), the weight integrand can now be written
as the Jacobian determinant of the map ΦΓ : Ck(H)→ [0, 2π)2k defined by22
ΦΓ(p1, . . . , pk) = (φ(p1, pLeft(1)), φ(p1, pRight(1)), . . . , φ(pk, pLeft(k)), φ(pk, pRight(k)))
considered as a function of the (xj, yj) through pj = xj + iyj.
Implementation 17. The command
> weight_integrands <graph-series-file>
takes as input a list of graphs Γ ∈ G̃2,k with (possibly undetermined) coefficients, and
sends to the standard output lines of the following form:
(* <graph encoding> <coefficient> *)
<weight integrand of the graph above>
where the weight integrands are written in Mathematica format, as Det[...].
We can take integration domain to be Hk, since for any i 6= j the set {(p1, . . . , pk) ∈
Ck : pi = pj} is a strict linear subspace of Ck, which has measure zero. The weight
integral is absolutely convergent [32], so by the Fubini–Tonelli theorem we may evaluate
it as an iterated integral in any order. We can use the residue theorem23 to integrate
out the Cartesian coordinates corresponding to the k real parts, halving the dimension.
It then remains to integrate the result (a function of the k imaginary parts) over Rk.
22Called a Gauss map by M. Polyak [36].
23G. Dito used the residue method for one graph [16] at k = 2, and remarked that that it becomes
unpractical for k ⩾ 3.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 277
Example 38. For the wedge graph Λ we have the Cartesian coordinates x+ iy in the
upper half-plane and the integrand (obtained using Implementation 17)
4y
f(x, y) = .
((x− 1)2 + y2)(x2 + y2)
To apply the residue theorem we interpret f(x, y) as a rational function in a single
complex variable x. Its poles are then ±iy and 1 ± iy, so the poles in the upper half-
plane are iy and 1 + iy (since y > 0). The residues at these poles are r1 = 2/(i + 2y)
and r2 = −2/(2y − i) respectively. Hence the residue theorem yields that the integral
of f(x, y) with respect to x over the real line is 2πi(r1 + r2) = 8π/(1 + 4y2). When we
integrate this over y > 0 and divide by (2π)2 we obtain 1/2, as desired.
This is of course a toy example. For higher k the expressions become larger, but also
one has to consider more carefully which poles are in the upper half-plane. From the
expression (12) for φ one can see that this issue depends on the relative position of the
coordinates on the imaginary axis (y and b in that formula).
For k = 3 with coordinates on H3 given by
a+ bi, c+ di, e+ fi,
let us agree to call a, c, e the real coordinates and b, d, f the imaginary coordinates. We
now split the integral into a sum of integrals over 3! = 6 regions, one for each possible
ordering of the imaginary coordinates:
b < d < f ; b < f < d; d < b < f ; d < f < b; f < b < d; f < d < b.
In each such region it is known for every (complexified) real coordinate which poles are
in the upper half-plane, so we can apply the residue theorem three times. The result can
be numerically integrated more effectively than the original expression, for one because
we have halved the dimension of the integration domain.
Remark 17. To integrate over t∫he reg∫ion of ∫H3 defined by b < d < f , one can choose
integration bou ∞ ∞ f∫ ∫nds as∫ follo∫ws: db df dd (and similarly for the other permu-0 b btations). For the region of H4 defined by b < d < f < h one can choose the integration
bounds ∞ ∞ h hdb dh dd df , and so on.
0 b b d
Implementation 18. The strategy above is implemented by the followingMathematica
code (for the order 4, but it can be adapted for others), whereW is the weight integrand.
W = an integrand, e.g. from list [10];
integrationvariables = {a, b, c, d, e, f, g, h};
imaginaryvariables =
integrationvariables[[2 #1]] & /@
Range[1, Length[integrationvariables]/2];
realvariables =
integrationvariables[[2 #1 - 1]] & /@
Range[1, Length[integrationvariables]/2];
basicAssumptions =
278 R. BURING AND A. V. KISELEV
Element[a, Reals] && Element[c, Reals] && Element[e, Reals] &&
Element[g, Reals] && b > 0 && d > 0 && f > 0 && h > 0;
ContourIntegrate[function_, variable_, assumptions_] :=
2*Pi*I*Total[
Map[
Function[{p}, (Numerator[Together[function]]/
D[Denominator[Together[function]], variable]) /. {variable ->
p}],
Select[
ReplaceList[variable,
Assuming[assumptions,
Flatten[FullSimplify[
Solve[Denominator[Together[function]] == 0, variable,
Complexes]]]]],
Function[{r}, Simplify[ComplexExpand[Im[r]] > 0, assumptions]]]]]
IteratedContourIntegrate[function_, variables_, assumptions_] :=
Fold[ContourIntegrate[Together[#1], #2, assumptions] &, function,
variables]
integrals = Map[
NIntegrate[
Simplify[
IteratedContourIntegrate[W, realvariables,
basicAssumptions && #1[[1]] < #1[[2]] < #1[[3]] < #1[[4]]]
TimeConstraint -> Infinity],
Evaluate[
Sequence @@
{{#1[[1]], 0, Infinity}, {#1[[3]], #1[[1]],
Infinity}, {#1[[2]], #1[[1]], #1[[3]]}, {#1[[4]], #1[[3]],
Infinity}}
],
Method -> {GlobalAdaptive, MaxErrorIncreases -> 10^4}
] &, Permutations[imaginaryvariables]]
Print[integrals]
Print[Total[integrals]]
Print[Total[integrals]/N[(2 Pi)^8]]
Remark 18. This strategy allows effective numerical integration of all weights up to
order 3. At the order 4, it works for some weights but not others: see Tables 2 and 3.
The call(s) to Map may be replaced by ParallelMap to parallelize the computation.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 279
r- r
Example 39. The second Bernoulli graph A @rRrA ?	@Rr [17] has the weight integrandAU
( ( ) ) ( ( ))
64bfd c (a− c)2 + b2 + d2(c− 2a) f2(e− 2c) + e (e− c)2 + d2
(a2 + b2) (f2 + (e− 1)2) (f2 + e2) (c2 + d2) ((a− c)2 + (b− d)2) ((a− c)2 + (b+ d)2) ((f + d)2 + (e− c)2)
The residue calculation followed by the numerical integration leads to the estimate
5.71871× 10−9 − 5.92495× 10−21i of the weight; this leads to the guess that it is zero
and in fact it is true.
Table 2. Verified values
Weight Approximation True value
w_4_1 −0.0069444401170± 0.000000906189 −1/144 ≈ −0.00694444
Table 3. Conjectured values
Weight Approximation Conjectured true value
w_4_103 −0.000086894703± 0.000000681076 −1/11520 ≈ 0.000086805
w_4_104 0.000347214860± 0.000000371598 1/2880 ≈ 0.000347222
w_4_112 −0.000347219933± 0.000000042901 −1/2880 ≈ −0.000347222
w_4_113 0.000694441623± 0.000000093136 1/1440 ≈ 0.000694444
w_4_133 0.000694443060± 0.000000078774 1/1440 ≈ 0.000694444
w_4_138 −0.001041664533± 0.000000095465 −1/960 ≈ −0.001041666
w_4_147 −0.000043376821± 0.000000095465 −1/23040 ≈ −0.000043402
w_4_148 0.000173611294± 0.000000015063 1/5760 ≈ 0.000173611
In particular, this table lists the approximate value of the master-parameters p4 =
w_4_103 and p5 = w_4_104. The relation w_4_133 = 2 · w_4_104 which was found in
Theorem 9 and listed in Table 7 of Appendix C is satisfied approximately. Furthermore,
the relation w_4_103 = 2 · w_4_147 seems to hold approximately.
280 R. BURING AND A. V. KISELEV
A.2. Claimed values of the 10 master-parameters. By using a different technique
B. Pym and E. Panzer have obtained the exact values of the ten master-parameters.
Claim ([34]). The values of ten master-parameters (which are the weights of ten graphs
in Figure 3 on p. 247) are given in Table 4 below.
Table 4. Recently suggested values of the master-parameters [34].
Master-parameter Value
p1 = w_4_100 1/1440
p2 = w_4_101 1/2880
p3 = w_4_102 1/5760
p4 = w_4_103 −1/11520
p5 = w_4_104 1/2880
p6 = w_4_107 13/2880
p7 = w_4_108 −17/2880
p8 = w_4_109 −1/1152
p9 = w_4_119 −1/1280
p10 = w_4_125 −1/960
Let it be emphasized that these ten values are conjectured via a use of software which
is currently under development.
Remark 19. The exact values of two master-parameters w_4_103 and w_4_104 reproduce
the values which had been conjectured in Table 3. We also note that all the weights
of graphs in ⋆ mod ō(ℏ4) are rational numbers. Thirdly, the values of non-master
parameters (namely, w_4_112, w_4_113, w_4_133, w_4_138, w_4_147, and w_4_148) in
Table 3, whenever recalculated on the basis of conjectured values from Table 4, do all
match the numerical approximations in Table 3, reproducing our conjectured rational
values in its rightmost column.
In conclusion, provided that all the ten values in Table 4 are true, this is the authentic
Kontsevich star-product up to ō(ℏ4):
(
f ⋆ g = f × g + ℏP ij∂ f∂ g + ℏ2 − 1∂ P ij∂)Pkℓi j ℓ j ∂i(f∂ 1 ij kℓ6 kg − ∂ℓP P ∂if∂k∂jg3
+ 1∂ P ijPkℓℓ ∂ ∂ f∂ g + 1P ijk i j Pkℓ∂k∂if∂ℓ∂jg + ℏ3 − 1∂m∂ℓP ij∂ kℓn∂jP Pmn∂if∂ g3 2 6 k
+ 1∂ ∂ P ijPkℓPmnn ℓ ∂ f∂ ∂ ∂ g − 1∂ ij kℓ mni m k j nP P P ∂k∂if∂m∂ℓ∂jg6 3
+ 1∂n∂ P ijℓ PkℓPmn∂ ∂ ∂ f∂ g + 1∂ P ijPkℓPmnm k i j n ∂m∂k∂if∂ ∂ g6 3 ℓ j
+ 1P ijPkℓPmn∂ ∂ ∂ f∂ ∂ ∂ g − 1∂ ∂ P ij kℓm k i n ℓ j m ℓ ∂nP Pmn∂6 6 if∂k∂jg
+ 1∂ ∂ P ij∂ PkℓPmnn ℓ j ∂if∂m∂kg − 1∂m∂ P ij∂ PkℓPmnℓ n ∂k∂if∂6 6 jg
− 1∂ P ij∂ ∂ PkℓPmnℓ n j ∂m∂if∂ g − 1P ijk ∂nPkℓ∂ mn6 6 ℓP ∂k∂if∂m∂jg )
− 1∂ ij kℓ mn 1 ij kℓ mn
6 n
P P ∂ℓP ∂k∂if∂m∂jg − ∂ℓP ∂nP P ∂k∂if∂m∂jg +6
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 281
(
+ ℏ4 − 1∂ ij kℓ mn pq 1 ij kℓ mn pq
6 q
P P P P ∂m∂k∂if∂p∂n∂ℓ∂jg + ∂qP P P P ∂p∂m∂k∂if∂n∂6 ℓ∂jg
+ 1 P ijPkℓPmnPpq∂p∂m∂k∂if∂q∂n∂ℓ∂jg + 1∂ ∂ P ijPkℓPmnPpq∂ ∂ f∂ ∂ ∂ ∂ g24 6 q n k i p m ℓ j
+ 1 ∂ P ij∂ PkℓPmnPpqn q ∂k∂if∂ 1p∂m∂ℓ∂jg + ∂ ∂ P ijq n PkℓPmnPpq∂p∂m∂k∂if∂ℓ∂18 6 jg
+ 1 ∂ P ijn ∂ kℓ mnqP P Ppq∂p∂m∂k∂if∂ℓ∂ 1jg − ∂q∂ ij kℓ mn pqn∂ℓP P P P ∂if∂p∂m∂18 30 k∂jg
− 2 ∂ ∂ P ij∂ PkℓPmnPpq∂ f∂ ∂ ∂ ∂ g − 1 ∂ P ij∂ ∂ PkℓPmnPpq∂ f∂ ∂ ∂ ∂ g
45 n ℓ q i p m k j 30 ℓ q n i p m k j
+ 1 ∂ P ij∂ Pkℓ∂ PmnPpqℓ n q ∂if∂p∂ 1 ij kℓ mn pq45 m∂k∂jg − P P ∂ P ∂ P ∂ ∂ ∂ f∂ ∂ ∂ g12 q n m k i p ℓ j
− 1∂ P ijPkℓPmn∂ Ppq∂ ∂ ∂ 1 ij kℓ mn pq
6 q n m k i
f∂p∂ℓ∂jg − ∂6 nP P ∂qP P ∂m∂k∂if∂p∂ℓ∂jg
− 1∂ P ijn ∂ kℓqP PmnPpq∂m∂k∂if∂p∂ℓ∂jg + 1 ∂q∂ ∂ P ijPkℓPmnPpqn ℓ ∂p∂m∂k∂9 30 if∂jg
+ 2 ∂ ∂ P ijn ℓ ∂qPkℓPmnPpq∂p∂m∂k∂if∂jg + 1 ∂ℓP ij∂ ∂ Pkℓq n PmnPpq∂p∂m∂k∂45 30 if∂jg
− 1 ∂ℓP ij∂ kℓnP ∂ Pmnq Ppq∂p∂ ∂ 1 ij kℓ mn pq45 m k∂if∂jg − ∂p∂nP P ∂6 qP P ∂k∂if∂m∂ℓ∂jg
+ 1P ij∂ kℓ mnq∂nP ∂ℓP Ppq∂ ∂ f∂ ∂ ∂ g + 1 ∂ P ij∂ Pkℓ∂ PmnPpqk i p m j q n ℓ ∂k∂if∂p∂6 18 m∂jg
− 1∂ ∂ ij kℓ mn pq 1 ij kℓ mn pq
9 p n
P ∂qP P P ∂k∂if∂m∂ℓ∂jg − ∂nP ∂pP ∂18 qP P ∂k∂if∂m∂ℓ∂jg
+ 1 ∂q∂n∂ℓP ijPkℓPmnPpq∂k∂if∂p∂m∂ g + 13∂ ∂ P ijj q n Pkℓ∂ℓPmnPpq∂30 90 k∂if∂p∂m∂jg
− 1 ∂ ∂ P ijPkℓn ℓ ∂ PmnPpqq ∂k∂if∂p∂ 1 ijm∂jg − ∂nP Pkℓ∂ mn pqq∂ℓP P ∂k∂if∂p∂ ∂45 30 m jg
+ 1 ∂ P ijPkℓ∂ Pmn∂ pqq ℓ nP ∂k∂if∂ ∂ ∂ g + 1p m j ∂ ij kℓ mn pq90 30 n∂ℓP ∂qP P P ∂k∂if∂p∂m∂jg
+ 1 ∂ P ij∂ Pkℓ∂ PmnPpqn q ℓ ∂k∂if∂ 1p∂m∂jg + ∂ℓP ij∂ kℓ mn pqq∂nP P P ∂k∂15 15 if∂p∂m∂jg
+ 1 ∂ P ijℓ ∂ Pkℓn ∂ mn pqqP P ∂k∂if∂ 1 ijp∂m∂jg − ∂p∂nP Pkℓ∂ Pmnq Ppq∂m∂k∂if∂ℓ∂90 6 jg
− 1P ij∂nPkℓ∂q∂ℓPmnPpq∂ ∂ ∂ f∂ ∂ g − 1 ∂ P ijPkℓ∂ Pmn∂ pq6 p k i m j 18 ℓ q nP ∂m∂k∂if∂p∂jg
− 1∂ ∂ P ij∂ kℓ mn pqp n qP P P ∂m∂k∂if∂ℓ∂jg − 1 ∂ ijnP ∂pPkℓ∂ mn pqqP P ∂m∂k∂if∂9 18 ℓ∂jg
− 1 ∂ ∂ ij kℓ mn pqq n∂ℓP P P P ∂m∂k∂if∂ ∂ g − 13p j ∂n∂ℓP ij∂ PkℓPmnPpqq ∂m∂k∂if∂p∂jg30 90
+ 1 ∂ ∂ P ij∂ PkℓPmnq ℓ n Ppq∂m∂k∂ f∂ 1 ij kℓ mn pq45 i p∂jg + ∂30 ℓP ∂q∂nP P P ∂m∂k∂if∂p∂jg
− 1 ∂ P ij∂ kℓℓ nP ∂qPmnPpq∂m∂k∂if∂p∂jg − 1 ∂ ∂ P ijPkℓPmn∂ Ppq∂90 30 q ℓ n m∂k∂if∂p∂jg
− 1 ∂ P ij∂ PkℓPmn∂ Ppq∂ ∂ ∂ f∂ ∂ g − 1 ∂ P ij kℓℓ q n m k i p j n P ∂q∂ℓPmnPpq∂p∂k∂if∂15 15 m∂jg
− 1 ∂ P ijPkℓ∂ Pmn∂ Ppq∂ ∂ ∂ f∂ ∂ g + 7 ∂ ∂ ∂ P ij∂ PkℓPmnPpq∂ f∂ ∂ ∂ g
90 q ℓ n m k i p j 90 p n ℓ q i m k j
− 1 ∂ ∂ P ij∂ ∂ PkℓPmnPpq∂ f∂ ∂ ∂ g + 1 ∂ ∂ P ij∂ Pkℓ mn pq
90 p ℓ q n i m k j p ℓ n
∂
30 q
P P ∂if∂m∂k∂jg
+ 2 ∂ P ij∂ ∂ Pkℓ∂ PmnPpqℓ p n q ∂if∂m∂k∂jg + 1 ∂p∂ℓP ij∂ PkℓPmnq ∂ pq45 30 nP ∂if∂m∂k∂jg
+ 1 ∂ ∂ P ijPkℓp ℓ ∂ Pmn∂ Ppq∂ f∂ ∂ ∂ g − 1 ijq n i m k j ∂ P ∂ Pkℓ∂ Pmn∂ Ppq∂ f∂ ∂ ∂ g45 90 ℓ p q n i m k j
− 1 ∂ P ij∂ Pkℓp q ∂ℓPmn∂nPpq∂ 1 ij kℓ mn pq90 if∂m∂k∂jg − ∂pP P ∂q∂ℓP ∂nP ∂90 if∂m∂k∂jg
+ 1 ∂ P ij∂ Pkℓ∂ ∂ PmnPpqm n q ℓ ∂if∂p∂ ∂ 1 ij kℓ mn pq30 k jg + ∂qP ∂jP ∂ℓP ∂nP ∂if∂p∂m∂kg90
+ 1 ∂ P ij∂ ∂ Pkℓ∂ mn pq 1 ij kℓ mn pq
90 n q j ℓ
P P ∂if∂p∂m∂kg + ∂ℓP ∂n∂jP ∂qP P ∂if∂p∂m∂90 kg
− 1 ∂ ijnP ∂jPkℓ∂ mn pq 1 ij kℓ mn pq30 q∂ℓP P ∂if∂p∂m∂kg − ∂q∂nP ∂ P ∂45 j ℓP P ∂if∂p∂m∂kg
282 R. BURING AND A. V. KISELEV
− 1 ∂ P ij∂ ∂ ∂ PkℓPmnPpqℓ q n j ∂if∂p∂m∂kg − 1 ∂ ijn∂ℓP ∂ kℓ mn pqq∂jP P P ∂60 45 if∂p∂m∂kg
− 1 ∂ ∂ P ij∂ Pkℓn ℓ j ∂ Pmnq Ppq∂if∂ ∂ ∂ g − 1p m k ∂ ij kℓ mn pq45 20 q∂n∂ℓP ∂jP P P ∂if∂p∂m∂kg
− 7 ∂p∂ ij kℓ mn pq 1n∂ℓP ∂qP P P ∂m∂k∂if∂jg + ∂p∂ℓP ij∂ ∂ PkℓPmn pq90 90 q n P ∂m∂k∂if∂jg
− 1 ∂ ∂ P ijp ℓ ∂nPkℓ∂ mn pqqP P ∂m∂k∂if∂jg − 2 ∂ℓP ij∂ ∂ Pkℓ∂ Pmn pqp n q P ∂m∂k∂if∂30 45 jg
− 1 ∂p∂ℓP ij∂qPkℓPmn∂ pqnP ∂m∂k∂if∂jg − 1 ∂ ij kℓp∂ℓP P ∂qPmn∂ pq30 45 nP ∂m∂k∂if∂jg
+ 1 ∂ P ij∂ Pkℓ∂ Pmn pq 1 ij kℓ mn pq
90 ℓ p q
∂nP ∂m∂k∂if∂jg + ∂ P ∂ P ∂ P ∂ P ∂ ∂ ∂ f∂ g90 p q ℓ n m k i j
+ 1 ∂ P ijPkℓ∂ ∂ Pmn∂ Ppq∂ 1 ij kℓp q ℓ n m∂k∂if∂jg − ∂mP ∂nP ∂ mn pqq∂ℓP P ∂90 30 p∂k∂if∂jg
+ 1 ∂ P ij∂ Pkℓq j ∂ℓPmn∂nPpq∂m∂k∂if∂ 1 ijpg + P ∂ ∂ Pkℓ∂ mn pq90 90 q j ℓP ∂nP ∂m∂k∂if∂pg
+ 1 P ij∂ Pkℓj ∂q∂ℓPmn∂nPpq∂ 1 ijm∂k∂if∂pg − ∂nP ∂ ∂ Pkℓ∂ PmnPpqq j ℓ ∂p∂90 30 k∂if∂mg
− 1 ∂ P ij∂ kℓ mn pq 1 ij kℓ mn pq
45 n j
P ∂q∂ℓP P ∂p∂k∂if∂mg − P ∂60 q∂n∂jP ∂ℓP P ∂p∂k∂if∂mg
− 1 P ij∂ ∂ Pkℓn j ∂q∂ Pmnℓ Ppq∂p∂k∂if∂mg − 1 P ij∂ Pkℓj ∂q∂ Pmn∂ Ppq∂45 45 ℓ n p∂k∂if∂mg
− 1 ∂ P ij∂ ∂ kℓ mn pq 1 ij kℓ mn pq
20 ℓ q n
∂jP P P ∂p∂m∂if∂kg − ∂40 p∂m∂ℓP ∂q∂n∂jP P P ∂if∂kg
− 1 ∂p∂ ij kℓ mn pq 1m∂ℓP ∂n∂jP ∂qP P ∂if∂kg + ∂ ijm∂ℓP ∂p∂n∂ kℓ mn pq72 72 jP ∂qP P ∂if∂kg
+ 1 ∂ ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ f∂ g − 1 ∂ ∂ P ij∂ ∂ ∂ Pkℓ∂ Pmn pqm ℓ p j q n i k p m q n j ℓ P ∂if∂kg360 60
− 1 ∂ ∂ ∂ P ijp n k ∂jPkℓ∂q∂ mnℓP Ppq∂if∂mg + 17 ∂mP ij∂p∂n∂jPkℓ∂q∂ℓPmnPpq∂60 720 if∂kg
− 17 ∂p∂n∂ ijkP ∂q∂jPkℓ∂ Pmnℓ Ppq∂ 1if∂mg − ∂p∂ ij kℓmP ∂q∂jP ∂ℓPmn∂ Ppqn ∂if∂720 180 kg
+ 1 ∂p∂nP ij∂ Pkℓj ∂q∂kPmn∂ℓPpq∂ f∂ g + 1 ∂ ∂ ij kℓ mn pq180 i m 360 p mP ∂jP ∂q∂ℓP ∂nP ∂if∂kg
− 1 ∂ P ij∂ ∂ Pkℓ∂ mn pqn p j q∂kP ∂ℓP ∂ f∂ g + 1 ∂ ij kℓ mn pq360 i m 160 mP ∂p∂jP ∂q∂ℓP ∂nP ∂if∂kg
+ 1 ∂ ∂ P ij∂ ∂ Pkℓ∂ mn pqp n q j kP ∂ℓP ∂if∂mg − 17 ∂ P ij∂ kℓ mn pq160 1440 p q∂m∂jP ∂ℓP ∂nP ∂if∂kg
+ 17 ∂q∂m∂kP ij∂ Pkℓ∂ Pmn∂ pq 1 ij kℓ mn pq1440 j ℓ nP ∂if∂pg − ∂pP ∂m∂jP ∂q∂ℓP ∂nP ∂if∂ g360 k
+ 1 ∂ ijq∂kP ∂m∂jPkℓ∂ mn pqℓP ∂nP ∂if∂pg − 13 ∂ ∂ P ij kℓ mn pqp k ∂q∂n∂jP ∂ℓP P ∂360 720 if∂mg
− 13 ∂ P ij∂ ∂ ∂ Pkℓ∂ ∂ PmnPpq∂ f∂ g − 1 ijk p n j q ℓ i m ∂p∂kP ∂n∂jPkℓ∂ mn pq720 60 q∂ℓP P ∂if∂mg
− 7 ∂p∂ P ijk ∂ kℓq∂jP ∂ Pmn∂ Ppq∂ 7 ij kℓ mn pq720 ℓ n if∂mg + ∂pP ∂jP ∂q∂kP ∂n∂ℓP ∂if∂720 mg
− 1 ∂ ij kℓp∂kP ∂jP ∂ ∂ mn pq 1 ij kℓ mn pq180 q ℓP ∂nP ∂if∂mg + ∂kP ∂p∂ P ∂160 j q∂ℓP ∂nP ∂if∂mg
+ 1 ∂ ∂ P ij∂ Pkℓm k j ∂q∂ℓPmn∂ pqnP ∂if∂pg + 13 ∂ ∂ P ijm k ∂q∂jPkℓ∂ mn pq160 1440 ℓP ∂nP ∂if∂pg
− 13 ∂ P ij∂ ∂ Pkℓ∂ ∂ mn pq 1 ij kℓ mn pq
1440 p n j q k
P ∂ℓP ∂if∂mg − ∂kP ∂q∂ ∂1440 m jP ∂ℓP ∂nP ∂if∂pg
+ 1 ∂pP ij∂q∂n∂ kℓ mn pq 1 ij kℓ mn pq1440 jP ∂kP ∂ℓP ∂if∂mg + ∂kP ∂m∂jP ∂q∂ℓP ∂ P ∂ f∂360 n i pg
+ 1 ∂ P ij∂ ∂ Pkℓ mn pq 13p q j ∂kP ∂n∂ℓP ∂if∂mg − ∂p∂m∂ℓP ij∂ kℓ mn pqq∂nP P P ∂240 360 if∂k∂jg
− 1 ∂p∂m∂ ijℓP ∂nPkℓ∂ PmnPpq∂ 1 ij kℓq if∂k∂jg + ∂m∂ℓP ∂p∂nP ∂ PmnPpqq ∂if∂ ∂ g720 30 k j
− 1 ∂ ij kℓ mn pq 19 ij kℓ mn pq
720 m
∂ℓP ∂pP ∂qP ∂nP ∂if∂k∂jg − ∂p∂mP ∂q∂nP ∂ℓP P ∂if∂k∂ g720 j
+ 1 ∂ ∂ P ijp m ∂ Pkℓ∂ ∂ PmnPpq∂ f∂ ∂ g + 13 ∂ P ij kℓ mn pqn q ℓ i k j ∂ ∂ P ∂ ∂ P P ∂ f∂ ∂ g180 360 m p n q ℓ i k j
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION 283
− 1 ∂ ∂ ij kℓ mn pq 1 ij kℓ mn pq
720 p m
P ∂qP ∂ℓP ∂nP ∂if∂k∂jg + ∂p∂mP P ∂q∂ℓP ∂ P ∂360 n if∂k∂jg
+ 1 ∂ P ijm ∂ kℓpP ∂q∂ℓPmn∂nPpq∂if∂k∂jg − 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ f∂ ∂ g720 80 p q m ℓ n i k j
− 1 ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ Ppqp m q ℓ n ∂ f∂ 1 ij kℓ mn pq180 i k∂jg − ∂ P ∂ ∂ P ∂ ∂ P P ∂ f∂ ∂ g36 n p j q ℓ i m k
+ 1 ∂ ∂ P ij∂ ∂ Pkℓ∂ PmnPpq∂ f∂ ∂ g − 1 ∂ ∂ P ij∂ Pkℓ mn pqp n q j ℓ i m k p n j ∂q∂ℓP P ∂if∂60 720 m∂kg
+ 1 ∂ ij kℓ mn pq 17 ij kℓ mn pq
45 ℓ
P ∂p∂n∂jP ∂qP P ∂if∂m∂kg + ∂ ∂ P ∂720 p ℓ n∂jP ∂qP P ∂if∂m∂kg
+ 13 ∂ ∂ P ij∂ ∂ ∂ PkℓPmnPpq∂ f∂ ∂ g + 1 ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ f∂ ∂ g
360 p ℓ q n j i m k 120 n j q k ℓ i p m
+ 1 ∂ ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq∂ f∂ ∂ g − 1 ∂ ∂ P ij∂ Pkℓ∂ ∂ PmnPpq∂ f∂ ∂ g
240 q k j ℓ n i p m 80 n k j q ℓ i p m
− 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppqp q j ℓ n ∂if∂m∂ 1kg − ∂ P ijm ∂ kℓq∂jP ∂ℓPmn∂ Ppqn ∂if∂72 90 p∂kg
− 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppqk q j ℓ n ∂ f∂ 1 ij kℓ mn pq180 i p∂mg − ∂ P ∂ P ∂ ∂ P ∂ P ∂ f∂ ∂ g360 p j q ℓ n i m k
+ 1 ∂mP ij∂ ∂ kℓq n∂jP ∂ℓPmnPpq∂if∂ 1 ij kℓ mn pq720 p∂kg − ∂ P ∂ ∂ P ∂ ∂ P P ∂ f∂ ∂ g180 k n j q ℓ i p m
+ 17 ∂ ∂ ∂ P ij∂ ∂ PkℓPmnPpq∂ f∂ ∂ g − 1 ijp n ℓ q j i m k ∂n∂ℓP ∂p∂ Pkℓ∂ PmnPpqj q ∂if∂180 72 m∂kg
+ 7 ∂ ∂ ∂ P ij∂ Pkℓ∂ PmnPpq∂ f∂ ∂ g + 7 ∂ ∂ P ij∂ ∂ PkℓPmn∂ pq
180 p n ℓ j q i m k 180 p ℓ q j n
P ∂if∂m∂kg
− 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppqℓ p j q n ∂if∂m∂kg + 1 ∂ ij kℓ mn pqp∂ℓP ∂jP ∂qP ∂nP ∂if∂m∂kg180 90
+ 1 ∂ ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq∂ f∂ ∂ g + 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ f∂ ∂ g
80 q m j ℓ n i p k 120 q m j ℓ n i p k
+ 1 ∂q∂mP ij∂ ∂ Pkℓ∂ PmnPpqn j ℓ ∂ 1 ij kℓ mn pq40 if∂p∂kg − ∂q∂n∂kP ∂jP ∂ℓP P ∂360 if∂p∂mg
− 11 ∂n∂ P ijk ∂ kℓq∂jP ∂ℓPmnPpq∂ 7 ij kℓ mn pq720 if∂p∂mg + ∂mP ∂240 n∂jP ∂q∂ℓP P ∂if∂p∂kg
− 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppqn q j k ℓ ∂if∂p∂ 1mg + ∂mP ij∂ Pkℓ∂ mn pqj q∂ℓP ∂nP ∂if∂p∂kg180 60
+ 1 ∂ ∂ P ij∂ ∂ Pkℓ∂ PmnPpq∂ f∂ ∂ g + 1 ∂ P ij∂ ∂ ∂ Pkℓ∂ PmnPpq∂ f∂ ∂ g
90 q k n j ℓ i p m 60 k q n j ℓ i p m
+ 13 ∂ ∂ ∂ P ij∂ ∂ PkℓPmnPpq∂ ∂ f∂ g + 1 ∂ ∂ ∂ ij kℓ mn pq
360 p m ℓ q n k i j 720 p m ℓ
P ∂nP ∂qP P ∂k∂if∂jg
− 1 ∂ ∂ P ij∂ ∂ Pkℓ∂ PmnPpq∂ ∂ f∂ g + 1 ∂ ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ f∂ g
30 m ℓ p n q k i j 720 m ℓ p q n k i j
+ 19 ∂ ∂ P ij∂ ∂ Pkℓ∂ PmnPpq∂ ∂ f∂ g − 1 ∂ ∂ P ij∂ Pkℓ∂ ∂ PmnPpq∂ ∂ f∂ g
720 p m q n ℓ k i j 180 p m n q ℓ k i j
− 13 ∂ P ij∂ ∂ Pkℓ∂ ∂ PmnPpq∂ ∂ f∂ g + 1m p n q ℓ k i j ∂p∂mP ij∂ kℓ mn pq360 720 qP ∂ℓP ∂nP ∂k∂if∂jg
− 1 ∂ ∂ P ijPkℓ∂ ∂ Pmn∂ Ppq∂ ∂ f∂ g − 1 ∂ P ij∂ Pkℓ mn pq
360 p m q ℓ n k i j
∂
720 m p q
∂ℓP ∂nP ∂k∂if∂jg
+ 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ f∂ g + 1 ∂ P ij∂ Pkℓ∂ ∂ Pmnp q m ℓ n k i j p m q ℓ ∂nPpq∂80 180 k∂if∂jg
− 1 ∂ ∂ P ij∂ ∂ Pkℓ∂ PmnPpq 1p n q j ℓ ∂k∂if∂mg − ∂p∂ ij kℓnP ∂jP ∂ ∂ Pmnq ℓ Ppq∂k∂if∂mg36 60
+ 1 ∂ P ij∂ ∂ Pkℓ∂ ∂ PmnPpq∂ ∂ f∂ g − 1 ∂ P ij∂ ∂ ∂ Pkℓ∂ PmnPpq∂ ∂ f∂ g
720 n p j q ℓ k i m 45 p q n j ℓ k i m
− 17 ∂ P ij∂ ∂ Pkℓ∂ ∂ PmnPpq∂ 13 ijp n j q ℓ k∂if∂mg − P ∂ ∂ ∂ kℓp n jP ∂q∂ℓPmnPpq∂720 360 k∂if∂mg
− 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn pqp q j ℓ ∂nP ∂k∂if∂mg − 1 ∂ P ijp ∂ kℓ mn pq120 240 jP ∂q∂ℓP ∂nP ∂k∂if∂mg
+ 1 P ij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ f∂ g + 1p j q ℓ n k i m ∂q∂mP ij∂jPkℓ∂ mn pq80 72 ℓP ∂nP ∂k∂if∂pg
+ 1 ∂ P ij∂ ∂ Pkℓ∂ Pmnm q j ℓ ∂ Ppqn ∂k∂if∂ g + 1 ∂ P ij∂ Pkℓp ∂ ∂ Pmn∂ Ppq∂ ∂ f∂ g90 180 m j q ℓ n k i p
+ 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ f∂ g − 1q m j ℓ n k i p P ij∂ kℓ mn pq360 720 q∂m∂jP ∂ℓP ∂nP ∂k∂if∂pg
284 R. BURING AND A. V. KISELEV
+ 1 P ij∂ ∂ Pkℓm j ∂q∂ Pmn∂ Ppq∂ ∂ f∂ g − 17ℓ n k i p ∂ ∂ ij kℓ mn pq180 180 p ℓP ∂q∂n∂jP P P ∂m∂if∂kg
− 1 ∂ ∂ P ij∂ ∂ Pkℓ∂ PmnPpqp ℓ n j q ∂m∂ 7if∂kg + ∂ P ij∂ ∂ ∂ Pkℓ∂ PmnPpq∂ ∂ f∂ g72 180 ℓ p n j q m i k
− 7 ∂ ∂ P ij∂ kℓ mn pq 1 ij kℓ mn pq
180 p ℓ q
∂jP P ∂nP ∂m∂if∂kg − ∂p∂ℓP ∂180 jP ∂qP ∂nP ∂m∂if∂kg
+ 1 ∂ P ijℓ ∂p∂jPkℓ∂ mnqP ∂nPpq∂ 1m∂if∂kg − ∂ P ij∂ ∂ Pkℓ∂ Pmnp q j ℓ ∂nPpq∂m∂if∂kg90 80
− 1 ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ 1 ij kℓ mn pq
120 p j q ℓ n m i
f∂kg + P ∂p∂jP ∂q∂ℓP ∂nP ∂m∂40 if∂kg
+ 1 ∂ P ij∂ ∂ ∂ Pkℓ∂ PmnPpq∂ ∂ f∂ g + 11 ∂ P ij∂ ∂ Pkℓ∂ ∂ PmnPpqm q n j ℓ p i k m n j q ℓ ∂p∂if∂kg360 720
− 7 ∂ ∂ P ijn k ∂q∂jPkℓ∂ mn pqℓP P ∂p∂ f∂ g − 1 ∂ P ij∂ kℓ mn pq240 i m 180 m jP ∂q∂ℓP ∂nP ∂p∂if∂kg
− 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ f∂ g + 1 P ij∂ ∂ Pkℓk q j ℓ n p i m m j ∂ ∂ Pmn pqq ℓ ∂nP ∂p∂if∂kg60 90
− 1 ∂ P ijk ∂q∂ ∂ kℓ mn pqn jP ∂ℓP P ∂p∂if∂mg − 1P ij∂ kℓ mn pq60 6 p∂nP ∂q∂ℓP P ∂k∂if∂m∂jg
+ 1 ∂ P ij∂ Pkℓ∂ Pmn∂ Ppqℓ j q n ∂m∂if∂p∂ 17kg + ∂ ∂ P ij∂ ∂ Pkℓ mnp m q n P Ppq∂72 360 k∂if∂ℓ∂jg
+ 1 ∂ ∂ P ijp m ∂nPkℓ∂ mn pq 1qP P ∂k∂if∂ℓ∂jg + ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq24 180 m p q n ∂k∂if∂ℓ∂jg
+ 2 ∂ ∂ ∂ P ij∂ PkℓPmnPpqp n ℓ q ∂ 2 ijk∂if∂m∂jg − ∂p∂n∂ℓP Pkℓ∂ PmnPpqq ∂k∂if∂45 45 m∂jg
− 1 ∂ ∂ P ij∂ kℓ mn pq 1 ij kℓ mn pq
30 n ℓ p
P ∂qP P ∂k∂if∂m∂jg + ∂8 p∂ℓP ∂q∂nP P P ∂k∂if∂m∂jg
− 1∂ ∂ P ijPkℓ∂ ∂ PmnPpq∂ ∂ f∂ ∂ g + 1 ∂ ∂ P ij∂ Pkℓ∂ Pmn pqp n q ℓ k i m j p ℓ n q P ∂k∂if∂m∂8 720 jg
− 1 ∂ ∂ P ij∂ Pkℓ∂ PmnPpq∂ ∂ f∂ ∂ g − 11 ∂ P ij∂ ∂ Pkℓ mn pqp n q ℓ k i m j ℓ p n ∂qP P ∂k∂if∂720 180 m∂jg
− 11 ∂ P ij∂ kℓ mn pq 1 ij kℓ mn pqn pP ∂q∂ℓP P ∂k∂if∂m∂jg + ∂p∂ℓP ∂qP P ∂nP ∂k∂if∂ ∂180 36 m jg
− 1 ∂ ∂ P ijPkℓq m ∂ℓPmn∂nPpq∂k∂if∂p∂jg + 1 ∂ ∂ P ijp ℓ Pkℓ∂ Pmn∂ Ppq∂36 90 q n k∂if∂m∂jg
− 1 ∂ ∂ P ijq m ∂nPkℓ∂ mn pq 1ℓP P ∂k∂if∂p∂jg − ∂ P ijℓ ∂ kℓ mnpP ∂qP ∂nPpq∂90 180 k∂if∂m∂jg
− 1 ∂qP ij∂mPkℓ∂ Pmn∂ Ppq∂ ∂ f∂ ∂ g + 1 ∂ P ij∂ ∂ Pkℓ mn pq180 ℓ n k i p j p q n ∂ℓP P ∂18 k∂if∂m∂jg
− 1 ∂pP ij∂ Pkℓn ∂q∂ mnℓP Ppq∂k∂if∂m∂jg + 1 ∂ ij kℓ mn pq18 144 pP ∂qP ∂ℓP ∂nP ∂k∂if∂m∂jg
− 1 ∂ ijmP ∂qPkℓ∂ Pmn∂ pq 1 ij kℓ mn pq144 ℓ nP ∂k∂if∂p∂jg − ∂pP P ∂q∂ℓP ∂nP ∂k∂if∂m∂jg90
+ 1 ∂ P ij∂ ∂ Pkℓ∂ PmnPpq∂ ∂ f∂ ∂ g − 1 ∂ P ij∂ Pkℓ mn pq
90 m q n ℓ k i p j
∂
60 m n q
∂ℓP P ∂k∂if∂p∂jg
+ 1 ∂ P ijPkℓ∂ ∂ Pmnm q ℓ ∂nPpq∂k∂if∂p∂jg − 1 ∂q∂nP ij∂ Pkℓ∂ PmnPpq∂60 240 j ℓ k∂if∂p∂mg
− 1 ∂ P ij∂ ∂ Pkℓ∂ PmnPpq∂ ∂ f∂ ∂ g − 13 ∂ P ij∂ ∂ kℓ mn pq
240 n q j ℓ p i m k 720 n q j
P ∂ℓP P ∂k∂if∂p∂mg
− 13 P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ f∂ ∂ g − 1 ∂ P ij∂ Pkℓ∂ ∂ PmnPpq∂ ∂ f∂ ∂ g
720 q j ℓ n m i p k 90 n j q ℓ k i p m
− 1 P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ f∂ ∂ g + 1 ∂ P ij∂ ∂ Pkℓ∂ Pmn pqq j ℓ n k i p m q n j ℓ P ∂k∂90 60 if∂p∂mg
+ 1 ∂ P ij∂ ∂ PkℓPmn∂ Ppq∂ ∂ f∂ ∂ g + 1 P ijℓ q j n m i p k ∂ ∂ ∂ Pkℓ∂ PmnPpq∂ ∂ f∂ ∂ g60 30 q n j ℓ k i p m
+ 1 ∂ P ij∂ ∂ ∂ PkℓPmnPpqℓ q n j ∂m∂if∂p∂ g − 1 P ijk ∂ kℓ mn pq30 90 n∂jP ∂q∂ℓP P ∂k∂if∂p∂mg
+ 1 ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ f∂ ∂ g + 13∂ ∂ P ij∂ ∂ PkℓPmn pq
360 q j ℓ n k i p m 90 q ℓ n j
P ∂m∂if∂p∂kg
+ 13 ∂ P ij∂ ∂ Pkℓ∂ Pmnℓ n j q Ppq∂m∂if∂p∂kg + 13 ∂q∂ P ijℓ ∂ kℓ mn pq180 180 jP P ∂nP )∂m∂if∂p∂kg
+ 1 ∂ P ij∂ Pkℓ∂ Pmnq j ℓ ∂nPpq∂m∂if∂p∂ 472 kg + ō(ℏ ). (13)
Out of 247 graphs at ℏ4, as many as 138 contain two-cycles (or “eyes”, as in Fig. 1 on
p. 229), cf. expansion (1) up to ō(ℏ3) on p. 225.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION i
Appendix B. C++ classes and methods
Class KontsevichGraph
Summary: a (signed) Kontsevich graph.
Data members (private):
size_t d_internal = 0;
size_t d_external = 0;
std::vector< std::pair<char, char> > d_targets;
int d_sign = 1;
Public typedefs:
typedef char Vertex;
typedef std::pair<Vertex, Vertex> VertexPair;
Constructors:
KontsevichGraph() = default;
KontsevichGraph(size_t internal, size_t external,
std::vector<VertexPair> targets,
int sign = 1, bool normalized = false);
Accessor methods:
std::vector<VertexPair> targets() const;
VertexPair targets(Vertex internal_vertex) const;
int sign() const;
int sign(int new_sign);
size_t internal() const;
size_t external() const;
Methods to obtain numerical information:
size_t vertices() const;
std::vector<Vertex> internal_vertices() const;
std::pair< size_t, std::vector<VertexPair> > abs() const;
size_t multiplicity() const;
size_t in_degree(KontsevichGraph::Vertex vertex) const;
std::vector<size_t> in_degrees() const;
std::vector<Vertex> neighbors_in(Vertex vertex) const;
KontsevichGraph mirror_image() const;
std::string as_sage_expression() const;
std::string encoding() const;
std::vector< std::tuple<KontsevichGraph, int, int> > permutations() const;
Methods that modify the graph:
void normalize();
KontsevichGraph& operator*=(const KontsevichGraph& rhs);
Methods that test for graph properties:
bool operator<(const KontsevichGraph& rhs) const;
bool is_zero() const;
ii R. BURING AND A. V. KISELEV
bool is_prime() const;
bool positive_differential_order() const;
bool has_cycles() const;
bool has_tadpoles() const;
bool has_multiple_edges() const;
bool has_max_internal_indegree(size_t max_indegree) const;
Static methods:
static std::set<KontsevichGraph> graphs(size_t internal,
size_t external = 2, bool modulo_signs = false,
bool modulo_mirror_images = false,
std::function<void(KontsevichGraph)> const& callback = nullptr,
std::function<bool(KontsevichGraph)> const& filter = nullptr);
Private methods:
friend std::ostream& operator<<(std::ostream &os, const KontsevichGraph& g);
friend std::istream& operator>>(std::istream& is, KontsevichGraph& g);
friend bool operator==(const KontsevichGraph &lhs, const KontsevichGraph& rhs);
friend bool operator!=(const KontsevichGraph &lhs, const KontsevichGraph& rhs);
Functions defined outside the class:
KontsevichGraph operator*(KontsevichGraph lhs, const KontsevichGraph& rhs);
std::ostream& operator<<(std::ostream &os, const KontsevichGraph::Vertex v);
Class KontsevichGraphSum<T>
• Template parameter T: type of the coefficients (e.g. KontsevichGraphSum<int>).
• Publically extends: std::vector< std::pair<T, KontsevichGraph> >.
Summary: a sum of Kontsevich graphs, with method to reduce modulo skew-symmetry.
Data members: inherited.
Public typedefs:
typedef std::pair<T, KontsevichGraph> Term;
Constructors (inherited):
using std::vector< std::pair<T, KontsevichGraph> >::vector;
Accessor methods:
using std::vector< std::pair<T, KontsevichGraph> >::operator[];
KontsevichGraphSum<T> operator[](std::vector<size_t> indegrees) const;
T operator[](KontsevichGraph) const;
Arithmetic operators:
KontsevichGraphSum<T> operator()(std::vector< KontsevichGraphSum<T> >) const;
KontsevichGraphSum<T>& operator+=(const KontsevichGraphSum<T>& rhs);
KontsevichGraphSum<T>& operator-=(const KontsevichGraphSum<T>& rhs);
KontsevichGraphSum<T>& operator=(const KontsevichGraphSum<T>&) = default;
Methods:
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION iii
std::vector< std::vector<size_t> > in_degrees(bool ascending = false) const;
KontsevichGraphSum<T> skew_symmetrization() const;
Methods that modify the graph sum:
void reduce_mod_skew();
Comparison operators:
bool operator==(const KontsevichGraphSum<T>& other) const;
bool operator==(int other) const;
bool operator!=(const KontsevichGraphSum<T>& other) const;
bool operator!=(int other) const;
Friend operators:
friend std::ostream& operator<< <>(std::ostream& os,
const KontsevichGraphSum<T>::Term& term);
friend std::ostream& operator<< <>(std::ostream& os,
const KontsevichGraphSum<T>& gs);
friend std::istream& operator>> <>(std::istream& is,
KontsevichGraphSum<T>& sum);
Functions defined outside the class:
KontsevichGraphSum<T> operator+(KontsevichGraphSum<T> lhs,
const KontsevichGraphSum<T>& rhs);
KontsevichGraphSum<T> operator-(KontsevichGraphSum<T> lhs,
const KontsevichGraphSum<T>& rhs);
KontsevichGraphSum<T> operator*(T lhs,
KontsevichGraphSum<T> rhs);
std::ostream& operator<<(std::ostream&, const std::pair<T, KontsevichGraph>&);
std::ostream& operator<<(std::ostream&, const KontsevichGraphSum<T>&);
std::istream& operator>>(std::istream&, KontsevichGraphSum<T>&);
Class KontsevichGraphSeries<T>
• Template parameter T: type of the coefficients (e.g. KontsevichGraphSeries<int>).
• Publically extends: std::map< size_t, KontsevichGraphSum<T> >
Summary: a formal power series expansion; sums of Kontsevich graphs as coefficients.
Data members: inherited, plus (private):
size_t d_precision = std::numeric_limits<std::size_t>::max();
Constructors (inherited):
using std::map< size_t, KontsevichGraphSum<T> >::map;
Accessor methods:
size_t precision() const;
size_t precision(size_t new_precision);
Arithmetic operators:
iv R. BURING AND A. V. KISELEV
KontsevichGraphSeries<T> operator()(std::vector< KontsevichGraphSeries<T> >)
const;
KontsevichGraphSeries<T>& operator+=(const KontsevichGraphSeries<T>& rhs);
KontsevichGraphSeries<T>& operator-=(const KontsevichGraphSeries<T>& rhs);
Methods:
KontsevichGraphSeries<T> skew_symmetrization() const;
KontsevichGraphSeries<T> inverse() const;
KontsevichGraphSeries<T> gauge_transform(const KontsevichGraphSeries<T>& gauge);
Comparison operators:
bool operator==(int other) const;
bool operator!=(int other) const;
Methods that modify the graph series:
void reduce_mod_skew();
Static methods:
static KontsevichGraphSeries<T> from_istream(std::istream& is,
std::function<T(std::string)> const& parser,
std::function<bool(KontsevichGraph, size_t)> const& filter = nullptr);
Friend methods:
friend std::ostream& operator<< <>(std::ostream& os,
const KontsevichGraphSeries<T>& series);
Functions defined outside the class:
KontsevichGraphSeries<T> operator+(KontsevichGraphSeries<T> lhs,
const KontsevichGraphSeries<T>& rhs);
KontsevichGraphSeries<T> operator-(KontsevichGraphSeries<T> lhs,
const KontsevichGraphSeries<T>& rhs);
std::ostream& operator<<(std::ostream&, const KontsevichGraphSeries<T>&);
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION v
Appendix C. Encoding of the entire ⋆-product modulo ō(ℏ4)
In the following two tables, containing the sets of basic graphs and the ⋆-product ex-
pansion respectively, encodings of graphs (see Implementation 1 on p. 229) are followed
by their coefficients.
Table 5. Basic sets of Kontsevich graphs, up to order 4, including zero graphs.
h^0: 2 4 1 0 1 0 2 2 3 2 4 w_4_35 2 4 1 0 3 0 4 1 5 1 2 w_4_92
2 0 1 1 2 4 1 0 1 0 2 2 3 3 4 w_4_36 2 4 1 0 3 0 4 1 5 2 3 w_4_93
h^1: 2 4 1 0 1 0 2 2 5 2 4 w_4_37 2 4 1 0 3 0 4 1 5 2 4 w_4_94
2 1 1 0 1 1/2 2 4 1 0 1 0 2 2 5 3 4 w_4_38 2 4 1 0 3 0 4 1 5 3 4 w_4_95
h^2: 2 4 1 0 1 0 2 3 5 3 4 w_4_39 2 4 1 0 3 0 4 2 3 1 2 w_4_96
2 2 1 0 1 0 2 1/12 2 4 1 0 1 0 4 0 3 2 3 w_4_40 2 4 1 0 3 0 4 2 3 1 3 w_4_97
2 2 1 0 3 1 2 -1/24 2 4 1 0 1 0 4 0 5 2 3 w_4_41 2 4 1 0 3 0 4 2 3 1 4 w_4_98
h^3: 2 4 1 0 1 0 4 0 5 2 4 w_4_42 2 4 1 0 3 0 4 2 5 1 2 w_4_99
2 3 0 0 1 0 1 2 3 0 2 4 1 0 1 0 4 1 3 2 3 w_4_43 2 4 1 0 3 0 4 2 5 1 3 w_4_100
2 3 1 0 1 0 2 0 2 1/24 2 4 1 0 1 0 4 1 5 2 3 w_4_44 2 4 1 0 3 0 4 2 5 1 4 w_4_101
2 3 1 0 1 0 2 0 3 0 2 4 1 0 1 0 4 1 5 2 4 w_4_45 2 4 1 0 3 0 4 3 5 1 2 w_4_102
2 3 1 0 1 0 2 1 2 0 2 4 1 0 1 0 4 2 3 0 4 w_4_46 2 4 1 0 3 0 4 3 5 1 3 w_4_103
2 3 1 0 1 0 2 1 3 -1/48 2 4 1 0 1 0 4 2 3 1 4 w_4_47 2 4 1 0 3 0 4 3 5 1 4 w_4_104
2 3 1 0 1 0 2 2 3 -1/48 2 4 1 0 1 0 4 2 3 2 3 w_4_48 2 4 1 0 3 1 2 0 3 0 3 w_4_105
2 3 1 0 1 0 4 2 3 0 2 4 1 0 1 0 4 2 3 2 4 w_4_49 2 4 1 0 3 1 2 0 3 1 2 w_4_106
2 3 1 0 1 2 4 2 3 0 2 4 1 0 1 0 4 2 3 3 4 w_4_50 2 4 1 0 3 1 2 0 3 1 4 w_4_107
2 3 1 0 3 0 2 1 2 0 2 4 1 0 1 0 4 2 5 2 3 w_4_51 2 4 1 0 3 1 2 0 3 2 3 w_4_108
2 3 1 0 3 0 4 1 2 0 2 4 1 0 1 0 4 2 5 2 4 w_4_52 2 4 1 0 3 1 2 0 3 2 4 w_4_109
2 3 1 0 3 0 4 1 3 0 2 4 1 0 1 0 4 2 5 3 4 w_4_53 2 4 1 0 3 1 2 0 3 3 4 w_4_110
2 3 1 0 3 1 2 0 3 -1/48 2 4 1 0 1 0 4 3 5 2 3 w_4_54 2 4 1 0 3 1 2 0 5 2 3 w_4_111
2 3 1 0 3 1 2 2 3 -1/48 2 4 1 0 1 0 4 3 5 2 4 w_4_55 2 4 1 0 3 1 2 0 5 2 4 w_4_112
2 3 1 0 3 1 4 2 3 0 2 4 1 0 1 2 4 2 3 2 3 w_4_56 2 4 1 0 3 1 2 0 5 3 4 w_4_113
2 3 1 0 3 2 4 1 3 0 2 4 0 0 1 2 4 2 3 3 4 0 2 4 1 0 3 1 2 2 3 2 3 w_4_114
h^4: 2 4 1 0 1 2 4 2 5 2 3 w_4_57 2 4 1 0 3 1 2 2 3 2 4 w_4_115
2 4 1 0 1 0 1 0 2 2 3 w_4_1 2 4 1 0 1 2 4 2 5 3 4 w_4_58 2 4 1 0 3 1 2 2 5 2 4 w_4_116
2 4 1 0 1 0 1 0 2 3 4 w_4_2 2 4 0 0 1 2 4 3 5 2 4 0 2 4 1 0 3 1 2 2 5 3 4 w_4_117
2 4 0 0 1 0 1 0 5 2 3 0 2 4 1 0 1 2 4 3 5 3 4 w_4_59 2 4 1 0 3 1 4 0 5 1 2 w_4_118
2 4 1 0 1 0 1 2 3 2 3 w_4_3 2 4 1 0 3 0 2 0 2 1 2 w_4_60 2 4 1 0 3 1 4 0 5 2 3 w_4_119
2 4 1 0 1 0 1 2 3 2 4 w_4_4 2 4 1 0 3 0 2 0 2 1 3 w_4_61 2 4 1 0 3 1 4 0 5 2 4 w_4_120
2 4 1 0 1 0 1 2 5 3 4 w_4_5 2 4 1 0 3 0 2 0 2 1 4 w_4_62 2 4 1 0 3 1 4 0 5 3 4 w_4_121
2 4 1 0 1 0 2 0 2 0 2 w_4_6 2 4 1 0 3 0 2 0 5 1 2 w_4_63 2 4 1 0 3 1 4 2 3 0 3 w_4_122
2 4 1 0 1 0 2 0 2 0 3 w_4_7 2 4 1 0 3 0 2 1 2 1 2 w_4_64 2 4 1 0 3 1 4 2 3 0 4 w_4_123
2 4 1 0 1 0 2 0 2 1 2 w_4_8 2 4 1 0 3 0 2 1 2 1 3 w_4_65 2 4 1 0 3 1 4 2 3 1 4 w_4_124
2 4 1 0 1 0 2 0 2 1 3 w_4_9 2 4 1 0 3 0 2 1 2 1 4 w_4_66 2 4 1 0 3 1 4 2 3 2 3 w_4_125
2 4 1 0 1 0 2 0 2 2 3 w_4_10 2 4 1 0 3 0 2 1 2 2 3 w_4_67 2 4 1 0 3 1 4 2 3 2 4 w_4_126
2 4 0 0 1 0 2 0 2 3 4 0 2 4 1 0 3 0 2 1 2 2 4 w_4_68 2 4 1 0 3 1 4 2 3 3 4 w_4_127
2 4 1 0 1 0 2 0 3 0 3 w_4_11 2 4 1 0 3 0 2 1 2 3 4 w_4_69 2 4 0 0 3 1 4 2 5 0 3 0
2 4 1 0 1 0 2 0 3 0 4 w_4_12 2 4 0 0 3 0 2 1 5 2 3 0 2 4 1 0 3 1 4 2 5 0 4 w_4_128
2 4 1 0 1 0 2 0 3 1 2 w_4_13 2 4 1 0 3 0 2 1 5 2 4 w_4_70 2 4 1 0 3 1 4 2 5 1 4 w_4_129
2 4 1 0 1 0 2 0 3 1 3 w_4_14 2 4 1 0 3 0 4 0 2 1 2 w_4_71 2 4 1 0 3 1 4 2 5 2 3 w_4_130
2 4 1 0 1 0 2 0 3 1 4 w_4_15 2 4 1 0 3 0 4 0 5 1 2 w_4_72 2 4 1 0 3 1 4 2 5 2 4 w_4_131
2 4 1 0 1 0 2 0 3 2 3 w_4_16 2 4 1 0 3 0 4 0 5 1 3 w_4_73 2 4 1 0 3 1 4 2 5 3 4 w_4_132
2 4 1 0 1 0 2 0 3 2 4 w_4_17 2 4 1 0 3 0 4 0 5 1 4 w_4_74 2 4 1 0 3 1 4 3 5 0 4 w_4_133
2 4 1 0 1 0 2 0 3 3 4 w_4_18 2 4 1 0 3 0 4 1 2 0 3 w_4_75 2 4 1 0 3 1 4 3 5 1 4 w_4_134
2 4 1 0 1 0 2 0 5 1 2 w_4_19 2 4 1 0 3 0 4 1 2 0 4 w_4_76 2 4 1 0 3 1 4 3 5 2 3 w_4_135
2 4 1 0 1 0 2 0 5 1 3 w_4_20 2 4 1 0 3 0 4 1 2 1 2 w_4_77 2 4 1 0 3 1 4 3 5 2 4 w_4_136
2 4 1 0 1 0 2 0 5 2 3 w_4_21 2 4 1 0 3 0 4 1 2 1 3 w_4_78 2 4 1 0 3 1 4 3 5 3 4 w_4_137
2 4 1 0 1 0 2 0 5 2 4 w_4_22 2 4 1 0 3 0 4 1 2 1 4 w_4_79 2 4 0 0 3 2 4 0 3 1 3 0
2 4 1 0 1 0 2 0 5 3 4 w_4_23 2 4 1 0 3 0 4 1 2 2 3 w_4_80 2 4 1 0 3 2 4 1 3 0 3 w_4_138
2 4 1 0 1 0 2 1 2 2 3 w_4_24 2 4 1 0 3 0 4 1 2 2 4 w_4_81 2 4 1 0 3 2 4 1 3 2 3 w_4_139
2 4 1 0 1 0 2 1 2 3 4 w_4_25 2 4 1 0 3 0 4 1 2 3 4 w_4_82 2 4 1 0 3 2 4 1 3 2 4 w_4_140
2 4 1 0 1 0 2 1 3 1 3 w_4_26 2 4 1 0 3 0 4 1 3 0 3 w_4_83 2 4 1 0 3 2 4 1 5 2 3 w_4_141
2 4 1 0 1 0 2 1 3 1 4 w_4_27 2 4 1 0 3 0 4 1 3 0 4 w_4_84 2 4 1 0 3 2 4 1 5 2 4 w_4_142
2 4 1 0 1 0 2 1 3 2 3 w_4_28 2 4 1 0 3 0 4 1 3 1 2 w_4_85 2 4 1 0 3 2 4 1 5 3 4 w_4_143
2 4 1 0 1 0 2 1 3 2 4 w_4_29 2 4 1 0 3 0 4 1 3 1 3 w_4_86 2 4 1 0 3 2 4 2 3 1 3 w_4_144
2 4 1 0 1 0 2 1 3 3 4 w_4_30 2 4 1 0 3 0 4 1 3 1 4 w_4_87 2 4 1 0 3 2 4 2 3 1 4 w_4_145
2 4 1 0 1 0 2 1 5 2 3 w_4_31 2 4 1 0 3 0 4 1 3 2 3 w_4_88 2 4 1 0 3 2 4 2 5 1 3 w_4_146
2 4 1 0 1 0 2 1 5 2 4 w_4_32 2 4 1 0 3 0 4 1 3 2 4 w_4_89 2 4 1 0 3 2 4 3 5 1 3 w_4_147
2 4 1 0 1 0 2 1 5 3 4 w_4_33 2 4 1 0 3 0 4 1 3 3 4 w_4_90 2 4 1 0 3 2 4 3 5 1 4 w_4_148
2 4 1 0 1 0 2 2 3 2 3 w_4_34 2 4 1 0 3 0 4 1 5 0 4 w_4_91 2 4 1 0 3 4 5 1 5 2 3 w_4_149
vi R. BURING AND A. V. KISELEV
Table 6. Kontsevich’s star product up to order 4.
h^0: 2 4 1 0 1 1 2 1 5 2 3 -16*w_4_21
2 0 1 1 2 4 1 0 1 0 2 0 5 2 4 16*w_4_22
h^1: 2 4 1 0 1 1 2 1 5 2 4 -16*w_4_22
2 1 1 0 1 1 2 4 1 0 1 0 2 0 5 3 4 16*w_4_23
h^2: 2 4 1 0 1 1 2 1 5 3 4 -16*w_4_23
2 2 1 0 1 0 1 1/2 2 4 1 0 1 0 2 1 2 2 3 16*w_4_24
2 2 1 0 1 0 2 1/3 2 4 1 0 1 0 2 1 2 2 4 -16*w_4_24
2 2 1 0 1 1 2 -1/3 2 4 1 0 1 0 2 1 2 3 4 16*w_4_25
2 2 1 0 3 1 2 -1/6 2 4 1 0 1 0 2 1 3 1 3 8*w_4_26
h^3: 2 4 1 0 1 0 4 1 2 0 4 -8*w_4_26
2 3 1 0 1 0 1 0 1 1/6 2 4 1 0 1 0 2 1 3 1 4 16*w_4_27
2 3 1 0 1 0 1 0 2 1/3 2 4 1 0 1 0 4 0 5 1 2 -16*w_4_27
2 3 1 0 1 0 1 1 2 -1/3 2 4 1 0 1 0 2 1 3 2 3 16*w_4_28
2 3 1 0 1 0 4 1 3 -1/6 2 4 1 0 1 0 4 1 2 2 4 -16*w_4_28
2 3 1 0 1 0 2 0 2 1/6 2 4 1 0 1 0 2 1 3 2 4 16*w_4_29
2 3 1 0 1 1 2 1 2 1/6 2 4 1 0 1 0 4 1 2 2 3 -16*w_4_29
2 3 1 0 1 0 2 2 3 -1/6 2 4 1 0 1 0 2 1 3 3 4 16*w_4_30
2 3 1 0 1 1 2 2 3 -1/6 2 4 1 0 1 0 4 1 2 3 4 16*w_4_30
2 3 1 0 1 0 4 1 2 -1/6 2 4 1 0 1 0 2 1 5 2 3 16*w_4_31
2 3 1 0 1 0 2 1 3 -1/6 2 4 1 0 1 0 4 2 5 1 2 -16*w_4_31
2 3 1 0 3 1 2 1 2 1/6 2 4 1 0 1 0 2 1 5 2 4 16*w_4_32
2 3 1 0 3 1 2 0 3 -1/6 2 4 1 0 1 0 4 2 3 1 2 -16*w_4_32
2 3 1 0 3 1 2 2 3 -1/6 2 4 1 0 1 0 2 1 5 3 4 16*w_4_33
h^4: 2 4 1 0 1 0 4 3 5 1 2 16*w_4_33
2 4 1 0 1 0 1 0 1 0 1 1/24 2 4 1 0 1 0 2 2 3 2 3 8*w_4_34
2 4 1 0 1 0 1 0 1 0 2 1/6 2 4 1 0 1 1 2 2 3 2 3 -8*w_4_34
2 4 1 0 1 0 1 0 1 1 2 -1/6 2 4 1 0 1 0 2 2 3 2 4 16*w_4_35
2 4 1 0 1 0 1 0 5 1 4 -1/12 2 4 1 0 1 1 2 2 3 2 4 -16*w_4_35
2 4 1 0 1 0 1 0 2 0 2 1/6 2 4 1 0 1 0 2 2 3 3 4 16*w_4_36
2 4 1 0 1 0 1 1 2 1 2 1/6 2 4 1 0 1 1 2 2 3 3 4 -16*w_4_36
2 4 1 0 1 0 1 0 2 2 4 -1/6 2 4 1 0 1 0 2 2 5 2 4 8*w_4_37
2 4 1 0 1 0 1 1 2 2 4 -1/6 2 4 1 0 1 1 2 2 5 2 4 -8*w_4_37
2 4 1 0 1 0 1 0 5 1 2 -1/6 2 4 1 0 1 0 2 2 5 3 4 16*w_4_38
2 4 1 0 1 0 1 0 2 1 4 -1/6 2 4 1 0 1 1 2 2 5 3 4 -16*w_4_38
2 4 1 0 1 0 4 1 3 1 3 1/6 2 4 1 0 1 0 2 3 5 3 4 8*w_4_39
2 4 1 0 1 0 4 1 3 0 4 -1/6 2 4 1 0 1 1 2 3 5 3 4 -8*w_4_39
2 4 1 0 1 0 4 1 3 3 4 -1/6 2 4 1 0 1 0 4 0 3 2 3 16*w_4_40
2 4 1 0 1 0 1 0 2 0 3 1/18 2 4 1 0 1 1 4 1 3 2 3 -16*w_4_40
2 4 1 0 1 0 1 0 2 1 3 -1/9 2 4 1 0 1 0 4 0 5 2 3 16*w_4_41
2 4 1 0 1 0 2 0 5 1 4 -1/18 2 4 1 0 1 1 4 1 5 2 3 -16*w_4_41
2 4 1 0 1 0 1 1 2 1 3 1/18 2 4 1 0 1 0 4 0 5 2 4 16*w_4_42
2 4 1 0 1 0 4 1 3 1 2 1/18 2 4 1 0 1 1 4 1 5 2 4 -16*w_4_42
2 4 1 0 3 1 2 0 5 1 4 1/72 2 4 1 0 1 0 4 1 3 2 3 16*w_4_43
2 4 1 0 1 0 1 0 2 2 3 16*w_4_1 2 4 1 0 1 0 4 1 3 2 4 -16*w_4_43
2 4 1 0 1 0 1 1 2 2 3 16*w_4_1 2 4 1 0 1 0 4 1 5 2 3 16*w_4_44
2 4 1 0 1 0 1 0 2 3 4 16*w_4_2 2 4 1 0 1 0 4 2 5 1 3 -16*w_4_44
2 4 1 0 1 0 1 1 2 3 4 16*w_4_2 2 4 1 0 1 0 4 1 5 2 4 16*w_4_45
2 4 1 0 1 0 1 2 3 2 3 4*w_4_3 2 4 1 0 1 0 4 2 3 1 3 -16*w_4_45
2 4 1 0 1 0 1 2 3 2 4 16*w_4_4 2 4 1 0 1 0 4 2 3 0 4 16*w_4_46
2 4 1 0 1 0 1 2 5 3 4 8*w_4_5 2 4 1 0 1 1 4 2 3 1 4 -16*w_4_46
2 4 1 0 1 0 2 0 2 0 2 8/3*w_4_6 2 4 1 0 1 0 4 2 3 1 4 16*w_4_47
2 4 1 0 1 1 2 1 2 1 2 -8/3*w_4_6 2 4 1 0 1 0 4 2 5 1 4 -16*w_4_47
2 4 1 0 1 0 2 0 2 0 3 16*w_4_7 2 4 1 0 1 0 4 2 3 2 3 16*w_4_48
2 4 1 0 1 1 2 1 2 1 3 -16*w_4_7 2 4 1 0 1 1 4 2 3 2 3 -16*w_4_48
2 4 1 0 1 0 2 0 2 1 2 8*w_4_8 2 4 1 0 1 0 4 2 3 2 4 16*w_4_49
2 4 1 0 1 0 2 1 2 1 2 -8*w_4_8 2 4 1 0 1 1 4 2 3 2 4 -16*w_4_49
2 4 1 0 1 0 2 0 2 1 3 16*w_4_9 2 4 1 0 1 0 4 2 3 3 4 16*w_4_50
2 4 1 0 1 0 4 1 2 1 2 -16*w_4_9 2 4 1 0 1 1 4 2 3 3 4 -16*w_4_50
2 4 1 0 1 0 2 0 2 2 3 16*w_4_10 2 4 1 0 1 0 4 2 5 2 3 16*w_4_51
2 4 1 0 1 1 2 1 2 2 3 -16*w_4_10 2 4 1 0 1 1 4 2 5 2 3 -16*w_4_51
2 4 1 0 1 0 2 0 3 0 3 8*w_4_11 2 4 1 0 1 0 4 2 5 2 4 16*w_4_52
2 4 1 0 1 1 2 1 3 1 3 -8*w_4_11 2 4 1 0 1 1 4 2 5 2 4 -16*w_4_52
2 4 1 0 1 0 2 0 3 0 4 16*w_4_12 2 4 1 0 1 0 4 2 5 3 4 16*w_4_53
2 4 1 0 1 1 2 1 3 1 4 -16*w_4_12 2 4 1 0 1 1 4 2 5 3 4 -16*w_4_53
2 4 1 0 1 0 2 0 3 1 2 16*w_4_13 2 4 1 0 1 0 4 3 5 2 3 16*w_4_54
2 4 1 0 1 0 2 1 2 1 4 -16*w_4_13 2 4 1 0 1 1 4 3 5 2 3 -16*w_4_54
2 4 1 0 1 0 2 0 3 1 3 16*w_4_14 2 4 1 0 1 0 4 3 5 2 4 16*w_4_55
2 4 1 0 1 0 4 1 2 1 4 -16*w_4_14 2 4 1 0 1 1 4 3 5 2 4 -16*w_4_55
2 4 1 0 1 0 2 0 3 1 4 16*w_4_15 2 4 1 0 1 2 4 2 3 2 3 16*w_4_56
2 4 1 0 1 0 4 1 5 1 2 -16*w_4_15 2 4 1 0 1 2 4 2 5 2 3 16/3*w_4_57
2 4 1 0 1 0 2 0 3 2 3 16*w_4_16 2 4 1 0 1 2 4 2 5 3 4 16*w_4_58
2 4 1 0 1 1 2 1 3 2 3 -16*w_4_16 2 4 1 0 1 2 4 3 5 3 4 16*w_4_59
2 4 1 0 1 0 2 0 3 2 4 16*w_4_17 2 4 1 0 3 0 2 0 2 1 2 16*w_4_60
2 4 1 0 1 1 2 1 3 2 4 -16*w_4_17 2 4 1 0 3 1 4 1 3 1 3 16*w_4_60
2 4 1 0 1 0 2 0 3 3 4 16*w_4_18 2 4 1 0 3 0 2 0 2 1 3 16*w_4_61
2 4 1 0 1 1 2 1 3 3 4 -16*w_4_18 2 4 1 0 3 1 4 1 3 1 4 16*w_4_61
2 4 1 0 1 0 2 0 5 1 2 16*w_4_19 2 4 1 0 3 0 2 0 2 1 4 16*w_4_62
2 4 1 0 1 0 2 1 2 1 3 -16*w_4_19 2 4 1 0 3 1 4 1 5 1 4 16*w_4_62
2 4 1 0 1 0 2 0 5 1 3 16*w_4_20 2 4 1 0 3 0 2 0 5 1 2 16*w_4_63
2 4 1 0 1 0 4 1 2 1 3 -16*w_4_20 2 4 1 0 3 1 4 1 3 1 2 16*w_4_63
2 4 1 0 1 0 2 0 5 2 3 16*w_4_21 2 4 1 0 3 0 2 1 2 1 2 8*w_4_642 4 1 0 3 1 4 1 3 0 3 8*w_4_64
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION vii
Table 6 (continued).
2 4 1 0 3 0 2 1 2 1 3 8*w_4_65 2 4 1 0 3 1 2 0 5 1 2 16*w_4_107
2 4 1 0 3 1 4 1 3 0 4 8*w_4_65 2 4 1 0 3 1 2 0 3 2 3 16*w_4_108
2 4 1 0 3 0 2 1 2 1 4 16*w_4_66 2 4 1 0 3 1 2 1 2 2 3 -16*w_4_108
2 4 1 0 3 0 4 1 5 1 4 16*w_4_66 2 4 1 0 3 1 2 0 3 2 4 16*w_4_109
2 4 1 0 3 0 2 1 2 2 3 16*w_4_67 2 4 1 0 3 1 2 1 2 3 4 16*w_4_109
2 4 1 0 3 1 4 1 3 3 4 16*w_4_67 2 4 1 0 3 1 2 0 3 3 4 16*w_4_110
2 4 1 0 3 0 2 1 2 2 4 16*w_4_68 2 4 1 0 3 1 2 1 2 2 4 16*w_4_110
2 4 1 0 3 1 4 1 3 2 3 -16*w_4_68 2 4 1 0 3 1 2 0 5 2 3 16*w_4_111
2 4 1 0 3 0 2 1 2 3 4 16*w_4_69 2 4 1 0 3 1 2 1 5 2 3 -16*w_4_111
2 4 1 0 3 1 4 1 3 2 4 -16*w_4_69 2 4 1 0 3 1 2 0 5 2 4 16*w_4_112
2 4 1 0 3 0 2 1 5 2 4 16*w_4_70 2 4 1 0 3 1 2 1 5 3 4 16*w_4_112
2 4 1 0 3 2 4 1 5 1 4 -16*w_4_70 2 4 1 0 3 1 2 0 5 3 4 16*w_4_113
2 4 1 0 3 0 4 0 2 1 2 16*w_4_71 2 4 1 0 3 1 2 1 5 2 4 16*w_4_113
2 4 1 0 3 1 4 1 5 1 3 16*w_4_71 2 4 1 0 3 1 2 2 3 2 3 8*w_4_114
2 4 1 0 3 0 4 0 5 1 2 16*w_4_72 2 4 1 0 3 1 2 2 3 2 4 16*w_4_115
2 4 1 0 3 1 4 1 5 1 2 16*w_4_72 2 4 1 0 3 1 2 2 3 3 4 -16*w_4_115
2 4 1 0 3 0 4 0 5 1 3 16*w_4_73 2 4 1 0 3 1 2 2 5 2 4 8*w_4_116
2 4 1 0 3 1 4 1 2 1 3 16*w_4_73 2 4 1 0 3 1 2 3 5 3 4 8*w_4_116
2 4 1 0 3 0 4 0 5 1 4 16*w_4_74 2 4 1 0 3 1 2 2 5 3 4 16*w_4_117
2 4 1 0 3 1 2 1 3 1 4 16*w_4_74 2 4 1 0 3 1 4 0 5 1 2 8*w_4_118
2 4 1 0 3 0 4 1 2 0 3 16*w_4_75 2 4 1 0 3 1 4 0 5 2 3 16*w_4_119
2 4 1 0 3 1 4 1 2 1 4 16*w_4_75 2 4 1 0 3 1 4 2 5 1 2 -16*w_4_119
2 4 1 0 3 0 4 1 2 0 4 16*w_4_76 2 4 1 0 3 1 4 0 5 2 4 16*w_4_120
2 4 1 0 3 1 4 1 2 1 2 16*w_4_76 2 4 1 0 3 1 4 3 5 1 2 -16*w_4_120
2 4 1 0 3 0 4 1 2 1 2 16*w_4_77 2 4 1 0 3 1 4 0 5 3 4 16*w_4_121
2 4 1 0 3 1 4 1 2 0 3 16*w_4_77 2 4 1 0 3 1 4 2 3 1 2 16*w_4_121
2 4 1 0 3 0 4 1 2 1 3 16*w_4_78 2 4 1 0 3 1 4 2 3 0 3 16*w_4_122
2 4 1 0 3 1 4 0 5 1 3 16*w_4_78 2 4 1 0 3 2 4 1 2 1 2 -16*w_4_122
2 4 1 0 3 0 4 1 2 1 4 16*w_4_79 2 4 1 0 3 1 4 2 3 0 4 16*w_4_123
2 4 1 0 3 0 4 1 5 1 3 16*w_4_79 2 4 1 0 3 2 4 1 2 1 3 -16*w_4_123
2 4 1 0 3 0 4 1 2 2 3 16*w_4_80 2 4 1 0 3 1 4 2 3 1 4 16*w_4_124
2 4 1 0 3 1 4 1 2 3 4 16*w_4_80 2 4 1 0 3 2 4 1 2 0 3 -16*w_4_124
2 4 1 0 3 0 4 1 2 2 4 16*w_4_81 2 4 1 0 3 1 4 2 3 2 3 16*w_4_125
2 4 1 0 3 1 4 1 2 2 3 -16*w_4_81 2 4 1 0 3 2 4 1 2 2 4 16*w_4_125
2 4 1 0 3 0 4 1 2 3 4 16*w_4_82 2 4 1 0 3 1 4 2 3 2 4 16*w_4_126
2 4 1 0 3 1 4 1 2 2 4 -16*w_4_82 2 4 1 0 3 2 4 1 2 3 4 16*w_4_126
2 4 1 0 3 0 4 1 3 0 3 8*w_4_83 2 4 1 0 3 1 4 2 3 3 4 16*w_4_127
2 4 1 0 3 1 2 1 3 1 3 8*w_4_83 2 4 1 0 3 2 4 1 2 2 3 -16*w_4_127
2 4 1 0 3 0 4 1 3 0 4 16*w_4_84 2 4 1 0 3 1 4 2 5 0 4 16*w_4_128
2 4 1 0 3 1 2 1 2 1 3 16*w_4_84 2 4 1 0 3 4 5 1 2 1 3 16*w_4_128
2 4 1 0 3 0 4 1 3 1 2 16*w_4_85 2 4 1 0 3 1 4 2 5 1 4 16*w_4_129
2 4 1 0 3 1 2 0 5 1 3 16*w_4_85 2 4 1 0 3 2 4 1 5 0 3 -16*w_4_129
2 4 1 0 3 0 4 1 3 1 3 16*w_4_86 2 4 1 0 3 1 4 2 5 2 3 16*w_4_130
2 4 1 0 3 1 2 0 3 1 3 16*w_4_86 2 4 1 0 3 4 5 1 2 2 4 -16*w_4_130
2 4 1 0 3 0 4 1 3 1 4 16*w_4_87 2 4 1 0 3 1 4 2 5 2 4 16*w_4_131
2 4 1 0 3 0 4 1 3 2 3 16*w_4_88 2 4 1 0 3 4 5 1 2 3 4 -16*w_4_131
2 4 1 0 3 1 2 1 3 3 4 -16*w_4_88 2 4 1 0 3 1 4 2 5 3 4 16*w_4_132
2 4 1 0 3 0 4 1 3 2 4 16*w_4_89 2 4 1 0 3 4 5 1 2 2 3 16*w_4_132
2 4 1 0 3 1 2 1 3 2 4 -16*w_4_89 2 4 1 0 3 1 4 3 5 0 4 16*w_4_133
2 4 1 0 3 0 4 1 3 3 4 16*w_4_90 2 4 1 0 3 2 4 1 3 1 2 16*w_4_133
2 4 1 0 3 1 2 1 3 2 3 -16*w_4_90 2 4 1 0 3 1 4 3 5 1 4 16*w_4_134
2 4 1 0 3 0 4 1 5 0 4 16*w_4_91 2 4 1 0 3 2 4 0 3 1 2 16*w_4_134
2 4 1 0 3 1 2 1 2 1 4 16*w_4_91 2 4 1 0 3 1 4 3 5 2 3 16*w_4_135
2 4 1 0 3 0 4 1 5 1 2 16*w_4_92 2 4 1 0 3 2 4 2 5 1 2 -16*w_4_135
2 4 1 0 3 0 4 1 5 2 3 16*w_4_93 2 4 1 0 3 1 4 3 5 2 4 16*w_4_136
2 4 1 0 3 4 5 1 2 1 4 -16*w_4_93 2 4 1 0 3 2 4 3 5 1 2 -16*w_4_136
2 4 1 0 3 0 4 1 5 2 4 16*w_4_94 2 4 1 0 3 1 4 3 5 3 4 16*w_4_137
2 4 1 0 3 2 4 1 5 1 2 -16*w_4_94 2 4 1 0 3 2 4 2 3 1 2 16*w_4_137
2 4 1 0 3 0 4 1 5 3 4 16*w_4_95 2 4 1 0 3 2 4 1 3 0 3 16*w_4_138
2 4 1 0 3 2 4 1 2 1 4 -16*w_4_95 2 4 1 0 3 2 4 1 3 1 3 -16*w_4_138
2 4 1 0 3 0 4 2 3 1 2 16*w_4_96 2 4 1 0 3 2 4 1 3 2 3 16*w_4_139
2 4 1 0 3 1 4 1 5 3 4 16*w_4_96 2 4 1 0 3 2 4 1 3 3 4 16*w_4_139
2 4 1 0 3 0 4 2 3 1 3 16*w_4_97 2 4 1 0 3 2 4 1 3 2 4 16*w_4_140
2 4 1 0 3 1 4 3 5 1 3 -16*w_4_97 2 4 1 0 3 2 4 1 5 2 3 16*w_4_141
2 4 1 0 3 0 4 2 3 1 4 16*w_4_98 2 4 1 0 3 4 5 1 5 2 4 -16*w_4_141
2 4 1 0 3 4 5 1 3 1 4 -16*w_4_98 2 4 1 0 3 2 4 1 5 2 4 16*w_4_142
2 4 1 0 3 0 4 2 5 1 2 16*w_4_99 2 4 1 0 3 2 4 1 5 3 4 16*w_4_143
2 4 1 0 3 1 4 1 5 2 3 -16*w_4_99 2 4 1 0 3 2 4 2 5 1 4 16*w_4_143
2 4 1 0 3 0 4 2 5 1 3 16*w_4_100 2 4 1 0 3 2 4 2 3 1 3 16*w_4_144
2 4 1 0 3 1 4 2 5 1 3 -16*w_4_100 2 4 1 0 3 4 5 1 3 3 4 -16*w_4_144
2 4 1 0 3 0 4 2 5 1 4 16*w_4_101 2 4 1 0 3 2 4 2 3 1 4 16*w_4_145
2 4 1 0 3 2 4 1 5 1 3 -16*w_4_101 2 4 1 0 3 4 5 1 5 3 4 -16*w_4_145
2 4 1 0 3 0 4 3 5 1 2 16*w_4_102 2 4 1 0 3 2 4 2 5 1 3 16*w_4_146
2 4 1 0 3 1 4 1 5 2 4 -16*w_4_102 2 4 1 0 3 4 5 1 3 2 4 -16*w_4_146
2 4 1 0 3 0 4 3 5 1 3 16*w_4_103 2 4 1 0 3 2 4 3 5 1 3 16*w_4_147
2 4 1 0 3 1 4 2 3 1 3 -16*w_4_103 2 4 1 0 3 4 5 1 3 2 3 -16*w_4_147
2 4 1 0 3 0 4 3 5 1 4 16*w_4_104 2 4 1 0 3 2 4 3 5 1 4 16*w_4_148
2 4 1 0 3 2 4 1 3 1 4 -16*w_4_104 2 4 1 0 3 4 5 1 5 2 3 16*w_4_149
2 4 1 0 3 1 2 0 3 0 3 8*w_4_105
2 4 1 0 3 1 2 1 2 1 2 8*w_4_105
2 4 1 0 3 1 2 0 3 1 2 16*w_4_106
2 4 1 0 3 1 2 0 3 1 4 16*w_4_107
viii R. BURING AND A. V. KISELEV
Table 7. Relations between weights of ℏ4-basic graphs: 149 via 10.
w_4_1==-1/144
w_4_2==-1/288
w_4_3==17/360 + 6*w_4_108
w_4_4==49/2880 - 3*w_4_104 - w_4_107 + (3*w_4_108)/2
w_4_5==-1/96 + 6*w_4_104 + 2*w_4_107
w_4_6==1/80
w_4_7==1/360
w_4_8==-1/240
w_4_9==-13/1440
w_4_10==-7/1440
w_4_11==1/240
w_4_12==-1/720
w_4_13==1/720
w_4_14==1/480
w_4_15==-1/1440
w_4_16==1/1440
w_4_17==-1/480
w_4_18==-1/360
w_4_19==-1/480
w_4_20==-1/240
w_4_21==-1/480
w_4_22==-1/720
w_4_23==1/1440
w_4_24==1/360
w_4_25==53/1440 + 3*w_4_100 + 12*w_4_103 - 15*w_4_104 - w_4_107 + 6*w_4_108 - 6*w_4_109
w_4_26==1/120
w_4_27==1/1440
w_4_28==-1/960 - (3*w_4_108)/2
w_4_29==-49/1440 - (3*w_4_100)/2 - 9*w_4_103 + (21*w_4_104)/2 + (3*w_4_107)/2 - (9*w_4_108)/2 + 3*w_4_109
w_4_30==1/72 + 6*w_4_103 - 6*w_4_104 + 3*w_4_108 - 3*w_4_109
w_4_31==61/2880 + (3*w_4_100)/2 + 6*w_4_103 - (15*w_4_104)/2 - w_4_107/2 + 3*w_4_108 - 3*w_4_109
w_4_32==1/1440
w_4_33==5/288 + 6*w_4_103 - 6*w_4_104 + 3*w_4_108 - 3*w_4_109
w_4_34==1/96 + w_4_108
w_4_35==-w_4_103
w_4_36==-13/2880 - w_4_100/2 + (3*w_4_104)/2 + w_4_107/2
w_4_37==0
w_4_38==1/1440 - w_4_100/2 + w_4_103 + (3*w_4_104)/2 + w_4_107/2 + w_4_108/2
w_4_39==0
w_4_40==0
w_4_41==1/1440
w_4_42==1/1440
w_4_43==37/1440 + 6*w_4_103 - 6*w_4_104 - w_4_107 + 3*w_4_108 - 3*w_4_109
w_4_44==17/360 + 15*w_4_103 - 18*w_4_104 - 2*w_4_107 + 6*w_4_108 - 6*w_4_109
w_4_45==7/1440 - 3*w_4_104 - w_4_107
w_4_46==-1/480
w_4_47==1/60 + 6*w_4_103 - 6*w_4_104 + 3*w_4_108 - 3*w_4_109
w_4_48==11/1440 - w_4_100/2 - w_4_103 + (5*w_4_104)/2 + w_4_107/2 + (3*w_4_108)/2
w_4_49==-w_4_104
w_4_50==-1/192 - w_4_108/2
w_4_51==-w_4_103
w_4_52==-1/1440 + w_4_100/2 - (3*w_4_104)/2 - w_4_107/2 - w_4_108/2
w_4_53==w_4_103
w_4_54==-1/576 + w_4_103 - w_4_104 - w_4_108/2
w_4_55==w_4_104
w_4_56==0
w_4_57==0
w_4_58==0
w_4_59==0
w_4_60==0
w_4_61==0
w_4_62==0
w_4_63==0
w_4_64==0
w_4_65==0
w_4_66==0
w_4_67==0
w_4_68==0
w_4_69==0
w_4_70==0
w_4_71==0
w_4_72==1/1440
w_4_73==1/1440
w_4_74==1/1440
w_4_75==-1/480
w_4_76==-1/720
w_4_77==1/180 + 3*w_4_103 - 3*w_4_104 - w_4_107
w_4_78==-1/144 - 3*w_4_103 + 3*w_4_104 + w_4_107
w_4_79==-1/1440
w_4_80==1/80 + w_4_100 - 3*w_4_104 + 3*w_4_108 - 2*w_4_109 - 2*w_4_125
w_4_81==1/480 - w_4_100/2 + 2*w_4_103 + w_4_104/2 + w_4_107/2 + w_4_108/2 + 2*w_4_125
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION ix
Table 7 (continued).
w_4_82==1/2880 - w_4_103 - w_4_104 + w_4_108/2 - w_4_109 - 2*w_4_125
w_4_83==-1/480
w_4_84==-1/720
w_4_85==1/180 - w_4_107
w_4_86==1/480
w_4_87==-1/1440
w_4_88==1/96 + 4*w_4_103 - 4*w_4_104 + 2*w_4_108 - 2*w_4_109
w_4_89==1/240 + (3*w_4_100)/2 + 3*w_4_103 - (9*w_4_104)/2 + w_4_107/2 + (3*w_4_108)/2 - 2*w_4_109
w_4_90==-1/192 - w_4_108/2
w_4_91==-1/720
w_4_92==17/1440 + 6*w_4_103 - 6*w_4_104 - 2*w_4_107
w_4_93==3/320 - w_4_100/2 + w_4_103 - (5*w_4_104)/2 + w_4_107/2 + 2*w_4_108 - 2*w_4_109 + w_4_119
w_4_94==1/1440 - w_4_100/2 - w_4_102 + w_4_103 - (3*w_4_104)/2 + w_4_107/2 + w_4_108/2 - w_4_109
w_4_95==-1/576 + w_4_103 - w_4_104 - w_4_108/2
w_4_96==0
w_4_97==0
w_4_98==0
w_4_99==-7/2880 - w_4_100/2 + w_4_103 + w_4_104/2 - w_4_107/2 - w_4_108 + w_4_109 - w_4_119
w_4_105==-1/160
w_4_106==13/1440
w_4_110==-1/288 - 2*w_4_103 + 2*w_4_104 - w_4_108 + w_4_109
w_4_111==-17/2880 - w_4_100/2 - 2*w_4_103 + (5*w_4_104)/2 - w_4_107/2 - w_4_108 + w_4_109
w_4_112==-7/576 - 4*w_4_103 + 4*w_4_104 - 2*w_4_108 + 2*w_4_109
w_4_113==-1/192 - 2*w_4_103 + 2*w_4_104 - w_4_108 + w_4_109
w_4_114==1/360 + w_4_108
w_4_115==23/5760 - w_4_100/2 + w_4_103 + (3*w_4_108)/4
w_4_116==0
w_4_117==-19/2880 + w_4_100 - 2*w_4_103 - w_4_108
w_4_118==-31/1440 - 12*w_4_103 + 12*w_4_104 + 4*w_4_107
w_4_120==-1/96 - w_4_100 - w_4_102 + 2*w_4_103 - 2*w_4_108 + w_4_109
w_4_121==-1/288 + 2*w_4_103 - 2*w_4_104 - w_4_108
w_4_122==-2*w_4_103
w_4_123==-7/2880 + w_4_100/2 - w_4_103 + w_4_104/2 - w_4_107/2 - w_4_108 + w_4_109
w_4_124==1/144 + w_4_100 + w_4_103 - 2*w_4_104 + w_4_108 - w_4_109
w_4_126==29/5760 + w_4_100/2 - w_4_103 + (5*w_4_108)/4 - w_4_109 - w_4_125
w_4_127==-1/640 + w_4_103 + w_4_104/2 - (3*w_4_108)/4 + w_4_109/2 + w_4_125
w_4_128==-1/144 + w_4_101 - 2*w_4_103 + 3*w_4_104 - w_4_108 + w_4_109
w_4_129==1/144 + w_4_101 + 2*w_4_103 - 3*w_4_104 + w_4_108 - w_4_109
w_4_130==7/1920 - w_4_100/2 + w_4_103 + w_4_107/2 + (3*w_4_108)/4 - w_4_109/2 + w_4_119 + w_4_125
w_4_131==23/5760 - w_4_100/4 + w_4_101/2 - w_4_102/2 + w_4_103/2 - (5*w_4_104)/4 + w_4_107/4 + w_4_108 - w_4_109 +
w_4_119/2 - w_4_125
w_4_132==-1/240 + w_4_101/2 + w_4_103/2 - w_4_108 + w_4_109/2 + w_4_125
w_4_133==2*w_4_104
w_4_134==0
w_4_135==-1/360 - w_4_100/4 - w_4_102/2 + w_4_104/4 + w_4_107/4 - w_4_108/4 + w_4_119/2
w_4_136==-7/1440 - w_4_100/2 - w_4_102 + w_4_103 - w_4_104/2 - w_4_108 + w_4_109/2
w_4_137==0
w_4_138==-1/144 - 2*w_4_103 + 2*w_4_104 - w_4_108 + w_4_109
w_4_139==1/1920 + w_4_103 - (3*w_4_104)/2 + w_4_108/4 - w_4_109/2
w_4_140==-1/1440 + w_4_100 + 2*w_4_103 - 5*w_4_104 - w_4_109
w_4_141==-w_4_100/4 - w_4_101/2 - w_4_102/2 - w_4_103/2 + w_4_104/4 + w_4_107/4 + w_4_108/4 - w_4_109/2 + w_4_119/2
w_4_142==1/5760 - w_4_102 + 2*w_4_103 - 3*w_4_104 - w_4_109
w_4_143==7/1440 + w_4_101/2 + (5*w_4_103)/2 - (5*w_4_104)/2 + (3*w_4_108)/4 - w_4_109
w_4_144==0
w_4_145==0
w_4_146==13/5760 + w_4_100/2 - w_4_101/2 + (3*w_4_103)/2 - 2*w_4_104 + w_4_108/4 - w_4_109/2
w_4_147==1/320 - w_4_101/2 + (3*w_4_103)/2 - w_4_104 + w_4_108/2 - w_4_109/2
w_4_148==11/1920 + 2*w_4_103 - w_4_104 + w_4_108 - w_4_109
w_4_149==-11/2880 + w_4_100/2 + w_4_104/2 - w_4_107/2 - w_4_108 + w_4_109 - w_4_119
x R. BURING AND A. V. KISELEV
Appendix D. Encoding of the associator of the ⋆-product modulo ō(ℏ4)
Encodings of graphs (see Implementation 1 on p. 229) are followed by their coeffi-
cients, in the following table containing the expansion of the associator (f ⋆ g) ⋆ h− f ⋆
(g ⋆ h).
The table below contains the output of the command
$ reduce_mod_jacobi assoc4_intermsof10_part100.txt
as described in Implementation 16.
The first part of the output lists the graph series S(1) − ♢, reduced modulo skew-
symmetry, wherein the coefficients of ♢ are still undetermined. The second part of
the ouput (after the blank line) specifies the coefficients such that S(1) = ♢. Every
coefficient in the second part is preceded by the encoding of the Leibniz graph that
specifies a differential operator acting on the Jacobi identity. Such a differential operator
expands into a sum of graphs that can be read in the first part of the output.
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION xi
Table 8. The associator of ⋆ up to order 4 in terms of 149 parameters.
h^0: 3 4 1 0 2 1 2 1 2 1 3 1/3 3 4 1 0 2 1 2 1 3 2 3 -1/3
h^1: 3 4 1 0 4 1 2 1 2 1 2 -1/3 # 2 2 2
h^2: # 2 2 3 3 4 1 0 1 0 1 2 3 2 3 1/3
# 1 1 1 3 4 1 0 1 0 2 1 2 2 3 -2/3 3 4 1 0 1 0 1 2 3 2 4 4/9
3 2 1 0 1 2 3 -2/3 3 4 1 0 2 0 2 1 2 1 3 2/3 3 4 1 0 1 0 3 1 2 2 3 -1/3
3 2 1 0 2 1 3 2/3 3 4 1 0 2 0 5 1 2 1 2 -2/3 3 4 1 0 1 0 3 1 2 2 4 -1/6
3 2 1 0 4 1 2 -2/3 # 1 3 2 3 4 1 0 1 0 2 1 3 2 5 1/3
h^3: 3 4 1 0 1 1 2 1 3 2 5 1/3 3 4 1 0 1 0 1 2 3 2 5 1/3
# 1 2 2 3 4 1 0 1 1 2 1 4 2 3 -2/9 3 4 1 0 1 0 2 1 2 3 4 1/3
3 3 1 0 1 1 2 2 3 -2/3 3 4 1 0 1 1 2 1 6 2 3 -1/3 3 4 1 0 1 0 2 1 4 2 3 -4/9
3 3 1 0 2 1 2 1 3 2/3 3 4 1 0 2 1 2 1 3 1 4 2/9 3 4 1 0 1 0 2 1 6 2 3 -1/3
3 3 1 0 4 1 2 1 2 -2/3 3 4 1 0 4 1 2 1 2 1 5 -2/9 3 4 1 0 1 0 5 2 3 1 2 -1/6
# 2 1 2 3 4 1 0 4 1 5 1 2 1 2 -1/3
3 3 1 0 1 0 2 2 3 -2/3 3 4 1 0 1 0 2 1 6 2 4 -1/63 4 1 0 4 1 2 1 2 1 3 1/3
3 3 1 0 2 0 2 1 3 2/3 3 4 1 0 1 0 5 2 6 1 2 1/6# 2 3 1
3 3 1 0 2 0 5 1 2 -2/3 3 4 1 0 1 0 2 1 4 2 5 -1/63 4 1 0 1 0 1 1 3 2 4 2/9
# 2 2 1 3 4 1 0 1 0 5 1 2 2 4 1/63 4 1 0 1 0 1 1 3 2 5 1/3
3 3 1 0 1 0 1 2 3 -2/3 3 4 1 0 2 0 3 1 2 1 3 1/33 4 1 0 1 0 1 1 6 2 3 -1/3
3 3 1 0 1 0 2 1 4 2/3 3 4 1 0 4 1 2 0 4 1 2 -1/33 4 1 0 1 0 2 1 3 1 4 -2/9
3 3 1 0 1 0 5 1 2 -2/3 3 4 1 0 2 0 5 1 2 1 3 4/93 4 1 0 1 0 5 1 2 1 3 2/9
# 1 2 1 3 4 1 0 4 1 2 0 6 1 2 -4/9
3 3 1 0 1 1 3 2 4 1/3 3 4 1 0 1 0 5 1 6 1 2 -1/3 3 4 1 0 2 0 5 1 6 1 2 -1/3
3 3 1 0 1 1 5 2 3 -1/3 3 4 1 0 1 0 5 1 2 1 4 1/3 3 4 1 0 1 0 5 1 2 2 5 1/3# 2 1 3
3 3 1 0 4 1 5 1 2 -1/3 3 4 1 0 4 0 5 1 2 1 2 -1/33 4 1 0 1 0 2 2 3 2 3 1/3
3 3 1 0 4 1 2 1 3 1/3 3 4 1 0 1 0 2 1 2 4 5 -1/33 4 1 0 1 0 2 2 3 2 4 2/9
# 1 1 1 3 4 1 0 2 0 5 1 2 1 4 1/3
3 3 1 0 4 1 3 2 4 1/6 3 4 1 0 1 0 2 2 3 2 5 1/3 3 4 1 0 2 0 5 1 3 1 2 1/6
3 3 1 0 1 2 3 3 4 -1/6 3 4 1 0 2 0 2 1 6 2 3 -1/6 3 4 1 0 2 0 3 1 2 1 4 1/6
3 3 1 0 1 2 5 3 4 -1/6 3 4 1 0 2 0 5 2 6 1 2 1/6 3 4 1 0 1 0 2 1 4 2 4 -1/3
3 3 1 0 4 1 2 3 4 -1/6 3 4 1 0 2 0 2 1 3 2 5 -1/6 # 1 3 1
3 3 1 0 4 2 3 1 4 -1/6 3 4 1 0 2 0 5 1 2 2 4 1/6 3 4 1 0 1 1 3 1 3 2 3 -1/6+8*w_4_6
3 3 1 0 4 1 5 2 4 1/6 3 4 1 0 2 0 2 1 3 2 4 -2/9 3 4 1 0 1 1 3 1 3 2 4 -1/3+16*w_4_7
3 3 1 0 4 3 5 1 2 -1/6 3 4 1 0 2 0 5 1 2 2 3 2/9 3 4 1 0 1 1 2 1 3 3 4 2/9
3 3 1 0 4 2 3 1 3 -1/6 3 4 1 0 2 0 5 1 2 2 5 1/3 3 4 1 0 1 1 2 1 3 4 5 1/9
# 1 1 2 3 4 1 0 2 0 2 1 2 3 5 -1/3 3 4 1 0 1 1 3 1 6 2 3 1/9+16*w_4_7
3 3 1 0 1 2 3 2 3 1/3 3 4 1 0 2 0 2 1 3 2 3 -1/3 3 4 1 0 1 1 3 1 6 2 4 1/9+16*w_4_12
3 3 1 0 1 2 3 2 4 1/3 # 3 2 1 3 4 1 0 1 1 2 1 4 3 4 2/9
3 3 1 0 2 1 5 2 3 -1/6 3 4 1 0 1 0 1 0 3 2 3 -1/3 3 4 1 0 1 1 5 2 3 1 5 -1/6+8*w_4_11
3 3 1 0 4 2 5 1 2 1/6 3 4 1 0 1 0 1 0 3 2 4 -2/93 4 1 0 1 0 1 0 3 2 5 -1/6 3 4 1 0 2 1 3 1 3 1 3 16/3*w_4_63 3 1 0 2 1 3 2 4 -1/6
3 3 1 0 4 1 2 2 3 1/6 3 4 1 0 1 0 1 0 2 3 5 1/3
3 4 1 0 4 1 2 1 4 1 4 -1/6+8*w_4_6
3 4 1 0 1 0 1 0 6 2 3 -1/6 3 4 1 0 2 1 3 1 3 1 4 32*w_4_73 3 1 0 4 1 2 2 4 1/3 3 4 1 0 1 0 2 0 3 1 4 2/9 3 4 1 0 4 1 2 1 3 1 4 1/9+16*w_4_73 3 1 0 2 1 2 3 4 -1/3 3 4 1 0 1 0 3 0 6 1 2 -2/9 3 4 1 0 4 1 2 1 4 1 5 16*w_4_73 3 1 0 2 1 3 2 3 -1/3
# 2 1 1 3 4 1 0 1 0 2 0 4 1 4 1/3
3 4 1 0 4 1 5 1 2 1 5 -1/3+16*w_4_7
3 4 1 0 1 0 5 1 2 0 5 -1/3 3 4 1 0 2 1 3 1 4 1 4 16*w_4_113 3 1 0 1 0 3 2 3 -1/3 3 4 1 0 4 1 2 1 3 1 3 -1/6+8*w_4_11
3 3 1 0 1 0 3 2 4 -1/6 3 4 1 0 1 0 5 0 6 1 2 -1/3
3 3 1 0 1 0 2 3 4 1/3 3 4 1 0 1 0 2 0 6 1 4 1/6
3 4 1 0 4 1 5 1 2 1 4 16*w_4_11
3 3 1 0 1 0 5 2 3 -1/6 3 4 1 0 1 0 2 0 4 1 5 1/6
3 4 1 0 2 1 3 1 4 1 5 32*w_4_12
3 4 1 0 4 1 2 1 3 1 5 16*w_4_12
3 3 1 0 2 0 3 1 3 1/3 # 3 1 2 3 4 1 0 4 1 5 1 2 1 3 1/9+16*w_4_12
3 3 1 0 4 1 2 0 4 -1/3 3 4 1 0 1 0 2 0 3 2 3 -1/3
3 3 1 0 4 0 5 1 2 -1/3 3 4 1 0 1 0 2 0 3 2 5 -1/6
3 4 1 0 4 1 5 1 6 1 2 16*w_4_12
3 4 1 0 1 1 2 1 4 3 5 -1/9
3 3 1 0 2 0 5 1 3 1/6 3 4 1 0 1 0 2 0 2 3 4 1/3 3 4 1 0 1 1 3 1 4 2 3 16*w_4_7
3 3 1 0 2 0 3 1 4 1/6 3 4 1 0 1 0 2 0 4 2 3 -2/93 4 1 0 1 0 2 0 6 2 3 -1/6 3 4 1 0 1 1 3 1 4 2 4 16*w_4_11h^4:
# 3 3 1 3 4 1 0 2 0 2 0 3 1 3 1/3
3 4 1 0 1 1 3 1 4 2 5 16*w_4_12
3 4 1 0 2 0 5 1 2 0 5 -1/3 3 4 1 0 1 1 5 1 6 2 3 16*w_4_123 4 1 0 1 0 1 0 1 2 3 -1/3 3 4 1 0 2 0 2 0 3 1 4 2/9 # 3 1 13 4 1 0 1 0 1 0 2 1 5 1/3 3 4 1 0 2 0 3 0 6 1 2 -2/9 3 4 1 0 1 0 3 0 3 2 3 -1/6-8*w_4_83 4 1 0 1 0 1 0 6 1 2 -1/3 3 4 1 0 2 0 5 0 6 1 2 -1/3 3 4 1 0 1 0 3 0 3 2 4 -1/3-16*w_4_9# 3 2 2 3 4 1 0 2 0 2 0 6 1 3 1/6 3 4 1 0 1 0 2 0 3 3 4 1/9-16*w_4_13 4 1 0 1 0 1 0 2 2 3 -2/3 3 4 1 0 2 0 2 0 3 1 5 1/6 3 4 1 0 1 0 2 0 3 4 5 -1/9-16*w_4_23 4 1 0 1 0 2 0 2 1 4 2/3 # 1 2 3 3 4 1 0 1 0 3 0 6 2 3 -1/9-16*w_4_193 4 1 0 1 0 2 0 6 1 2 -2/3
# 3 1 3 3 4 1 0 1 1 2 2 3 2 3 1/3 3 4 1 0 1 0 3 0 6 2 4 -1/9-16*w_4_20
3 4 1 0 1 0 2 0 2 2 3 -1/3 3 4 1 0 1 1 2 2 3 2 4 2/9 3 4 1 0 1 0 2 0 4 3 4 1/3+16*w_4_1
3 4 1 0 2 0 2 0 2 1 3 1/3 3 4 1 0 1 1 2 2 3 2 5 1/3 3 4 1 0 1 0 5 2 3 0 5 -1/6+8*w_4_26
3 4 1 0 2 0 2 0 6 1 2 -1/3 3 4 1 0 2 1 2 1 6 2 3 -1/6 3 4 1 0 2 0 3 0 3 1 3 -8*w_4_8+8*w_4_6
# 2 3 2 3 4 1 0 4 2 5 1 2 1 2 1/6 3 4 1 0 4 1 2 0 4 0 4 -1/6+8/3*w_4_6
3 4 1 0 1 0 1 1 2 2 3 -2/3 3 4 1 0 2 1 2 1 3 2 5 -1/6 3 4 1 0 2 0 3 0 3 1 4 -16*w_4_9+16*w_4_7
3 4 1 0 1 0 2 1 2 1 4 2/3 3 4 1 0 4 1 2 1 2 2 3 1/6 3 4 1 0 2 0 3 0 4 1 3 -16*w_4_13+16*w_4_7
3 4 1 0 1 0 5 1 2 1 2 -2/3 3 4 1 0 2 1 2 1 3 2 4 -2/9 3 4 1 0 2 0 3 0 6 1 3 -16*w_4_19+16*w_4_7
# 1 3 3 3 4 1 0 4 1 2 1 2 2 5 2/9 3 4 1 0 4 0 5 1 2 0 5 -1/3+16*w_4_7
3 4 1 0 1 1 2 1 2 2 3 -1/3 3 4 1 0 4 1 2 1 2 2 4 1/3 3 4 1 0 2 0 3 0 4 1 4 16*w_4_11-16*w_4_143 4 1 0 2 1 2 1 2 3 4 -1/3 3 4 1 0 2 0 5 1 3 0 5 8*w_4_11+8*w_4_26
3 4 1 0 4 0 5 1 2 0 4 -1/6+8*w_4_11
xii R. BURING AND A. V. KISELEV
Table 8 (part 2).
3 4 1 0 2 0 3 0 4 1 5 -16*w_4_15+16*w_4_12 3 4 1 0 1 0 1 2 3 3 4 -16*w_4_1
3 4 1 0 2 0 3 0 6 1 4 16*w_4_12-16*w_4_20 3 4 1 0 1 0 2 1 4 3 5 -16*w_4_2
3 4 1 0 2 0 5 0 6 1 3 16*w_4_12+16*w_4_27 3 4 1 0 1 0 3 1 3 2 3 16*w_4_8
3 4 1 0 4 0 5 0 6 1 2 16*w_4_12 3 4 1 0 1 0 3 1 3 2 5 16*w_4_13
3 4 1 0 1 0 2 0 4 3 5 -16*w_4_2 3 4 1 0 1 0 5 1 6 2 3 16*w_4_15
3 4 1 0 1 0 3 0 4 2 3 -16*w_4_13 3 4 1 0 1 0 3 1 3 2 4 16*w_4_19
3 4 1 0 1 0 3 0 4 2 4 -16*w_4_14 3 4 1 0 1 0 5 2 3 1 4 16*w_4_20
3 4 1 0 1 0 3 0 4 2 5 -16*w_4_15 # 1 2 2
3 4 1 0 1 0 5 0 6 2 3 16*w_4_27 3 4 1 0 1 1 3 2 3 2 3 -1/6+8*w_4_6
# 1 1 3 3 4 1 0 1 1 3 2 3 2 4 -1/3+16*w_4_7
3 4 1 0 1 2 3 2 3 2 3 -1/6+8/3*w_4_6 3 4 1 0 1 1 3 2 4 2 4 -1/6+8*w_4_11
3 4 1 0 1 2 3 2 3 2 4 -1/3+16*w_4_7 3 4 1 0 4 1 3 1 2 2 4 1/6
3 4 1 0 1 2 3 2 4 2 4 -1/6+8*w_4_11 3 4 1 0 1 1 3 2 3 2 5 -1/9+16*w_4_7
3 4 1 0 2 1 2 2 3 3 4 1/3+16*w_4_1 3 4 1 0 1 1 3 2 4 2 5 -1/9+16*w_4_12
3 4 1 0 2 1 2 2 4 3 4 1/9-16*w_4_1 3 4 1 0 1 1 2 2 3 3 5 -1/6
3 4 1 0 2 1 2 2 3 4 5 16*w_4_2 3 4 1 0 1 1 5 2 3 2 3 1/3+16*w_4_7
3 4 1 0 2 1 2 2 4 3 5 1/9+16*w_4_2 3 4 1 0 1 1 2 2 6 3 5 -1/6
3 4 1 0 2 1 3 2 3 2 3 -8*w_4_8+8*w_4_6 3 4 1 0 1 1 5 2 4 2 3 1/9
3 4 1 0 4 1 2 2 4 2 4 -1/6-8*w_4_8 3 4 1 0 1 1 2 2 4 3 5 -1/18
3 4 1 0 2 1 5 2 3 2 3 -16*w_4_9+16*w_4_7 3 4 1 0 1 1 5 2 6 2 3 1/6+16*w_4_12
3 4 1 0 4 2 5 1 2 2 5 -1/3-16*w_4_9 3 4 1 0 1 1 2 2 6 3 4 -1/6
3 4 1 0 2 1 3 2 3 2 5 -16*w_4_13+16*w_4_7 3 4 1 0 1 1 5 2 3 2 4 1/6+16*w_4_12
3 4 1 0 4 1 2 2 4 2 5 -16*w_4_13 3 4 1 0 4 1 2 1 2 3 4 -1/6
3 4 1 0 2 1 5 2 3 2 5 16*w_4_11-16*w_4_14 3 4 1 0 4 2 3 1 2 1 4 -1/6
3 4 1 0 4 2 5 1 2 2 4 -16*w_4_14 3 4 1 0 2 1 3 1 6 2 5 -1/9
3 4 1 0 2 1 5 2 6 2 3 -16*w_4_15+16*w_4_12
3 4 1 0 4 2 5 2 6 1 2 -16*w_4_15 3 4 1 0 4 1 2 1 6 2 5 1/9
3 4 1 0 2 1 3 2 3 2 4 -16*w_4_19+16*w_4_7 3 4 1 0 2 1 2 1 3 3 4 1/3+16*w_4_1
3 4 1 0 4 1 2 2 3 2 4 -1/9-16*w_4_19 3 4 1 0 2 1 2 1 4 3 4 -1/9-16*w_4_1
3 4 1 0 2 1 5 2 3 2 4 16*w_4_12-16*w_4_20 3 4 1 0 4 1 2 1 2 4 5 16*w_4_13 4 1 0 2 1 2 1 3 4 5 16*w_4_2
3 4 1 0 4 2 5 1 2 2 3 -1/9-16*w_4_20
3 4 1 0 2 1 3 2 4 2 4 8*w_4_11+8*w_4_26 3 4 1 0 2 1 2 1 4 3 5 -1/9+16*w_4_2
3 4 1 0 4 1 2 2 3 2 3 -1/6+8*w_4_26 3 4 1 0 4 1 2 1 2 3 5 -1/3-16*w_4_2
3 4 1 0 2 1 3 2 4 2 5 16*w_4_12+16*w_4_27 3 4 1 0 2 1 3 1 3 2 3 8*w_4_8+8*w_4_6
3 4 1 0 4 1 2 2 3 2 5 16*w_4_27 3 4 1 0 4 1 2 1 4 2 4 16*w_4_8
3 4 1 0 1 2 3 2 4 2 5 16*w_4_12 3 4 1 0 2 1 3 1 3 2 4 16*w_4_9+16*w_4_7
# 2 2 1 3 4 1 0 4 1 2 1 4 2 3 1/9+16*w_4_9
3 4 1 0 1 0 5 1 4 2 3 1/9 3 4 1 0 4 1 2 1 4 2 5 1/6+16*w_4_9
3 4 1 0 1 0 5 1 4 2 5 1/6 3 4 1 0 2 1 3 1 4 2 3 16*w_4_13+16*w_4_7
3 4 1 0 1 0 5 1 3 2 3 1/6+16*w_4_9 3 4 1 0 4 1 2 1 3 2 4 -1/9+16*w_4_13
3 4 1 0 1 0 5 1 3 2 4 1/6+16*w_4_20 3 4 1 0 4 1 5 1 2 2 5 16*w_4_13
3 4 1 0 1 0 5 1 3 2 5 1/6+16*w_4_14 3 4 1 0 2 1 3 1 4 2 4 16*w_4_11+16*w_4_14
3 4 1 0 1 0 3 1 4 2 3 1/6+16*w_4_19 3 4 1 0 4 1 2 1 3 2 3 -1/3+16*w_4_14
3 4 1 0 1 0 3 1 4 2 4 1/6-16*w_4_26 3 4 1 0 4 1 5 1 2 2 4 1/6+16*w_4_14
3 4 1 0 1 0 3 1 4 2 5 1/6-16*w_4_27 3 4 1 0 2 1 3 1 4 2 5 16*w_4_15+16*w_4_123 4 1 0 4 1 2 1 3 2 5 16*w_4_15
3 4 1 0 1 0 1 2 3 3 5 -1/6 3 4 1 0 4 1 5 1 2 2 3 1/9+16*w_4_15
3 4 1 0 1 0 1 2 3 4 5 -1/3-16*w_4_2 3 4 1 0 2 1 3 1 6 2 3 16*w_4_19+16*w_4_7
3 4 1 0 1 0 3 1 2 3 5 1/9 3 4 1 0 4 1 2 1 6 2 4 1/6+16*w_4_19
3 4 1 0 1 0 3 1 2 4 5 -1/18 3 4 1 0 4 2 5 1 2 1 5 16*w_4_19
3 4 1 0 1 0 3 1 6 2 3 -1/9+16*w_4_13 3 4 1 0 2 1 3 1 6 2 4 16*w_4_12+16*w_4_20
3 4 1 0 1 0 3 1 6 2 4 -1/9-16*w_4_27 3 4 1 0 4 1 2 1 6 2 3 16*w_4_20
3 4 1 0 1 0 2 1 3 3 4 -1/9-16*w_4_1 3 4 1 0 4 2 5 1 6 1 2 1/6+16*w_4_20
3 4 1 0 1 0 2 1 3 4 5 1/9-16*w_4_2 3 4 1 0 2 1 5 2 3 1 5 8*w_4_11-8*w_4_26
3 4 1 0 1 0 5 2 3 1 3 1/9+16*w_4_9 3 4 1 0 4 2 5 1 2 1 4 1/6-16*w_4_26
3 4 1 0 1 0 5 2 6 1 3 1/9+16*w_4_15 3 4 1 0 2 1 5 1 6 2 3 16*w_4_12-16*w_4_27
3 4 1 0 2 0 5 1 4 1 3 -1/9 3 4 1 0 4 1 5 2 6 1 2 1/6-16*w_4_27
3 4 1 0 4 1 2 0 6 1 5 1/9 3 4 1 0 4 2 5 1 2 1 3 -1/9-16*w_4_27
3 4 1 0 1 0 1 2 6 3 5 -1/6 3 4 1 0 4 1 5 2 4 1 2 1/6
3 4 1 0 1 0 2 1 4 3 4 1/3+16*w_4_1 3 4 1 0 1 1 2 2 4 3 4 1/9
3 4 1 0 1 0 5 2 3 1 5 -1/3+16*w_4_14 3 4 1 0 4 3 5 1 2 1 2 -1/6
3 4 1 0 1 0 5 1 2 4 5 -1/6 3 4 1 0 4 2 3 1 2 1 3 -1/6
3 4 1 0 1 0 5 2 4 1 5 -1/6 3 4 1 0 1 1 5 2 3 2 5 16*w_4_11
3 4 1 0 2 0 3 1 3 1 3 8*w_4_8+8*w_4_6 # 2 1 2
3 4 1 0 4 1 2 0 4 1 4 -1/6+8*w_4_6 3 4 1 0 1 0 3 2 3 2 3 1/6+8*w_4_8
3 4 1 0 2 0 3 1 3 1 4 16*w_4_19+16*w_4_7 3 4 1 0 1 0 3 2 3 2 4 1/3+16*w_4_19
3 4 1 0 2 0 3 1 3 1 5 16*w_4_13+16*w_4_7 3 4 1 0 1 0 3 2 4 2 4 1/6-8*w_4_26
3 4 1 0 2 0 5 1 3 1 3 16*w_4_9+16*w_4_7 3 4 1 0 2 0 5 1 4 2 5 1/6
3 4 1 0 4 1 2 0 4 1 3 1/3+16*w_4_7 3 4 1 0 1 0 3 2 3 2 5 1/9+16*w_4_13
3 4 1 0 4 0 5 1 2 1 5 -1/9+16*w_4_7 3 4 1 0 1 0 3 2 4 2 5 1/9-16*w_4_27
3 4 1 0 4 1 2 0 6 1 4 -1/3+16*w_4_7 3 4 1 0 1 0 2 2 3 3 4 -1/3-16*w_4_1
3 4 1 0 2 0 3 1 4 1 4 8*w_4_11-8*w_4_26 3 4 1 0 1 0 2 2 3 3 5 -1/6
3 4 1 0 2 0 5 1 3 1 5 16*w_4_11+16*w_4_14 3 4 1 0 1 0 5 2 3 2 3 1/3+16*w_4_9
3 4 1 0 4 0 5 1 2 1 4 16*w_4_11 3 4 1 0 1 0 2 2 6 3 5 -1/6
3 4 1 0 4 1 5 1 2 0 4 -1/6+8*w_4_11 3 4 1 0 1 0 5 2 4 2 3 1/9
3 4 1 0 2 0 3 1 4 1 5 16*w_4_12-16*w_4_27 3 4 1 0 1 0 2 2 4 3 5 -1/6-16*w_4_2
3 4 1 0 2 0 5 1 3 1 4 16*w_4_12+16*w_4_20 3 4 1 0 1 0 5 2 6 2 3 1/6+16*w_4_15
3 4 1 0 2 0 5 1 6 1 3 16*w_4_15+16*w_4_12 3 4 1 0 1 0 2 2 6 3 4 -1/6
3 4 1 0 4 0 5 1 2 1 3 1/6+16*w_4_12 3 4 1 0 1 0 5 2 3 2 4 1/6+16*w_4_20
3 4 1 0 4 1 2 0 6 1 3 1/6+16*w_4_12 3 4 1 0 2 0 5 1 2 4 5 -1/6
3 4 1 0 4 0 5 1 6 1 2 -1/9+16*w_4_12 3 4 1 0 2 0 5 2 4 1 5 -1/6
3 4 1 0 1 0 5 1 6 2 5 1/6 3 4 1 0 2 0 5 2 4 1 3 -1/9
3 4 1 0 1 0 5 4 6 1 2 -1/6 3 4 1 0 4 1 2 0 6 2 5 1/9
3 4 1 0 1 0 5 3 6 1 2 -1/6 3 4 1 0 2 0 3 1 2 3 5 16*w_4_1
3 4 1 0 1 0 5 2 4 1 4 -1/6 3 4 1 0 2 0 5 1 2 3 5 -1/3-16*w_4_1
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION xiii
Table 8 (part 3).
3 4 1 0 2 0 3 1 2 4 5 -1/6-16*w_4_2 3 4 1 0 2 1 5 3 4 1 5 32*w_4_46
3 4 1 0 2 0 5 1 2 3 4 16*w_4_2 3 4 1 0 4 3 5 1 2 1 4 -16*w_4_46
3 4 1 0 2 0 3 1 3 2 3 32*w_4_8 3 4 1 0 4 5 6 1 2 1 4 16*w_4_46
3 4 1 0 4 1 2 0 4 2 4 1/6+8*w_4_8 3 4 1 0 4 1 3 1 4 2 4 16*w_4_60-16*w_4_83
3 4 1 0 2 0 3 1 3 2 4 16*w_4_9+16*w_4_19 3 4 1 0 4 1 3 1 4 2 3 16*w_4_61-16*w_4_84
3 4 1 0 2 0 3 1 3 2 5 16*w_4_9+16*w_4_13 3 4 1 0 4 1 3 1 4 2 5 -16*w_4_74+16*w_4_62
3 4 1 0 4 1 2 0 4 2 3 1/3+16*w_4_9 3 4 1 0 4 1 5 1 4 2 3 -16*w_4_63+16*w_4_62
3 4 1 0 2 0 3 1 4 2 3 16*w_4_19+16*w_4_13 3 4 1 0 4 2 5 1 6 1 5 16*w_4_63-16*w_4_62
3 4 1 0 2 0 5 1 3 2 3 16*w_4_9+16*w_4_13 3 4 1 0 4 1 5 1 3 2 3 -16*w_4_76+16*w_4_71
3 4 1 0 4 0 5 1 2 2 5 1/9+16*w_4_13 3 4 1 0 4 1 5 1 3 2 5 16*w_4_71-16*w_4_75
3 4 1 0 2 0 3 1 4 2 4 -16*w_4_26+16*w_4_14 3 4 1 0 4 1 5 1 3 2 4 -16*w_4_73+16*w_4_71
3 4 1 0 2 0 5 1 3 2 5 32*w_4_14 3 4 1 0 4 1 5 1 6 2 4 16*w_4_73-16*w_4_71
3 4 1 0 4 0 5 1 2 2 4 16*w_4_14 3 4 1 0 4 1 5 2 3 1 3 -16*w_4_76+16*w_4_73
3 4 1 0 2 0 3 1 4 2 5 16*w_4_15-16*w_4_27 3 4 1 0 4 2 5 1 3 1 5 16*w_4_73-16*w_4_75
3 4 1 0 2 0 5 1 3 2 4 16*w_4_15+16*w_4_20 3 4 1 0 4 1 5 1 6 2 5 16*w_4_74-16*w_4_62
3 4 1 0 4 0 5 1 2 2 3 1/6+16*w_4_15 3 4 1 0 4 1 5 2 4 1 3 -1/18-16*w_4_63+16*w_4_74
3 4 1 0 2 0 3 1 6 2 3 16*w_4_19+16*w_4_13 3 4 1 0 4 2 3 1 3 1 5 16*w_4_74-16*w_4_91
3 4 1 0 2 0 5 2 3 1 3 16*w_4_9+16*w_4_19 3 4 1 0 4 1 5 2 3 1 4 -16*w_4_73+16*w_4_75
3 4 1 0 4 1 2 0 6 2 4 1/3+16*w_4_19 3 4 1 0 4 2 5 1 3 1 3 -16*w_4_76+16*w_4_75
3 4 1 0 2 0 3 1 6 2 4 16*w_4_20-16*w_4_27 3 4 1 0 4 1 5 2 6 1 4 -16*w_4_71+16*w_4_75
3 4 1 0 2 0 5 2 6 1 3 16*w_4_15+16*w_4_20 3 4 1 0 4 1 5 2 3 1 5 16*w_4_76-16*w_4_75
3 4 1 0 4 1 2 0 6 2 3 1/6+16*w_4_20 3 4 1 0 4 2 5 1 3 1 4 16*w_4_76-16*w_4_73
3 4 1 0 2 0 5 2 3 1 5 -16*w_4_26+16*w_4_14 3 4 1 0 4 2 5 1 6 1 4 16*w_4_76-16*w_4_71
3 4 1 0 4 2 5 1 2 0 4 1/6-8*w_4_26 3 4 1 0 4 1 5 2 4 1 4 -16*w_4_60+16*w_4_83
3 4 1 0 2 0 5 1 6 2 3 16*w_4_15-16*w_4_27 3 4 1 0 4 2 3 1 3 1 3 -8*w_4_105+8*w_4_83
3 4 1 0 2 0 5 2 3 1 4 16*w_4_20-16*w_4_27 3 4 1 0 4 1 5 2 4 1 5 1/6-16*w_4_61+16*w_4_84
3 4 1 0 4 0 5 2 6 1 2 1/9-16*w_4_27 3 4 1 0 4 2 5 1 4 1 5 -16*w_4_61+16*w_4_84
3 4 1 0 2 0 5 1 6 2 5 1/6 3 4 1 0 4 1 5 2 6 1 5 1/6+16*w_4_91-16*w_4_62
3 4 1 0 2 0 5 4 6 1 2 -1/6 3 4 1 0 4 2 5 1 4 1 3 -1/18-16*w_4_63+16*w_4_91
3 4 1 0 2 0 5 3 6 1 2 -1/6 3 4 1 0 4 2 3 1 4 1 5 -16*w_4_74+16*w_4_91
3 4 1 0 2 0 5 2 4 1 4 -1/6 3 4 1 0 4 2 3 1 4 1 4 8*w_4_105-8*w_4_83
3 4 1 0 1 0 2 2 4 3 4 16*w_4_1 3 4 1 0 4 2 5 1 4 1 4 1/6+16*w_4_105-16*w_4_60
3 4 1 0 1 0 2 2 3 4 5 -16*w_4_2 3 4 1 0 4 1 2 1 4 3 5 -1/9
3 4 1 0 1 0 5 2 3 2 5 16*w_4_14 3 4 1 0 1 1 5 2 4 3 4 1/18+16*w_4_40
# 1 2 1 3 4 1 0 1 1 2 3 4 3 5 -1/6
3 4 1 0 1 1 3 2 3 3 4 1/6+16*w_4_10 3 4 1 0 1 1 5 2 3 3 4 16*w_4_17
3 4 1 0 1 1 3 2 4 3 4 1/6+16*w_4_16 3 4 1 0 1 1 5 3 6 2 3 16*w_4_21
3 4 1 0 1 1 3 2 6 3 4 1/6+16*w_4_21 3 4 1 0 1 1 5 3 4 2 3 16*w_4_22
3 4 1 0 4 1 3 1 3 2 3 -1/6-16*w_4_105+16*w_4_60 3 4 1 0 1 1 5 4 6 2 3 -16*w_4_23
3 4 1 0 4 1 3 1 3 2 4 -1/6+16*w_4_61-16*w_4_84 3 4 1 0 1 1 5 2 6 3 4 16*w_4_41
3 4 1 0 4 1 3 1 3 2 5 -1/6-16*w_4_91+16*w_4_62 3 4 1 0 1 1 5 3 6 2 4 16*w_4_41
3 4 1 0 1 1 3 2 3 3 5 1/9+16*w_4_10 3 4 1 0 1 1 5 3 4 2 4 16*w_4_423 4 1 0 1 1 5 3 4 2 5 16*w_4_46
3 4 1 0 1 1 3 2 3 4 5 1/9 3 4 1 0 1 1 5 3 6 2 5 16*w_4_46
3 4 1 0 1 1 3 2 4 3 5 1/9+16*w_4_17 # 2 1 1
3 4 1 0 1 1 3 2 4 4 5 1/9+16*w_4_18 3 4 1 0 1 0 3 2 3 3 4 1/6-16*w_4_24
3 4 1 0 4 1 3 1 2 3 5 1/9-16*w_4_40 3 4 1 0 1 0 3 2 4 3 4 1/6-16*w_4_28
3 4 1 0 4 1 3 1 2 4 5 -1/18-16*w_4_40 3 4 1 0 1 0 3 2 6 3 4 1/6-16*w_4_31
3 4 1 0 4 1 3 1 6 2 3 1/18+16*w_4_63-16*w_4_91 3 4 1 0 4 1 3 0 4 2 3 1/6-16*w_4_106+16*w_4_61
3 4 1 0 4 1 3 1 6 2 4 1/18+16*w_4_63-16*w_4_74 3 4 1 0 4 1 3 0 4 2 4 1/6+16*w_4_60-16*w_4_86
3 4 1 0 1 1 3 2 6 3 5 1/18+16*w_4_22 3 4 1 0 4 1 3 0 4 2 5 1/6-16*w_4_107+16*w_4_62
3 4 1 0 1 1 3 2 6 4 5 1/18+16*w_4_23 3 4 1 0 1 0 3 2 3 3 5 -1/9+16*w_4_24
3 4 1 0 1 1 5 2 3 3 5 -1/3+16*w_4_16 3 4 1 0 1 0 3 2 3 4 5 -1/9-16*w_4_25
3 4 1 0 1 1 2 3 4 4 5 -1/6 3 4 1 0 1 0 3 2 4 3 5 -1/9-16*w_4_29
3 4 1 0 1 1 5 2 3 4 5 -1/6-16*w_4_18 3 4 1 0 1 0 3 2 4 4 5 -1/9-16*w_4_30
3 4 1 0 1 1 5 2 4 3 5 -1/9+16*w_4_40 3 4 1 0 2 0 5 1 4 3 4 1/18+16*w_4_40-16*w_4_43
3 4 1 0 1 1 5 2 6 3 5 -1/6+16*w_4_42 3 4 1 0 2 0 5 1 4 3 5 1/18+16*w_4_40+16*w_4_43
3 4 1 0 2 1 3 1 3 3 4 32*w_4_10 3 4 1 0 4 1 3 0 6 2 3 1/18+16*w_4_63-16*w_4_107
3 4 1 0 4 1 2 1 4 3 4 -1/9-16*w_4_10 3 4 1 0 4 1 3 0 6 2 4 1/18+16*w_4_63-16*w_4_85
3 4 1 0 4 1 2 1 4 4 5 1/6+16*w_4_10 3 4 1 0 1 0 3 2 6 3 5 -1/18-16*w_4_32
3 4 1 0 2 1 3 1 4 3 4 32*w_4_16 3 4 1 0 1 0 3 2 6 4 5 -1/18-16*w_4_33
3 4 1 0 4 1 2 1 3 3 4 1/3-16*w_4_16 3 4 1 0 1 0 2 3 4 3 4 1/6-8*w_4_3
3 4 1 0 4 1 5 1 2 4 5 -1/6-16*w_4_16
3 4 1 0 2 1 3 1 4 3 5 32*w_4_17 3 4 1 0 1 0 5 2 3 3 5 -1/3+16*w_4_28
3 4 1 0 4 1 2 1 3 4 5 16*w_4_17 3 4 1 0 1 0 2 3 4 4 5 -1/6+16*w_4_4
3 4 1 0 4 1 5 1 2 3 5 -1/9-16*w_4_17 3 4 1 0 1 0 5 2 3 4 5 -1/6-16*w_4_30
3 4 1 0 2 1 3 1 4 4 5 32*w_4_18 3 4 1 0 1 0 5 2 4 3 5 -1/6+16*w_4_43
3 4 1 0 4 1 2 1 3 3 5 1/6+16*w_4_18 3 4 1 0 1 0 5 2 6 3 5 -1/6-16*w_4_45
3 4 1 0 4 1 5 1 2 3 4 -1/9-16*w_4_18 3 4 1 0 2 0 3 1 3 3 4 -16*w_4_24+16*w_4_10
3 4 1 0 2 1 3 1 6 3 4 32*w_4_21 3 4 1 0 2 0 3 1 3 3 5 16*w_4_24+16*w_4_10
3 4 1 0 4 1 2 1 6 3 4 -16*w_4_21 3 4 1 0 4 1 2 0 4 3 4 -1/3-16*w_4_10
3 4 1 0 4 5 6 1 2 1 5 1/6+16*w_4_21 3 4 1 0 2 0 3 1 4 3 4 -16*w_4_28+16*w_4_16
3 4 1 0 2 1 3 1 6 3 5 32*w_4_22 3 4 1 0 2 0 5 1 3 3 5 16*w_4_28+16*w_4_16
3 4 1 0 4 1 2 1 6 4 5 16*w_4_22 3 4 1 0 4 0 5 1 2 4 5 -16*w_4_16
3 4 1 0 4 3 5 1 2 1 5 -1/18-16*w_4_22 3 4 1 0 2 0 3 1 4 3 5 -16*w_4_29+16*w_4_17
3 4 1 0 2 1 3 1 6 4 5 32*w_4_23 3 4 1 0 2 0 5 1 3 3 4 16*w_4_29+16*w_4_17
3 4 1 0 4 1 2 1 6 3 5 16*w_4_23 3 4 1 0 4 0 5 1 2 3 5 -1/6-16*w_4_17
3 4 1 0 4 3 5 1 6 1 2 -1/18-16*w_4_23 3 4 1 0 2 0 3 1 4 4 5 16*w_4_18-16*w_4_30
3 4 1 0 2 1 5 1 4 3 4 32*w_4_40 3 4 1 0 2 0 5 1 3 4 5 -16*w_4_18-16*w_4_30
3 4 1 0 2 1 5 1 6 3 4 32*w_4_41 3 4 1 0 4 0 5 1 2 3 4 -1/6-16*w_4_18
3 4 1 0 4 1 5 3 6 1 2 -16*w_4_41 3 4 1 0 2 0 3 1 6 3 4 16*w_4_21-16*w_4_31
3 4 1 0 4 5 6 1 2 1 3 16*w_4_41 3 4 1 0 2 0 5 3 6 1 3 16*w_4_21+16*w_4_31
3 4 1 0 2 1 5 1 6 3 5 32*w_4_42 3 4 1 0 4 1 2 0 6 3 4 -1/6-16*w_4_21
3 4 1 0 4 1 5 4 6 1 2 -16*w_4_42 3 4 1 0 2 0 3 1 6 3 5 16*w_4_22-16*w_4_32
3 4 1 0 4 3 5 1 2 1 3 1/6-16*w_4_42 3 4 1 0 2 0 5 3 4 1 3 16*w_4_22+16*w_4_323 4 1 0 4 1 2 0 6 4 5 1/6+16*w_4_22
xiv R. BURING AND A. V. KISELEV
Table 8 (part 4).
3 4 1 0 2 0 3 1 6 4 5 16*w_4_23-16*w_4_33 3 4 1 0 2 1 5 2 3 3 5 -16*w_4_28+16*w_4_16
3 4 1 0 2 0 5 4 6 1 3 -16*w_4_23-16*w_4_33 3 4 1 0 4 2 5 1 2 4 5 -1/6+16*w_4_28
3 4 1 0 4 1 2 0 6 3 5 16*w_4_23 3 4 1 0 2 1 3 2 4 3 5 16*w_4_29+16*w_4_17
3 4 1 0 4 0 3 1 2 3 5 -16*w_4_40 3 4 1 0 4 1 2 2 3 4 5 16*w_4_29
3 4 1 0 2 0 5 1 6 3 4 -16*w_4_44+16*w_4_41 3 4 1 0 2 1 5 2 3 3 4 -16*w_4_29+16*w_4_17
3 4 1 0 2 0 5 3 6 1 4 16*w_4_44+16*w_4_41 3 4 1 0 4 2 5 1 2 3 5 1/9+16*w_4_29
3 4 1 0 4 0 5 3 6 1 2 -16*w_4_41 3 4 1 0 2 1 3 2 4 4 5 16*w_4_18+16*w_4_30
3 4 1 0 2 0 5 1 6 3 5 -16*w_4_45+16*w_4_42 3 4 1 0 4 1 2 2 3 3 5 1/6+16*w_4_30
3 4 1 0 2 0 5 3 4 1 4 16*w_4_45+16*w_4_42 3 4 1 0 2 1 5 2 3 4 5 -16*w_4_18+16*w_4_30
3 4 1 0 4 0 5 4 6 1 2 -16*w_4_42
3 4 1 0 2 0 5 3 4 1 5 -16*w_4_47+16*w_4_46 3 4 1 0 4 2 5 1 2 3 4 1/9+16*w_4_30
3 4 1 0 2 0 5 3 6 1 5 16*w_4_47+16*w_4_46 3 4 1 0 2 1 3 2 6 3 4 16*w_4_21+16*w_4_313 4 1 0 4 1 2 2 6 3 4 -16*w_4_31
3 4 1 0 4 3 5 1 2 0 4 -1/6-16*w_4_46 3 4 1 0 2 1 5 3 6 2 3 16*w_4_21-16*w_4_31
3 4 1 0 4 0 3 1 3 2 3 16*w_4_60-16*w_4_64
3 4 1 0 4 0 5 1 4 2 4 16*w_4_60-16*w_4_86 3 4 1 0 4 5 6 1 2 2 5 1/6-16*w_4_31
3 4 1 0 4 0 3 1 3 2 4 16*w_4_61-16*w_4_65 3 4 1 0 2 1 3 2 6 3 5 16*w_4_22+16*w_4_32
3 4 1 0 4 0 5 1 4 2 5 16*w_4_61-16*w_4_87 3 4 1 0 4 1 2 2 6 4 5 16*w_4_32
3 4 1 0 4 0 3 1 3 2 5 -16*w_4_66+16*w_4_62 3 4 1 0 2 1 5 3 4 2 3 16*w_4_22-16*w_4_32
3 4 1 0 4 0 5 1 4 2 3 16*w_4_62-16*w_4_85 3 4 1 0 4 3 5 1 2 2 5 1/18+16*w_4_32
3 4 1 0 4 0 3 1 6 2 3 -16*w_4_66+16*w_4_63 3 4 1 0 2 1 3 2 6 4 5 16*w_4_23+16*w_4_33
3 4 1 0 4 0 5 1 3 2 3 -16*w_4_77+16*w_4_71 3 4 1 0 4 1 2 2 6 3 5 16*w_4_33
3 4 1 0 4 0 5 1 3 2 4 16*w_4_71-16*w_4_78 3 4 1 0 2 1 5 4 6 2 3 -16*w_4_23+16*w_4_33
3 4 1 0 4 0 5 1 3 2 5 -16*w_4_79+16*w_4_71 3 4 1 0 4 3 5 2 6 1 2 1/18+16*w_4_33
3 4 1 0 4 0 5 1 6 2 3 -16*w_4_92+16*w_4_72 3 4 1 0 2 1 5 2 4 3 4 1/18+16*w_4_40+16*w_4_43
3 4 1 0 4 1 5 0 6 2 3 -16*w_4_118+16*w_4_72 3 4 1 0 4 2 3 1 2 3 5 1/6-16*w_4_43
3 4 1 0 4 0 5 2 6 1 3 -16*w_4_92+16*w_4_72 3 4 1 0 2 1 5 2 4 3 5 1/18+16*w_4_40-16*w_4_43
3 4 1 0 4 0 5 1 6 2 4 -16*w_4_79+16*w_4_73 3 4 1 0 4 2 3 1 2 4 5 16*w_4_43
3 4 1 0 4 1 5 0 6 2 4 16*w_4_73-16*w_4_78 3 4 1 0 2 1 5 2 6 3 4 16*w_4_44+16*w_4_41
3 4 1 0 4 0 5 2 3 1 3 16*w_4_73-16*w_4_77 3 4 1 0 4 2 5 3 6 1 2 -16*w_4_44
3 4 1 0 4 0 5 1 6 2 5 1/18-16*w_4_66+16*w_4_74 3 4 1 0 2 1 5 3 6 2 4 -16*w_4_44+16*w_4_41
3 4 1 0 4 1 5 0 6 2 5 16*w_4_74-16*w_4_107 3 4 1 0 4 5 6 1 2 2 3 -16*w_4_44
3 4 1 0 4 0 5 2 4 1 3 16*w_4_74-16*w_4_85 3 4 1 0 2 1 5 2 6 3 5 16*w_4_45+16*w_4_42
3 4 1 0 4 0 5 2 3 1 4 -16*w_4_78+16*w_4_75 3 4 1 0 4 2 5 4 6 1 2 -16*w_4_45
3 4 1 0 4 1 5 2 3 0 4 -16*w_4_77+16*w_4_75 3 4 1 0 2 1 5 3 4 2 4 -16*w_4_45+16*w_4_42
3 4 1 0 4 0 5 2 6 1 4 -16*w_4_79+16*w_4_75 3 4 1 0 4 3 5 1 2 2 3 1/6+16*w_4_45
3 4 1 0 4 0 5 2 3 1 5 -16*w_4_79+16*w_4_76 3 4 1 0 2 1 5 3 4 2 5 16*w_4_47+16*w_4_46
3 4 1 0 4 1 5 2 3 0 5 16*w_4_76-16*w_4_78 3 4 1 0 4 3 5 1 2 2 4 -16*w_4_47
3 4 1 0 4 2 5 1 3 0 4 16*w_4_76-16*w_4_77 3 4 1 0 2 1 5 3 6 2 5 -16*w_4_47+16*w_4_46
3 4 1 0 4 0 5 2 4 1 4 -16*w_4_86+16*w_4_83 3 4 1 0 4 5 6 1 2 2 4 -16*w_4_47
3 4 1 0 4 1 5 2 4 0 4 1/12+8*w_4_83-8*w_4_64 3 4 1 0 4 2 5 2 4 1 4 -16*w_4_60+16*w_4_64
3 4 1 0 4 0 5 2 4 1 5 -16*w_4_87+16*w_4_84 3 4 1 0 4 2 5 2 4 1 5 -16*w_4_61+16*w_4_65
3 4 1 0 4 1 5 2 4 0 5 1/6+16*w_4_84-16*w_4_65 3 4 1 0 4 1 5 2 6 2 5 16*w_4_66-16*w_4_62
3 4 1 0 4 2 3 0 4 1 3 -16*w_4_106+16*w_4_84 3 4 1 0 4 2 5 2 4 1 3 16*w_4_66-16*w_4_63
3 4 1 0 4 0 5 2 6 1 5 1/18-16*w_4_66+16*w_4_91 3 4 1 0 4 1 5 2 3 2 3 -16*w_4_76+16*w_4_77
3 4 1 0 4 1 5 2 6 0 5 16*w_4_91-16*w_4_85 3 4 1 0 4 2 5 1 3 2 5 16*w_4_77-16*w_4_75
3 4 1 0 4 2 3 0 4 1 5 -16*w_4_107+16*w_4_91 3 4 1 0 4 2 5 2 3 1 4 -16*w_4_73+16*w_4_77
3 4 1 0 4 2 3 0 4 1 4 16*w_4_105-16*w_4_86 3 4 1 0 4 2 5 2 6 1 4 16*w_4_77-16*w_4_71
3 4 1 0 4 2 5 1 4 0 4 1/12+8*w_4_105-8*w_4_64 3 4 1 0 4 1 5 2 3 2 4 -16*w_4_73+16*w_4_78
3 4 1 0 1 0 2 3 4 3 5 -16*w_4_4 3 4 1 0 4 2 5 1 3 2 3 -16*w_4_76+16*w_4_78
3 4 1 0 1 0 2 3 6 4 5 -16*w_4_5 3 4 1 0 4 2 5 1 6 2 4 -16*w_4_71+16*w_4_78
3 4 1 0 2 0 3 1 3 4 5 -16*w_4_25 3 4 1 0 4 2 5 2 3 1 5 16*w_4_78-16*w_4_75
3 4 1 0 1 0 5 2 3 3 4 16*w_4_29 3 4 1 0 4 1 5 2 3 2 5 16*w_4_79-16*w_4_75
3 4 1 0 1 0 5 3 6 2 3 16*w_4_31 3 4 1 0 4 2 5 1 3 2 4 16*w_4_79-16*w_4_73
3 4 1 0 1 0 5 3 4 2 3 16*w_4_32 3 4 1 0 4 1 5 2 6 2 4 16*w_4_79-16*w_4_71
3 4 1 0 1 0 5 4 6 2 3 -16*w_4_33 3 4 1 0 4 2 5 2 3 1 3 16*w_4_79-16*w_4_76
3 4 1 0 1 0 5 2 4 3 4 -16*w_4_43 3 4 1 0 4 1 5 2 4 2 3 -1/18-16*w_4_63+16*w_4_85
3 4 1 0 1 0 5 2 6 3 4 -16*w_4_44 3 4 1 0 4 2 3 1 3 2 5 -16*w_4_91+16*w_4_85
3 4 1 0 1 0 5 3 6 2 4 16*w_4_44 3 4 1 0 4 2 3 1 6 2 4 -16*w_4_74+16*w_4_85
3 4 1 0 1 0 5 3 4 2 4 16*w_4_45 3 4 1 0 4 2 5 2 6 1 5 -16*w_4_62+16*w_4_85
3 4 1 0 1 0 5 3 4 2 5 -16*w_4_47 3 4 1 0 4 1 5 2 4 2 4 -1/6-16*w_4_60+16*w_4_86
3 4 1 0 1 0 5 3 6 2 5 16*w_4_47 3 4 1 0 4 2 3 1 3 2 3 -16*w_4_105+16*w_4_86
# 1 1 2 3 4 1 0 4 2 3 1 4 2 4 16*w_4_86-16*w_4_83
3 4 1 0 4 1 3 2 3 2 3 -1/12-8*w_4_105+8*w_4_64 3 4 1 0 4 2 5 1 4 2 4 -16*w_4_60+16*w_4_86
3 4 1 0 4 1 3 2 3 2 4 -1/6-16*w_4_84+16*w_4_65 3 4 1 0 4 1 5 2 4 2 5 -16*w_4_61+16*w_4_87
3 4 1 0 4 1 3 2 4 2 4 -1/12-8*w_4_83+8*w_4_64 3 4 1 0 4 2 3 1 3 2 4 16*w_4_87-16*w_4_84
3 4 1 0 4 1 3 2 3 2 5 -1/18+16*w_4_66-16*w_4_91 3 4 1 0 4 1 5 2 6 2 3 16*w_4_92-16*w_4_723 4 1 0 4 2 5 2 6 1 3 16*w_4_92-16*w_4_72
3 4 1 0 4 1 3 2 4 2 5 -1/18+16*w_4_66-16*w_4_74 3 4 1 0 4 2 3 1 4 2 3 16*w_4_106-16*w_4_84
3 4 1 0 1 2 3 2 3 3 4 1/3+16*w_4_10 3 4 1 0 4 2 5 1 4 2 5 -1/6+16*w_4_106-16*w_4_61
3 4 1 0 1 2 3 2 6 3 5 1/6+16*w_4_22 3 4 1 0 4 2 3 1 4 2 5 -16*w_4_74+16*w_4_107
3 4 1 0 1 2 3 2 4 4 5 1/6+16*w_4_18 3 4 1 0 4 2 5 1 4 2 3 -1/18-16*w_4_63+16*w_4_107
3 4 1 0 1 2 3 2 4 3 5 1/6+16*w_4_17 3 4 1 0 4 2 3 1 6 2 3 16*w_4_107-16*w_4_91
3 4 1 0 1 2 3 2 6 3 4 1/6+16*w_4_21 3 4 1 0 4 2 5 1 6 2 5 -1/6+16*w_4_107-16*w_4_62
3 4 1 0 1 2 5 3 4 2 5 1/6+16*w_4_46 3 4 1 0 4 2 5 1 6 2 3 16*w_4_118-16*w_4_72
3 4 1 0 2 1 2 3 4 3 4 -1/6+8*w_4_3 3 4 1 0 1 2 3 2 4 3 4 16*w_4_16
3 4 1 0 2 1 2 3 4 3 5 -1/6+16*w_4_4 3 4 1 0 1 2 3 2 6 4 5 16*w_4_23
3 4 1 0 2 1 2 3 4 4 5 -16*w_4_4 3 4 1 0 1 2 5 2 4 3 4 16*w_4_40
3 4 1 0 2 1 2 3 6 4 5 16*w_4_5 3 4 1 0 1 2 5 2 6 3 4 16*w_4_41
3 4 1 0 2 1 3 2 3 3 4 16*w_4_24+16*w_4_10 3 4 1 0 1 2 5 2 6 3 5 16*w_4_42
3 4 1 0 4 1 2 2 4 3 4 1/9-16*w_4_24 # 1 1 1
3 4 1 0 2 1 3 2 3 3 5 -16*w_4_24+16*w_4_10 3 4 1 0 4 1 3 2 3 3 4 1/6+16*w_4_108+16*w_4_67
3 4 1 0 4 1 2 2 4 4 5 1/6-16*w_4_24 3 4 1 0 4 1 3 2 4 3 4 1/6-16*w_4_67+16*w_4_90
3 4 1 0 2 1 3 2 3 4 5 16*w_4_25 3 4 1 0 4 1 3 2 6 3 4 1/6+16*w_4_111
3 4 1 0 4 1 2 2 4 3 5 1/9+16*w_4_25 3 4 1 0 4 1 3 2 3 3 5 1/18-16*w_4_110+16*w_4_68
3 4 1 0 2 1 3 2 4 3 4 16*w_4_28+16*w_4_16 3 4 1 0 4 1 3 2 3 4 5 1/18-16*w_4_109+16*w_4_69
3 4 1 0 4 1 2 2 3 3 4 1/3-16*w_4_28 3 4 1 0 4 1 3 2 4 3 5 1/18+16*w_4_89+16*w_4_69
3 4 1 0 4 1 3 2 4 4 5 1/18+16*w_4_68+16*w_4_88
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION xv
Table 8 (part 5).
3 4 1 0 4 1 3 2 6 3 5 1/36+16*w_4_70-16*w_4_113 3 4 1 0 4 3 5 1 6 2 3 16*w_4_94-16*w_4_120
3 4 1 0 4 1 3 2 6 4 5 1/36-16*w_4_112+16*w_4_70 3 4 1 0 4 2 5 1 6 4 5 16*w_4_121-16*w_4_96
3 4 1 0 1 2 3 3 4 3 4 -1/6+8*w_4_34 3 4 1 0 4 3 5 2 3 1 5 16*w_4_95-16*w_4_121
3 4 1 0 1 2 3 3 4 4 5 1/6+16*w_4_36 3 4 1 0 4 2 5 3 4 1 4 16*w_4_122+16*w_4_103
3 4 1 0 1 2 5 3 4 3 4 -1/6+16*w_4_48 3 4 1 0 4 2 5 4 6 1 4 -16*w_4_122+16*w_4_97
3 4 1 0 1 2 5 3 4 4 5 1/6+16*w_4_50 3 4 1 0 4 2 5 3 4 1 5 16*w_4_123-16*w_4_124
3 4 1 0 2 1 3 3 4 3 4 16*w_4_34 3 4 1 0 4 5 6 1 6 2 4 16*w_4_123-16*w_4_98
3 4 1 0 4 1 2 3 4 3 4 -1/6+8*w_4_34 3 4 1 0 4 3 5 2 3 1 4 16*w_4_123-16*w_4_124
3 4 1 0 2 1 3 3 4 3 5 32*w_4_35 3 4 1 0 4 5 6 1 4 2 5 16*w_4_98-16*w_4_124
3 4 1 0 4 1 2 3 4 4 5 -16*w_4_35 3 4 1 0 4 2 5 3 6 1 5 16*w_4_128-16*w_4_129
3 4 1 0 2 1 3 3 4 4 5 32*w_4_36 3 4 1 0 4 3 5 1 6 2 4 -16*w_4_128+16*w_4_101
3 4 1 0 4 1 2 3 4 3 5 -1/6-16*w_4_36 3 4 1 0 4 3 5 2 6 1 4 -16*w_4_129+16*w_4_101
3 4 1 0 2 1 3 3 6 3 5 16*w_4_37 3 4 1 0 4 5 6 1 4 2 3 16*w_4_128-16*w_4_129
3 4 1 0 4 1 2 4 6 4 5 8*w_4_37 3 4 1 0 4 2 5 4 6 1 5 -16*w_4_134+16*w_4_133
3 4 1 0 2 1 3 3 6 4 5 32*w_4_38 3 4 1 0 4 3 5 2 4 1 5 -16*w_4_133+16*w_4_104
3 4 1 0 4 1 2 3 6 4 5 16*w_4_38 3 4 1 0 4 3 5 1 4 2 3 16*w_4_134-16*w_4_133
3 4 1 0 2 1 3 4 6 4 5 16*w_4_39 3 4 1 0 4 3 5 1 4 2 5 -16*w_4_134+16*w_4_104
3 4 1 0 4 1 2 3 6 3 5 8*w_4_39 3 4 1 0 4 3 5 2 4 1 4 32*w_4_138
3 4 1 0 2 1 5 3 4 3 4 32*w_4_48 3 4 1 0 4 5 6 1 4 2 4 16*w_4_138
3 4 1 0 4 3 5 1 2 3 5 -1/6+16*w_4_48 3 4 1 0 1 2 3 3 4 3 5 16*w_4_35
3 4 1 0 2 1 5 3 4 3 5 32*w_4_49 3 4 1 0 1 2 3 3 6 3 5 8*w_4_37
3 4 1 0 4 3 5 1 2 4 5 16*w_4_49 3 4 1 0 1 2 3 3 6 4 5 16*w_4_38
3 4 1 0 2 1 5 3 4 4 5 32*w_4_50 3 4 1 0 1 2 3 4 6 4 5 8*w_4_393 4 1 0 1 2 5 3 4 3 5 16*w_4_49
3 4 1 0 4 3 5 1 2 3 4 -1/6-16*w_4_50 3 4 1 0 1 2 5 3 6 3 4 16*w_4_51
3 4 1 0 2 1 5 3 6 3 4 32*w_4_51 3 4 1 0 1 2 5 3 6 3 5 16*w_4_52
3 4 1 0 4 5 6 1 2 3 5 -16*w_4_51 3 4 1 0 1 2 5 3 6 4 5 16*w_4_53
3 4 1 0 2 1 5 3 6 3 5 32*w_4_52 3 4 1 0 1 2 5 4 6 3 4 16*w_4_54
3 4 1 0 4 5 6 1 2 4 5 -16*w_4_52 3 4 1 0 1 2 5 4 6 3 5 16*w_4_55
3 4 1 0 2 1 5 3 6 4 5 32*w_4_53 3 4 1 0 4 2 5 3 6 1 4 16*w_4_100
3 4 1 0 4 5 6 1 2 3 4 16*w_4_53 3 4 1 0 4 3 5 1 4 2 4 16*w_4_138
3 4 1 0 2 1 5 4 6 3 4 32*w_4_54
3 4 1 0 4 3 5 3 6 1 2 -16*w_4_54
3 4 1 0 2 1 5 4 6 3 5 32*w_4_55
3 4 1 0 4 3 5 4 6 1 2 -16*w_4_55
3 4 1 0 4 1 5 2 3 3 4 16*w_4_80+16*w_4_81
3 4 1 0 4 2 5 1 3 3 5 -16*w_4_80+16*w_4_82
3 4 1 0 4 1 5 2 3 3 5 16*w_4_81+16*w_4_82
3 4 1 0 4 2 5 1 3 4 5 -16*w_4_80-16*w_4_81
3 4 1 0 4 1 5 2 3 4 5 -16*w_4_80+16*w_4_82
3 4 1 0 4 2 5 1 3 3 4 16*w_4_81+16*w_4_82
3 4 1 0 4 1 5 2 4 3 4 1/18+16*w_4_68+16*w_4_88
3 4 1 0 4 2 3 1 3 3 5 -16*w_4_110-16*w_4_88
3 4 1 0 4 1 5 2 4 3 5 1/18+16*w_4_89+16*w_4_69
3 4 1 0 4 2 3 1 3 4 5 -16*w_4_89-16*w_4_109
3 4 1 0 4 1 5 2 4 4 5 1/6-16*w_4_67+16*w_4_90
3 4 1 0 4 2 3 1 3 3 4 16*w_4_108+16*w_4_90
3 4 1 0 4 1 5 2 6 3 4 16*w_4_99+16*w_4_93
3 4 1 0 4 2 5 3 6 1 3 -16*w_4_93+16*w_4_119
3 4 1 0 4 1 5 2 6 3 5 16*w_4_102+16*w_4_94
3 4 1 0 4 2 5 4 6 1 3 -16*w_4_94+16*w_4_120
3 4 1 0 4 1 5 2 6 4 5 16*w_4_95-16*w_4_96
3 4 1 0 4 2 5 3 4 1 3 16*w_4_95-16*w_4_121
3 4 1 0 4 1 5 3 4 2 3 -16*w_4_121+16*w_4_96
3 4 1 0 4 3 5 1 3 2 5 16*w_4_95-16*w_4_96
3 4 1 0 4 1 5 3 4 2 4 16*w_4_103+16*w_4_97
3 4 1 0 4 3 5 1 3 2 3 16*w_4_122-16*w_4_97
3 4 1 0 4 1 5 3 4 2 5 16*w_4_98-16*w_4_124
3 4 1 0 4 3 5 1 3 2 4 16*w_4_123-16*w_4_98
3 4 1 0 4 1 5 3 6 2 3 16*w_4_99+16*w_4_119
3 4 1 0 4 5 6 1 3 2 5 16*w_4_99+16*w_4_93
3 4 1 0 4 1 5 3 6 2 4 32*w_4_100
3 4 1 0 4 5 6 1 3 2 3 16*w_4_100
3 4 1 0 4 1 5 3 6 2 5 -16*w_4_129+16*w_4_101
3 4 1 0 4 5 6 1 3 2 4 -16*w_4_128+16*w_4_101
3 4 1 0 4 1 5 4 6 2 3 16*w_4_102+16*w_4_120
3 4 1 0 4 3 5 2 6 1 3 16*w_4_102+16*w_4_94
3 4 1 0 4 1 5 4 6 2 4 16*w_4_103+16*w_4_97
3 4 1 0 4 3 5 2 3 1 3 16*w_4_122+16*w_4_103
3 4 1 0 4 1 5 4 6 2 5 -16*w_4_134+16*w_4_104
3 4 1 0 4 3 5 2 4 1 3 -16*w_4_133+16*w_4_104
3 4 1 0 4 2 3 1 4 3 4 16*w_4_108+16*w_4_90
3 4 1 0 4 2 5 1 4 4 5 -1/6-16*w_4_108-16*w_4_67
3 4 1 0 4 2 3 1 4 3 5 16*w_4_89+16*w_4_109
3 4 1 0 4 2 5 1 4 3 5 1/18-16*w_4_109+16*w_4_69
3 4 1 0 4 2 3 1 4 4 5 16*w_4_110+16*w_4_88
3 4 1 0 4 2 5 1 4 3 4 1/18-16*w_4_110+16*w_4_68
3 4 1 0 4 2 3 1 6 3 4 32*w_4_111
3 4 1 0 4 5 6 1 6 2 5 1/6+16*w_4_111
3 4 1 0 4 2 3 1 6 3 5 16*w_4_112-16*w_4_113
3 4 1 0 4 3 5 1 6 2 5 1/36-16*w_4_112+16*w_4_70
3 4 1 0 4 2 3 1 6 4 5 -16*w_4_112+16*w_4_113
3 4 1 0 4 3 5 2 6 1 5 1/36+16*w_4_70-16*w_4_113
3 4 1 0 4 2 5 1 6 3 4 16*w_4_99+16*w_4_119
3 4 1 0 4 5 6 1 6 2 3 -16*w_4_93+16*w_4_119
3 4 1 0 4 2 5 1 6 3 5 16*w_4_102+16*w_4_120
xvi R. BURING AND A. V. KISELEV
Table 9. Sample output of reduce_mod_jacobi.
3 4 1 0 1 0 3 2 6 3 4 -24+c_1_1221_211 3 4 1 0 4 1 5 3 6 2 3 -8+c_1_1023_111
3 4 1 0 4 1 3 2 6 3 4 -8+c_1_1240_111 3 4 1 0 4 5 6 1 3 2 5 -16+c_1_540_111
3 4 1 0 1 0 3 2 3 4 5 -48+c_1_1221_211 3 4 1 0 4 1 5 3 6 2 4 32-c_1_1242_111
-c_1_513_211 -c_1_540_111
3 4 1 0 1 0 3 2 4 3 5 24-c_1_1221_211 3 4 1 0 4 5 6 1 3 2 3 16-c_1_1023_111
3 4 1 0 4 1 3 2 4 3 5 24-c_1_1240_111 +c_1_1245_111
-c_1_540_111 3 4 1 0 4 1 5 4 6 2 3 -16+c_1_1242_111
3 4 1 0 1 2 3 3 4 4 5 -8-c_1_1228_111 3 4 1 0 4 3 5 2 6 1 3 -8-c_1_1245_111
3 4 1 0 1 2 5 3 4 3 4 -8-c_1_1228_111 3 4 1 0 4 2 3 1 4 3 5 24+c_1_538_111
3 4 1 0 2 0 3 1 4 3 5 24-c_1_1005_211 -c_1_1019_111
3 4 1 0 2 0 5 1 3 3 4 -24-c_1_516_211 3 4 1 0 4 2 3 1 6 3 4 -16+c_1_1019_111
3 4 1 0 2 0 3 1 6 3 4 -24+c_1_1005_211 3 4 1 0 4 5 6 1 6 2 5 -8+c_1_536_111
3 4 1 0 2 0 5 3 6 1 3 24+c_1_516_211 3 4 1 0 4 2 5 1 6 3 4 -8+c_1_1021_111
3 4 1 0 2 1 3 2 3 4 5 48+c_1_1230_112 3 4 1 0 4 5 6 1 6 2 3 8+c_1_538_111
-c_1_1008_112 3 4 1 0 4 2 5 1 6 3 5 -16+c_1_540_111
3 4 1 0 4 1 2 2 4 3 5 48-c_1_525_112 3 4 1 0 4 3 5 1 6 2 3 8-c_1_1023_111
-c_1_1239_112 3 4 1 0 4 2 5 3 4 1 5 -8+c_1_1021_111
3 4 1 0 2 1 3 2 4 3 5 -24-c_1_1230_112 3 4 1 0 4 5 6 1 6 2 4 8+c_1_538_111
3 4 1 0 4 1 2 2 3 4 5 -24+c_1_1239_112 3 4 1 0 4 3 5 2 3 1 4 -8+c_1_1023_111
3 4 1 0 2 1 5 2 3 3 4 24-c_1_1008_112 3 4 1 0 4 5 6 1 4 2 5 -16+c_1_540_111
3 4 1 0 4 2 5 1 2 3 5 -24+c_1_525_112 3 4 1 0 2 0 3 1 3 4 5 -48+c_1_1005_211-c_1_516_211
3 4 1 0 2 1 3 2 6 3 4 24+c_1_1230_112 3 4 1 0 1 0 5 2 3 3 4 -24-c_1_513_211
3 4 1 0 4 1 2 2 6 3 4 -24+c_1_1239_112 3 4 1 0 1 0 5 3 6 2 3 24+c_1_513_211
3 4 1 0 2 1 5 3 6 2 3 -24+c_1_1008_112 3 4 1 0 1 2 3 3 6 4 5 -8-c_1_1228_111
3 4 1 0 4 5 6 1 2 2 5 -24+c_1_525_112 3 4 1 0 1 2 5 3 6 3 5 8+c_1_1228_111
3 4 1 0 2 1 3 3 4 4 5 -16-c_1_1012_111 3 4 1 0 4 2 5 3 6 1 4 16+c_1_538_111-c_1_1021_111
3 4 1 0 4 1 2 3 4 3 5 8+c_1_529_111 3 4 1 0 4 1 3 2 3 4 5 -c_1_1023_111+c_1_1240_111
3 4 1 0 2 1 3 3 6 4 5 -16-c_1_1012_111 3 4 1 0 4 2 5 1 4 3 5 c_1_536_111-c_1_1021_111
3 4 1 0 4 1 2 3 6 4 5 -8-c_1_529_111
3 4 1 0 2 1 5 3 4 3 4 -16-c_1_1012_111 3 4 1 0 1 2 3 0 3 5 4 c_1_513_211==-24
3 4 1 0 4 3 5 1 2 3 5 -8-c_1_529_111 3 4 1 0 2 1 3 0 3 5 4 c_1_516_211==-24
3 4 1 0 2 1 5 3 6 3 5 16+c_1_1012_111 3 4 1 1 2 2 3 0 3 5 4 c_1_525_112==24
3 4 1 0 4 5 6 1 2 4 5 -8-c_1_529_111 3 4 1 1 2 3 5 0 3 5 4 c_1_529_111==-8
3 4 1 0 4 1 5 2 3 3 4 8-c_1_1023_111 3 4 1 1 4 2 3 0 3 5 4 c_1_536_111==8
3 4 1 0 4 2 5 1 3 3 5 -16+c_1_540_111 3 4 1 1 4 2 5 0 3 5 4 c_1_538_111==-8
3 4 1 0 4 1 5 2 3 3 5 -8-c_1_538_111 3 4 1 1 5 2 3 0 3 5 4 c_1_540_111==16
3 4 1 0 4 2 5 1 3 4 5 -8+c_1_1021_111 3 4 1 0 2 0 3 1 3 5 4 c_1_1005_211==24
3 4 1 0 4 1 5 2 3 4 5 -16+c_1_1242_111 3 4 1 0 2 2 3 1 3 5 4 c_1_1008_112==24
3 4 1 0 4 2 5 1 3 3 4 -8-c_1_1245_111 3 4 1 0 2 3 5 1 3 5 4 c_1_1012_111==-16
3 4 1 0 4 1 5 2 4 3 5 24-c_1_536_111 3 4 1 0 4 2 3 1 3 5 4 c_1_1019_111==16
-c_1_1242_111 3 4 1 0 4 2 5 1 3 5 4 c_1_1021_111==8
3 4 1 0 4 2 3 1 3 4 5 -24-c_1_1245_111 3 4 1 0 5 2 3 1 3 5 4 c_1_1023_111==8
+c_1_1019_111 3 4 1 0 1 0 3 2 3 5 4 c_1_1221_211==24
3 4 1 0 4 1 5 2 6 3 4 -16+c_1_1242_111 3 4 1 0 1 3 5 2 3 5 4 c_1_1228_111==-8
3 4 1 0 4 2 5 3 6 1 3 8+c_1_1245_111 3 4 1 0 2 1 3 2 3 5 4 c_1_1230_112==-24
3 4 1 0 4 1 5 2 6 3 5 -8-c_1_538_111 3 4 1 0 4 1 2 2 3 5 4 c_1_1239_112==24
3 4 1 0 4 2 5 4 6 1 3 -8+c_1_1021_111 3 4 1 0 4 1 3 2 3 5 4 c_1_1240_111==8
3 4 1 0 4 1 5 3 4 2 5 -16+c_1_1242_111 3 4 1 0 4 1 5 2 3 5 4 c_1_1242_111==16
3 4 1 0 4 3 5 1 3 2 4 8+c_1_1245_111 3 4 1 0 5 1 3 2 3 5 4 c_1_1245_111==-8
COMPUTER-ASSISTED PROOF SCHEMES IN DEFORMATION QUANTIZATION xvii
Appendix E. Gauge transformation that removes
4 master-parameters out of 10
Encodings of graphs (see Implementation 1 on p. 229) built over one sink vertex are
followed by their coefficients, in the following table containing the gauge transformation
which was claimed to exist in Theorem 14.
Table 10. Gauge transformation that removes 4 master-parameters out of 10.
h^0:
1 0 1 1
h^4:
1 4 1 0 2 0 3 1 4 0 3 16*w_4_101
1 4 1 0 2 0 3 1 4 1 3 8*w_4_101
1 4 1 0 2 0 3 1 4 2 3 8*w_4_101
1 4 1 0 2 1 3 0 4 1 2 -8*w_4_101
1 4 1 0 2 1 3 0 4 2 3 8*w_4_101
1 4 1 0 2 1 3 1 4 0 2 -8*w_4_101
1 4 1 0 2 1 3 2 4 0 2 -8*w_4_101
1 4 1 0 2 0 3 0 4 1 3 -16*w_4_102
1 4 1 0 2 0 3 1 4 1 3 -8*w_4_102
1 4 1 0 2 0 3 2 4 1 2 -8*w_4_102
1 4 1 0 2 0 3 2 4 1 3 -16*w_4_102
1 4 1 0 2 1 3 0 4 1 2 -8*w_4_102
1 4 1 0 2 1 3 0 4 1 3 -8*w_4_102
1 4 1 0 2 0 3 0 4 1 2 16*w_4_119
1 4 1 0 2 0 3 1 4 1 2 16*w_4_119
1 4 1 0 2 0 3 1 4 1 3 8*w_4_119
1 4 1 0 2 0 3 2 4 1 2 8*w_4_119
1 4 1 0 2 1 3 0 4 1 2 8*w_4_119
1 4 1 0 2 3 4 0 4 1 2 -8*w_4_119
1 4 1 0 2 0 3 0 1 1 2 -32*w_4_125
1 4 1 0 2 0 3 1 2 1 2 16*w_4_125
1 4 1 0 2 0 3 1 2 1 3 -16*w_4_125
1 4 1 0 2 0 3 1 2 2 3 16*w_4_125
1 4 1 0 2 0 3 1 4 1 2 16*w_4_125
1 4 1 0 2 0 3 1 4 1 3 -16*w_4_125
1 4 1 0 2 0 3 1 4 2 3 16*w_4_125

Chapter 12
Formality morphism as the
mechanism of ⋆-product
associativity: how it works
This chapter is based on the peer-reviewed journal publication R. Buring and A.V. Kise-
lev, Collected works Inst. Math., Kyiv 16:1, 22–43, 2019. (Preprint arXiv:1907.00639
[q-alg] – 16 p.) This paper follows the talk given by the dissertant at conference Sym-
metry & Integrability of Equations of Mathematical Physics (December 22–23, IM NASU
Kyiv, Ukraine, 2018).
Commentary. In reference to Part I of the dissertation, the material of this chapter is
used in Chapter 2 (§2.5) and Chapter 3. This chapter concludes the exposition about
star products from Chapters 10 and 11.
303
FORMALITY MORPHISM AS THE MECHANISM OF
⋆-PRODUCT ASSOCIATIVITY: HOW IT WORKS
RICARDO BURING1) AND ARTHEMY V. KISELEV2)
‘Symmetries & integrability of equations of mathematical physics’,
(22–24 December 2018, IM NASU Kiev, Ukraine)
Abstract. The formality morphism F = {Fn, n ⩾ 1} in Kontsevich’s deformation quan-
tization is a collection of maps from tensor powers of the differential graded Lie
algebra (dgLa) of multivector fields to the dgLa of polydifferential operators on fi-
nite-dimensional affine manifolds. Not a Lie algebra morphism by its term F1 alone,
the entire set F is an L∞-morphism instead. It induces a map of the Maurer–Cartan
elements, taking Poisson bi-vectors to deformations µA →7 ⋆A[[ℏ]] of the usual multipli-
cation of functions into associative noncommutative ⋆-products of power series in ℏ.
The associativity of ⋆-products is then realized, in terms of the Kontsevich graphs
which encode polydifferential operators, by differential consequences of the Jacobi
identity. The aim of this paper is to illustrate the work of this algebraic mechanism
for the Kontsevich ⋆-products (in particular, with harmonic propagators). We inspect
how the Kontsevich weights are correlated for the orgraphs which occur in the asso-
ciator for ⋆ and in its expansion using Leibniz graphs with the Jacobi identity at a
vertex.
Introduction. The Kontsevich formality morphism F relates two differential graded Lie
algebras (dgLa). Its domain of definition is the shifted-graded vector space T ↓[1] (Mrpoly )
of multivectors on an affine real finite-dimensional manifold Mr; the graded Lie algebra
structure is the Schouten bracket [[ , ]] and the differential is set to (the bracket with)
zero by definition. On the other hand, the target space of the formality morphism F is
the graded vector space D↓[1]poly(Mr) of polydifferential operators on Mr; the graded Lie
algebra structure is the Gerstenhaber bracket [ , ]G and the differential dH = [µA, ·] is
induced by using the multiplication µ ∞ r rA in the algebra A := C (M ) of functions on M .
It is readily seen that w.r.t. the above notation, Poisson bi-vectors P satisfying the
Jacobi identity [[P,P]] = 0 on Mr are the Maurer–Cartan elements (indeed, (d ≡ 0)(P)+
1
2 [[P,P]] = 0). Likewise, for a (non)commutative star-product ⋆ = µA[[ℏ]] + 〈tail =: B〉,
which deforms the usual multiplication µ = µA[[ℏ]] in A[[ℏ]] = C∞(Mr) ⊗R R[[ℏ]] by a
tail B w.r.t. a formal parameter ℏ, the requirement that ⋆ be associative again is the
Date: 1 July 2019.
2010 Mathematics Subject Classification. 05C22, 16E45, 53D55, secondary 53D17, 68R10, 81R60.
1) Address: Institut für Mathematik, Johannes Gutenberg–Universität, Staudingerweg 9, D-55128
Mainz, Germany. E-mail (corresponding author): rburing@uni-mainz.de .
2) Address: Bernoulli Institute for Mathematics, Computer Science and Artificial Intelli-
gence, University of Groningen, P.O.Box 407, 9700AK Groningen, The Netherlands. E-mail:
A.V.Kiselev@rug.nl .
304
FORMALITY MORPHISM AS THE MECHANISM OF ⋆-PRODUCT ASSOCIATIVITY 305
Maurer–Cartan equation,
[µ, B]G + 12 [B, B]G = 0 ⇐⇒
1
2 [µ + B, µ + B]G = 0.
Here, the leading order equality [µ, µ]G = 0 expresses the given associativity of the
product µ itself.
The Kontsevich formality mapping F = {Fn : T⊗npoly → Dpoly, n ⩾ 1} in [14, 15] is an
L∞-morphism which induces a map that takes Maurer–Cartan elements P, i.e. formal
Poisson bi-vectors P̃ = ℏP+ ō(ℏ) on Mr, to Maurer–Cartan elements1, i.e. the tails B in
solutions ⋆ of the associativity equation on A[[ℏ]].
The theory required to build the Kontsevich map F is standard, well reflected in the
literature (see [14, 15], as well as [9, 11] and references therein); a proper choice of signs
is analysed in [2, 18]. The framework of homotopy Lie algebras and L∞-morphisms,
introduced by Schlessinger–Stasheff [17], is available from [16], cf. [10] in the context of
present paper.
So, the general fact of (existence o
P (f) factoriz)ation,Assoc(⋆)( )( f , g, h) = ^ P, [[P,P]] ( f , g, h), f , g, h ∈ A[[ℏ]], (1)
is known to the expert community. Indeed, this factorization is immediate from the
construction of L∞-morphism in [15, §6.4]. We shall inspect how this mechanism works
in practice, i.e. how precisely the ⋆-product is made associative in its perturbative
expansion whenever the b(i-vector P is Poisson, thus satisfying the Jacobi identityJac(P) := 12 [[P,P]] = 0. To the sam)e extent as our paper [6] justifies a similar fac-torization, [[P,Q(P)]] = ^ P, [[P,P]] , of the Poisson cocycle condition for universal
deformations Ṗ = Q(P) of Poisson structures2, we presently motivate the findings in [5]
for ⋆ mod ō(ℏ3), proceeding to the next order ⋆ mod ō(ℏ4) from [7] (and higher or-
ders, recently available from [3]).3 Let us emphasize that the theoretical constructions
and algorithms (contained in the computer-assisted proof scheme under study and in
the tools for graph weight calculation) would still work at arbitrarily high orders of
expansion ⋆ mod ō(ℏk) as k → ∞. Explicit factorization (1) up to ō(ℏk) helps us build
the star-product ⋆ mod ō(ℏk) by using a self-starting iterative process, because the
Jacobi identity for P is the only obstruction to the associativity of ⋆. Specifically,
the Kontsevich weights of graphs on fewer vertices (yet with a number of edges such
that they do not show up in the perturbative expansion of ⋆) dictate the coefficients
of Leibniz orgraphs in operator ^ at higher orders in ℏ. These weights in the r.-h.s.
of (1) constrain the higher-order weights of the Kontsevich orgraphs in the expansion
of ⋆-product itself. This is important also in the context of a number-theoretic open
problem about the (ir)rational value (const ∈ Q \ {0}) · ζ(3)2/π6 + (const ∈ Q) of a graph
weight at ℏ7 in ⋆ (see [12] and [3]).
Our paper is structured as follows. First, we fix notation and recall some basic facts
from relevant theory. Secondly, we provide three examples which illustrate the work
1In fact, the morphism F is a quasi-isomorphism (see [15, Th. 6.3]), inducing a bijection between
the sets of gauge-equivalence classes of Maurer–Cartan elements.
2Universal w.r.t. all Poisson brackets on all finite-dimensional affine manifolds, such infinitesimal
deformations were pioneered in [14]; explicit examples of these flows Ṗ = Q(P) are given in [4, 8, 6].
3Note that both the approaches – to noncommutative associative ⋆-products and deformations of
Poisson structures – rely on the same calculus of oriented graphs by Kontsevich [13, 14, 15].
306 R. BURING AND A.V.KISELEV
of formality morphism in solving Eq. (1). Specifically, we read the operators ^k = ^
mod ō(ℏk) satisfying
P mod k ( )Assoc(⋆)( )( f , g, h) ō(ℏ ) = ^k P, [[P,P]] ( f , g, h) (1′)
at k = 2, 3, and 4. This corresponds to the expansions ⋆ mod ō(ℏk) in [15], [5], and [7],
respectively. One can then continue with k = 5, 6; these expansions are in [3]. Indepen-
dently, one can probe such factorizations using other stable formality morphisms: for
instance, the ones which correspond to a different star-product, the weights in which
are determined by a logarithmic propagator instead of the harmonic one (see [1]).
1. Two differential graded Lie algebra structures
Let Mr be an r-dimensional affine real manifold (we set k = R for simplicity). In the
algebra A := C∞(Mr) of smooth functions, denote by µA (or equivalently, by the dot ·)
the usual commutative, associative, bi-linear multiplication. The space of formal power
series in ℏ over A will be A[[ℏ]] and the ℏ-linear multiplication in it is µ (instead of µA[[ℏ]]).
Consider two differential graded Lie algebra stuctures. First, we have that the shifted-
graded space T ↓[1] rpoly(M ) of multivector fields on Mr is equipped with the shifted-graded
skew-symmetric Schouten bracket [[ , ]] (itself bi-linear by construction and satisfying the
shifted-graded Jacobi identity); the differential is set to zero. Secondly, the vector space
D↓[1] rpoly(M ) of polydifferential operators (linear in each argument but not necessarily skew
over the set of arguments or a derivation in any of them) is graded by using the number
of arguments m: by definition, let deg(θ(m arguments)) := m−1. For instance, deg(µA) =
1. The Lie algebra structure on D↓[1] (Mrpoly ) is the Gerstenhaber bracket [ , ]G; for two
homogeneous operators Φ1 and Φ it equals [Φ ,Φ ] = Φ ◦ ⃗Φ − (−)degΦ1·degΦ22 1 2 G 1 2 Φ2 ◦ ⃗Φ1,
where the directed, non-associative insertion product is, by definition
∑k1
(Φ ◦ ⃗Φ )(a , . . . , a ) = (−)ik ( )21 2 0 k1+k2 Φ1 a0⊗. . .⊗ai−1⊗Φ2(ai⊗. . .⊗ai+k2)⊗ai+k2+1⊗. . .⊗ak1+k2 .
i=0
In the above, Φ : A⊗(ki+1)i → A so that a j ∈ A. Like [[·, ·]], the Gerstenhaber bracket
satisfies the shifted-graded Jacobi identity. The Hochshild differential on D↓[1] rpoly(M ) is
dH = [µA, ·] 2G; indeed, its square vanishes, dH = 0, due to the Jacobi identity for [ , ]G
into which one plugs the equality [µA, µA]G = 0.
Example 1. The associativity of the product µA in the algebra of functions A = C∞(Mr)
is the statement that
µ(1)A (µ
(2)
A (a0, a1), a2) + (−1)(i=1)·(deg µA{=1)µ(1)A (a0, µ(2)A (a1, a2))(deg µ(1)− (−) =1)·(deg µ(2)=1) µ(1)(µ(1)(a , a ), a ) −{µ(2)(a , µ(1) }A A A A 0 1 2 A 0 A (a1, a2)) }
= 2 (a0 · a1) · a2 − a0 · (a1 · a2) = 0.
So, the associator Assoc(µA)(a0, a 11, a2) = 2 [µA, µA]G (a0, a1, a2) = 0 for any a j ∈ A.
FORMALITY MORPHISM AS THE MECHANISM OF ⋆-PRODUCT ASSOCIATIVITY 307
2. The Maurer–Cartan elements
In every differential graded Lie algebra with a Lie bracket [ , ], the Maurer–Cartan
(MC) elements are solutions of degree 1 for the Maurer–Cartan equation
dα + 12 [α, α] = 0, (2)
where d is the differential (equal, we recall, to zero identically on T ↓[1] (Mrpoly ) and dH =
[µA, ·]G on D↓[1] rpoly(M ). Likewise, the Lie algebra structure[·, ·] is the Schouten bracket
[[·, ·]] and Gerstenhaber bracket [·, ·]G, respectively.)
Now tensor the degree-one parts of both dgLa structures with ℏ·k[[ℏ]], i.e. with formal
power series starting at ℏ1, and, preserving the notation (that is, extending the brackets
and the differentials by ℏ-linearity), consider the same Maurer–Cartan equation (2). Let
us study its formal power series solutions α = ℏ1α1 + · · · .
So far, in the Poisson world we have that the Maurer–Cartan bi-vectors are formal
Poisson structures 0+ℏP1+ō(ℏ) satisfying (2), which is [[ℏP1 + ō(ℏ), ℏP1 + ō(ℏ)]] = 0 with
zero differential. In the world of associative structures, the Maurer–Cartan elements
are the tails B in expansions ⋆ = µ + B, so that the associativity equation [⋆,⋆]G = 0
reads (for [µ, µ]G = 0)
[µ, B] 1G + 2 [B, B]G = 0,
which is again (2).
3. The L∞-morphisms
Our goal is to have (and use) a morphism T ↓[1] rpoly(M ) → D
↓[1] r
poly(M ) which would induce
a map that takes Maurer–Cartan elements in the Poisson world to Maurer–Cartan
elements in the associative world.
The leading term F1, i.e. the first approximation to the morphism which we consider,
is the Hochschild–Kostant–Rosenberg (H∑KR) map (obviously, extended by linearity),
F : ξ1 ∧ ∧ →7
1
. . . ξ (−)σm ξσ(1) ⊗ . . . ⊗ ξm! ∈S σ(m),σ m
which takes a split multi-vector to a polydifferential operator (in fact, an m-vector).
More explicitly, we have that ( ∑ ∏ )m
F1 : (ξ1 ∧ . . . ∧
1
ξm) →7 a1 ⊗ . . . ⊗ am 7→ (−)σ ξσ(i)(ai) , (3)m! σ∈Sm i=1
here a ∈ A := C∞j (Mr). For zero-vectors h ∈ A, one has F1 : h 7→ (1 7→ h).
Claim 1 ([15, §4.6.2]). The leading term, map F1, is not a Lie algebra morphism (which,
if it were, would take the Schouten bracket of multivectors to the Gerstenhaber bracket
of polydifferential operators).
Proof (by counterexample). Take two bi-vectors; their Schouten bracket is a tri-vector,
but the Gerstenhaber bracket of two bi-vectors is a differential operator which has
homogeneous components of differential orders (2,1,1) and (1,1,2). And in general,
those components do not vanish. □
308 R. BURING AND A.V.KISELEV
The construction of not a single map F1 but of an entire collection F = {Fn, n ⩾ 1}
of maps does nevertheless yield a well-defined mapping of the Maurer–Cartan elements
from the two differential graded Lie algebras.4
Theorem 2 ([15, Main Theorem]). There exists a collection of linear maps F ↓[1]= {Fn : T r ⊗npoly(M ) →
D↓[1]poly(M
r), n ⩾ 1} such that F1 is the H(KR map (3) and F is a)n L∞-morphism of thetwo differen)tial graded Lie algebras: T ↓[1] (Mr (poly ), [[·, ·]], d = 0 → D↓[1] rpoly(M ), [·, ·]G,
dH = [µA, ·]G . Namely,
(1) each component Fn is homogeneous of own grading 1 − n,
(2) each morphism Fn is graded skew-symmetric, i.e.
Fn(. . . , ξ, η, . . .) = −(−)deg(ξ)·deg(η)Fn(. . . , η, ξ, . . .)
for ξ, η homogeneous,
(3) for each n ⩾ 1 and (homogeneous) multivectors ξ1, . . ., ξ ∈ T ↓[1] rn poly(M ), we have
that (cf. [11, §3.6]) ∑n
d (F (ξ , . . . , ξ )) − (−)n−1H n 1 ∑n ∑ (−)
uFn(ξ1, . . . , dξi, . . . , ξn)
i=1 [ ]
+ 12 p+q=n (−)
pn+t F∑p(ξσ(1), . . . , ξσ((p)),F∈S q(ξσ(p+1), . . . , ξσ(n))σ Gp,q>0 p,q )
= (−)n (−)sFn−1 [ξi, ξ j], ξ1, . . . , ξ̂i, . . . , ξ̂ j, . . . , ξ . (4)i n< j
In the above formula, σ runs through the set of (p, q)-shuffles, i.e. all permuta-
tions σ ∈ S n such that σ(1) < . . . < σ(p) and independently σ(p+1) < . . . < σ(n);
the exponents t and s are the numbers of transpositions of odd elements which
we count when passing (t) from (Fp, Fq, ξ1, . . ., ξn) to (Fp, ξσ(1), . . ., ξσ(p), Fq,
ξσ(p+1), . . ., ξσ(n)), and (s) from (ξ1, . . ., ξn) to (ξi, ξ j, ξ1, . . ., ξ̂1, . . ., ξ̂ j, . . ., ξn).5
Remark 1. Let n := 1, then equality (4) in Theorem 2 is
dH ◦ F − (−)1−1 · (−)u=0 from (d,ξ1)→7 (d,ξ1)1 F1 ◦ d = 0 ⇐⇒ dH ◦ F1 = F1 ◦ d,
whence F1 is a morphism of complexes.
• (Let n :=) 2[, then for any] homog(eneous m)ultiv(ectors ξ1 andF )ξ2,[[ξ , ξ ]] − F (ξ ),F (ξ ) = d F (ξ , ξ ) +F (d = 0)(ξ ), ξ +(−)deg ξ ( )11 1 2 1 1 1 2 G H 2 1 2 2 1 2 F2 ξ1, (d = 0)(ξ2) ,
so that in our case F1 is “almost” a Lie algebra morphism but for the discrepancy
which is controlled by the differential of the (value of the) succeeding map F2 in the
sequence F = {Fn, n ⩾ 1}. Big formula (4) shows in precisely which sense this is also the
case for higher homotopies Fn, n ⩾ 2 in the L∞-morphism F . Indeed, an L∞-morphism
is a map between dgLas which, in every term, almost preserves the bracket up to a
homotopy dH ◦ {. . .} provided by the next term.
4The name ‘Formality’ for the collection F of maps is motivated by Theorem 4.10 in [15] and by
the main theorem in loc. cit.
5The exponent u is not essential for us now because the differential d on T ↓[1] (Mrpoly ) is set equal to
zero identically, so that the entire term with u does not contribute (recall Fn is linear).
FORMALITY MORPHISM AS THE MECHANISM OF ⋆-PRODUCT ASSOCIATIVITY 309
Even though neither F1 nor the entire collection F = {Fn, n ⩾ 1} is a dgLa morphism,
their defining property (4) guarantees that F gives us a well defined mapping of the
Maurer–Cartan elements (which, we recall, are formal Poisson bi-vectors and tails B of
associative (non)commutative multiplcations ⋆ = µ + B on A[[ℏ]], respectively).
Corollary 3. The natural ℏ-linear extension of F , now acting on the space of formal
power series in ℏ with coefficients in∑T ↓[1] rpoly(M ) and with zero free term by the rule
ξ 7→ 1 Fn(ξ, . . . , ξ),n⩾1 n!
t∑akes the Maurer–Cartan elements P̃ = ℏP + ō(ℏ) to the Maurer–Cartan elements B =1
n⩾1 n!Fn(P̃, . . . , P̃) = ℏP̃ + ō(ℏ). (Note that the HKR map F1, extended by ℏ-linearity,
still is an identity mapping on multivectors, now viewed as special polydifferential
operators.)
In plain terms, for a bivector P itself Poisson, formal Poisson structures P̃ = ℏP+ ō(ℏ)
satisfying [[P̃, P̃]] = 0 are mapped by F to the tails B = ℏP + ō(ℏ) such that ⋆ = µ + B
is associative and its leading order deformation term is a given Poisson structure P.
Proof (of Corollary 3). Let us presently consider the restricted case when P̃ = ℏP, with-
out any higher order tail ō(ℏ). The M∑aurer–Cartan equation in D↓[1] rpoly(M ) ⊗ ℏk[[ℏ]] is
[µ, B∑] 1G + 2 [B, B]G = 0, where B = 1n⩾1 n!Fn(P̃, . . . , P̃) and we let P̃ = ℏP, so thatn
B = ℏn⩾1 n!Fn(P, . . ., P). Let us plug this formal power series in the l.-h.s. of the above
equation. Equating the coefficients at powers ℏn and multiplying by n!, we obtain the
expression ∑ n! [ ]
[µ,Fn(P, . . . ,P)] 1G + 2 ∑p+q=n Fp!q! p(P, . . . ,P),Fq(P, . . . ,P) G.p,q>0It is readily seen that now the sum σ∈S in (4) over the set of (p, q)-shuffles of n = p+qp,q
identical copies of an object P just counts the number of ways to pick p copies going first
in an ordered string of length n. To balance the signs, we note at once that by item 2
in Theorem 2, see above, F (α)p(. . . ,P ,P(α+1), . . .) = +Fp(. . . ,P(α+1),P(α), . . .) because bi-
vector’s shifted degree is +1, so that no (p, q)-shuffles of (P, . . . ,P) contribute with
any sign factor. The only sign contribution that remains stems from the symbol Fq of
grading 1−q transported along p copies of odd-degree bi-vector P; this yields t = (1−p)·q
and (−)pn+t = (−)p·(p+q) · (−)(1−q)·p = (−)p·(p+1) = +.
The left-hand side of the Maurer–Cartan equation (2) is, by the above, expressed
by the left-hand side of (4) which the L∞-morphism F satisfies. In the right-hand
side of (4), we now obtain (with, actually, whatever sign factors) the values of linear
mappings Fn−1 at twice the Jacobiator [[P̃, P̃]] as one of the arguments. All these
values are therefore zero, which implies that the right-hand side of the Maurer–Cartan
equation (2) vanishes, so that the tail B indeed is a Maurer–Cartan element in the
Hochschild cochain complex (in other words, the star-product ⋆ = µ+ B is associative).
This completes the proof in the restricted case when P̃ = ℏP. Formal power series
bi-vectors P̃ = ℏP + ō(ℏ) refer to the same count of signs as above, yet the calcula-
tion of multiplicities at ℏn (for all possible lexicographically ordered p- and q-tuples of
n arguments) is an extensive exercise in combinatorics. □
310 R. BURING AND A.V.KISELEV
Corollary 4. Because the right-hand side of (2) in the above reasoning is determined by
the right-hand side of (4), we read off an explicit formula of the operator ^ that solves
the factorization problem ( )
Assoc(⋆)(P)( f , g, h) = ^ P, [[P,P]] ( f , g, h), f , g, h ∈ A[[ℏ]]. (1)
Indeed, the operator is ∑ ℏn
^ = 2 · · cn · F
(
n−1 [[P,P]],P, . . . ,P
)
. (5)
n⩾1 n!
But what are the coefficients cn ∈ R equal to? Let us find it out.
4. Explicit construction of the formality morphism F
The first explicit formula for the formality morphism F which we study in this paper
was discovered by Kontsevich in [15, §6.4], providing an expansion of every term Fn
using weighted decorated graph{s: ∑ ∑ }
F = Fn = WΓ · Um Γ .⩾0 Γ∈Gn,m
Here Γ belongs to the set Gn,m of oriented graphs on n internal vertices (i.e. arrowtails),
m sinks (from which no arrows start), and 2n + m − 2 ⩾ 0 edges, such that at every
internal vertex there is an ordering of outgoing edges. By decorating each edge with
a summation index that runs from 1 to r, by viewing each edge as a derivation ∂/∂xα
of the arrowhead vertex content, by placing n multivectors from an ordered tuple of
arguments of Fn into the respective vertices, now taking the sum over all indices of the
resulting products of the content of vertices, and skew-symmetrizing over the n-tuple
of (shifted-)graded multivectors, we realize each graph at hand as a polydifferential
operator T ↓[1] r ⊗n ↓[1] rpoly(M ) → Dpoly(M ) whose arguments are multivectors. Note that the
value Fn(ξ1, . . . , ξn) itself is, by construction, a differential operator w.r.t. the contents
of sinks of the graph Γ. All of this is discussed in detail in [13, 14, 15] or [4, 5, 7].
The formula for the harm(∏onic weights)WΓ ∈ R is giv∫en in∧[15, §6.2]; it isn 1
WΓ = ·
1
#Star(k)! (2π)2n+m−2
dϕe,
+
k=1 C̄n,m e∈EΓ
where # Star(k) is the number of edges starting from vertex k, dφe is the “harmonic
angle” differential 1-form associated to the edge e, and the integration domain C̄+n,m is
the connected component of C̄n,m which is the closure of configurations where points q j,
1 ⩽ j ⩽ m on R are placed in increa(sing order: q)1 < · · · < qm. For convenience, let usalso define ∏n
wΓ = #Star(k)! ·WΓ.
k=1
The convenience is that by summing over labelled graphs Γ, we actually sum over
the equivalence classes [Γ] (i.e. over unlabeled graphs) with multiplicities (wΓ/WΓ) ·
n!/#Aut(Γ). The division by the volume #Aut(Γ) of the symmetry group eliminates the
repetitions of graphs which differ only by a labeling of vertices but, modulo such, do
not differ by the labeling of ordered edge tuples (issued from the vertices which are
matched by a symmetry).
FORMALITY MORPHISM AS THE MECHANISM OF ⋆-PRODUCT ASSOCIATIVITY 311
Let us remember that the integrand in the formula of WΓ is defined in terms of the
harmonic propagator; other propagators (e.g. logarithmic, or other members of the
family interpolating between harmonic and logarithmic [1]) would give other formality
morphisms. A path integral realization of the ⋆-product itself and of the components Fn
in the formality morphism is proposed in [10].
To calculate the graph weights WΓ in practice, we employ methods which were out-
lined in [7], as well as [12, App. E] (about the cyclic weight relations), and [3] that
puts those real values in the context of Riemann multiple zeta functions and polylog-
arithms.6 Examples of such decorated oriented graphs Γ and their weights WΓ will be
given in the next section.
4.1. Sum over equivalence classes. The sum in Kontsevich’s formula is over labeled
graphs: internal vertices are numbered from 1 to n, and the edges starting from each
internal vertex k are numbered from 1 to #Star(k). Under a re-labeling σ : Γ 7→ Γσ of
internal vertices and edges it is seen from the definitions that the operator UΓ and the
weight WΓ enjoy the same skew-symmetry property (as remarked in [15, §6.5]), whence
WΓ · UΓ = WΓσ · UΓσ . It follows that the sum over labeled graphs can be replaced by
a sum over∏equivalence classes [Γ] of graphs, modulo labeling of internal vertices andedges. For this it remains to count the size of an equivalence class: the edges can belabeled in nk=1 #Star(k)! ways, while the n internal vertices can be labeled in n!/#Aut(Γ)
ways.
Example 2. The double wedge on two ground vertices has only one possible labeling of
vertices, due to the aut(o∏morphism tha) t interchanges the wedges.We denote by M nΓ = k=1 #Star(k)! · n!/#Aut(Γ) the multiplicity of the graph Γ, and
let Ḡn,m be the set of equivalence classes [Γ] modulo labeling of Γ ∈ Gn,m. The formula
for the formality morphism {can th∑en be ∑rewritten as }
F = Fn = M ·W · U ;m⩾0 [Γ]∈Ḡ Γ Γ Γn,m
here the Γ in MΓ ·WΓ · UΓ is any representative of [Γ]. Any ambiguity in signs (due to
the choice of representative) in the latter two factors is cancelled in their product. Note
that the factor (∏n )k=1 #Star(k)! in MΓ kills the corresponding factor in WΓ, as remarked
in [15, §6.5].
4.2. The coefficient of a graph in the ⋆-product. The ⋆-product associated to a Poisson
structure P is give∑n by Corollary 3:ℏn ∑ ℏn ∑
⋆ = µ + Fn(P, . . . ,P) = µ + MΓ ·WΓ · UΓ(P, . . . ,P).n! n!
n⩾1 n⩾1 [Γ]∈Ḡn,2
For a graph Γ ∈ Gn,2 such that each internal vertex has two outgoing edges (these are the
only graphs that contribute, because we insert bi-vectors) we have M nΓ = 2 ·n!/#Aut(Γ).
In total, the coefficient of UΓ(P, . . . ,P) at ℏn is 2n/#Aut(Γ) ·WΓ = wΓ/#Aut(Γ). The skew-
symmetrization without prefactor of bi-vector coefficients in UΓ(P, . . . ,P) provides an
extra factor 2n.
6It is the values wΓ instead of WΓ which are calculated by software [3].
312 R. BURING AND A.V.KISELEV
Example 3 (at ℏ1). The coefficient of the wedge graph is 1/2 and the operator is 2P,
hence we recover P.
4.3. The coefficient of a Leibniz graph in the associator. The factorizing operator ^ for
Assoc(⋆) is given by C∑orollary 4:ℏn
^ = 2 · · cn · F
(
n−1 [[P,P]],P, . . . ,P
)
∑ n!n⩾1
· ℏ
n ∑
· · (= 2 cn MΓ ·WΓ · UΓ [[P,P]],P ), . . . ,P .n!
n⩾1 [Γ]∈Ḡn−1,3
For a graph Γ ∈ Gn−1,3 where one internal vertex has three outgoing edges and the
rest have two, we have MΓ = 3! · 2n−2 · (n − 1)!/#Aut(Γ). In total, the coefficient of
UΓ([[P,P]],P, . .[. ,P) at ℏn is1 ]· · · · n−2 · − · W [Γ · c ]n · w2 cn 3! 2 (n 1)! Γ= 2n! #Aut(Γ) n #Aut(Γ)
The skew-symmetrization without prefactor of bi- and tri-vector coefficients in the op-
erator UΓ([[P,P]],P, . . . ,P) provides an extra factor 3! · 2n−2.
Example 4 (at ℏ2). The coefficient of the tripod graph is c 12 · 3! and the operator is
3! · [[P,P]], hence we recover c2[[P,P]] = 23 Jac(P). (The right-hand side is known from
the associator, e.g. from [5].) This yields c2 = 1/3. In addition, we see that the HKR
map F1 acts here by the identity on [[P,P]].
In the next section, we shall find that at ℏn, the coefficients of our Leibniz graphs
(with Jac(P)[inserted] in[stead]of [[P,P]]) are[[P, P]] · c w w cP 3! · 2n−2 · 2 · n · Γ = 2n · Γ , so 3! · 2n · n = 2n.Jac( ) n #Aut(Γ) #Aut(Γ) n
We deduce that cn = n/3! = n/6 in all our experiments.
Conjecture. For all n ⩾ 2, the coefficients in (5) are cn = n/3! = n/6 (hence, the
coefficients of markers Γ for equivalence classes [Γ] of the Leibniz graphs in (5) are
2n ·wΓ/#Aut(Γ)), although it still remains to be explained how exactly this follows from
the L∞ condition (4).
5. Examples
Let P be a Poisson bi-vector∑on an affine manifold Mr. We inspect the asssociativityof the star-product n⋆ = µ + ℏn⩾1 n!Fn(P, . . ., P) given by Corollary 3 by illustrating
the work of the factorization mechanism from Corollary 4. The powers of deformation
parameter ℏ provide a natural filtration ℏ2 · A(2) + ℏ3 · A(3) + ℏ4 · A(4) + ō(ℏ4) so that we
verify the vanishing of Assoc(⋆)(P)(·, ·, ·) mod ō(ℏ4) for ⋆ mod ō(ℏ4) order by order.
At ℏ0 there is nothing to do (indeed, the usual multiplication is associative). All
contribution to the associator of ⋆ at ℏ1 cancels out because the leading deformation
term ℏP in the star-product ⋆ = µ + ℏP + ō(ℏ) is a bi-derivation. The order ℏ2 was
discussed in Example 4 in §4.3.
FORMALITY MORPHISM AS THE MECHANISM OF ⋆-PRODUCT ASSOCIATIVITY 313
Remark 2. In all our reasoning at any order ℏn⩾2, the Jacobiator in Leibniz graphs is
expanded (w.r.t. the three cyclic permutations of its arguments) into the Kontsevich
graphs, built of wedges, in such a way that the internal edge, connecting two Poisson
bi-vectors in Jac(P), is proclaimed Left by construction. Specifically, the algorithm to
expand each Leibniz graphs is as follows:
(1) Split the trivalent vertex with ordered targets (a, b, c) into two wedges: the first
wedge stands on a and b (in that order), and the second wedge stands on the
first wedge-top and c (in that order), so that the internal edge of the Jacobiator
is marked Left, preceding the Right edge towards c.
(2) Re-direct the edges (if any) which had the tri-valent vertex as their target, to
one of the wedge-tops; take the sum over all possible combinations (this is the
iterated Leibniz rule).
(3) Take the sum over cyclic permutations of the targets of the edges which (initially)
have (a, b, c) as their targets (this is the expansion of the Jacobiator).
5.1. The order ℏ3. To factorize the next order expansion of the associator, Assoc(⋆)(P)
mod ō(ℏ3) = ℏ2 · A(2) + ℏ3 · A(3) + ō(ℏ3), at ℏ3 in the operator ^ in the right-hand side
of (1), we use graphs on n − 1 = 2 vertices, m = 3 sinks, and 2(n − 1) +m − 2 = 5 edges.
At ℏ3, two internal vertices in the Leibniz graphs in the r.-h.s. of factorization (1) are
manifestly different: one vertex, containg the bi-vector P, is a source of two outgoing
edges, and the other, with [[P,P]], of three. Therefore, the automorphism groups of
such Leibniz graphs (under relabellings of internal vertices of the same valency but
with the sinks fixed) can only be trivial, i.e. one-element. (This will not necessarily be
the case of Leibniz graphs on (n − 2) + 1 internal vertices at ℏ⩾4: compare Examples 8
vs 9 on p. 316 below, where the weight of a graph is divided further by the size of its
automorphism group.)
The coefficient of ℏ3 in the factorizin∑g operator ^,1 (
coeff(^, ℏ3) = 2 · · c · M ·W · U [[P,P]],P ), . . . ,P ,
3! 3 Γ Γ Γ
[Γ]∈Ḡ2,3
expands into a sum of ⩽ 24 admissible oriented graphs. Indeed, there are six essentially
different oriented graph topologies, filtered by the number of sinks on which the tri-
vector [[P,P]] and bi-vector P stand; the ordering of sinks in the associator then yields
3 + 3 + 3 × 2 + 3 × 2 + 3 = 24 oriented graphs. (None of them is a zero orgraph.) As
we recall from [5], only thirteen of them actually occur with nonzero coefficients in the
term A(3) ∼ ℏ3 in Assoc(⋆)(P)), the remaining eleven have zero weights.7 The weights
of 15 relevant oriented Leibniz graphs from [5] are listed in Table 1.8
7Yet, these seemingly ‘unnecessary’ graphs can contribute to the cyclic weight relations (see [12,
App. E]): zero values of some of such graph weights can simplify the system of linear relations between
nonzero weights.
8To get the values, one uses the software [3] by Banks–Panzer–Pym or, independently, exact symbolic
or approximate numeric methods from [7], also taking into account the cyclic weight relations from [12,
App. E].
314 R. BURING AND A.V.KISELEV
Table 1. Weights wΓ of oriented Leibniz graphs Γ in coeff(^, ℏ3).
(S ) = [01; 012] 1 (S ) = [12; 012] 1 (S ) = [20; 012] −1f 221 12 g 122 12 h 212 12
(I ) = [02; 312] 1 (I ) = [12; 032] 1 (S ) = [24; 012] −1f 112 48 g 112 48 h 112 24
(S ) 1 −1 −1f 211 = [04; 012] 24 (Ig)211 = [10; 032] 48 (Ih)211 = [20; 013] 48
(I f )111 = [04; 312] 148 (Ih)111 = [24; 013]
−1
48 (Ig)111 = [14; 032] 0
(Sg)111 = [14; 012] 0 (I f )121 = [01; 312] 124 (Ih)121 = [21; 013]
−1
24
Here we let(by definition #r  #r  #r  
I f := ∂ j Jac(P
)
)(Pi j, g, h) ∂ f = r r	r@ r − r rr	
Ri " !"H r!−
@Rr
 	r@R @( ) Rj ? j ?r
 

 Hj
 j
" = 0.	
  rr 	r@Rr? !
Likewise, Ig := ∂ j Jac(P)( f ,Pi j, h) ·r∂#ig and
(
Ih := r∂#j Jac(P)( f , g ,rP#
i j) · ∂ih, resp ectively.9We also set
AHj r r r ri r 	r@ r iA
Hj L HjiA
S := Pi j 
R @Rf ∂ j Jac(P)(∂i f , g, h) =A"@ @!−A
	
"
rH A r
AU	R R AU r 
r
 Hjr!−"AU	r 	r@Rr!= 0.
Similarly, we let S := Pi j∂ Jac(P)( f , ∂ g, h) = 0 and S := Pi jg j i h ∂ j Jac(P)( f , g, ∂ih) = 0.
Note that after all the Leibniz rules are reworked, each of the six graphs I f , . . ., S h
–with the Jacobiator Jac(P) = 12 [[P,P]] at the tri-valent vertex – splits into several
homogeneous components, like (I f )111 or (S h)212; taken alone, each of the components
encodes a zero polydifferential operator of respective orders.
Claim 5. Multiplied by a common factor ([[P,P )]]/ Jac(P) · 2k−1 = 2 · 4 = 8, the Leibniz
graph weights from Table 1 at ℏ3 fully reproduce the factorization which was found in
the main Claim in [5], namely:
A(3) = 2 (S ) (3) 2 (3) 2221 3 f 221, A122 = 3 (S g)122, A212 = −3 (S h)212,
A(3) 1 − (3) ((= (I I ) , A = 1 I + 1 1 )111 6 f h 111 112 6 f 6 Ig − 3S h)112,
A(3) 1121 = 3 (I
(3) 1 1 1
f − Ih)121, A211 = 3S f − 6 Ig − 6 Ih 211.
Otherwise speaking, the sum of these Leibniz oriented graphs with these weights (times
2 · 4 = 8), when expanded into the sum of 39 weighted Kontsevich graphs (built only of
wedges), equals identically the ℏ3-proportional term in the associator Assoc(⋆)(P)( f , g, h).
Proof scheme. The encodings of weighted Kontsevich-graph expansions of the homo-
geneous components of the weighted Leibniz graphs I f , . . ., S h, which show up in the
associator at ℏ3 and which are processed according to the algorithm in Remark 2, are
listed in Appendix A. Reducing that collection modulo skew symmetry at internal ver-
tices, we reproduce, as desired, the entire term A(3) in the expansion ℏ2·A(2)+ℏ3·A(3)+ō(ℏ3)
of the associator Assoc(⋆)(P) mod ō(ℏ3). □
9In [5], the indices i and j were interchanged in the definitions of both Ig and Ih (compare the
expression of I f ); that typo is now corrected in the above formulae.
FORMALITY MORPHISM AS THE MECHANISM OF ⋆-PRODUCT ASSOCIATIVITY 315
Three examples, corresponding to the leftmost column of equalities in Claim 5, illus-
trate this scheme at order ℏ3. The three cases differ in that for A(3)221 in Example 5, there
is just one Leibniz graph without any arrows acting on the Jacobiator vertex. In the
other Example 6 for A(3)121, there are two Leibniz graphs still without Leibniz-rule actions
on the Jacobiators in them, so that we aim to show how similar terms are collected.10
Finally, in Example 7 about A(3)111 there are two Leibniz graphs with one Leibniz rule
action per either graph: an arrow targets the two internal vertices in the Jacobiator.
Example 5. Take the Leibniz graph (Sf )221 = [01; 012]. Its weight is 1/12. Multiplying
the Leibniz graph by 8 t)imes its weight and expanding the Jacobiator (there are noLeibniz rules to expand) yields the sum of three Kontsevich graphs: 2(3 [01; 01; 42] +
[01; 12; 40] + [01; 20; 41] . This is identically equal to the differential order (2, 2, 1)
homogeneous part A(3)221 of Assoc(⋆)(P) at ℏ3. For instance, these terms are listed in [7,
App. D].
Example 6. Take the Leibniz graphs (I f )121 = [01; 312] and (Ih)121 = [21; 013]. Their
weights are 1/24 and −1/24, respectively; multiply them by 8. Expanding the Jacobiator
in the linear combination 13 (I f−Ih)121 yields the sum of Kontsevich gr)aphs 1(3 [01; 31; 42]+
[01; 12; 43] + [01; 23; 41] − [21; 01; 43] − [21; 13; 40] − [21; 30; 41] . The two Leibniz
graphs have a Kontsevich graph in common: [01; 12; 43] = [21; 01; 43] (recall that
internal vertex labels can be permuted at no cost and the swap L ⇄ R at a wedge costs
a minus sign). This gives one cancellation; the remaining four terms equal A(3)121 as listed
in [7, App. D].
Example 7. Take the Leibniz graphs (I f )111 = [04; 312] and (Ih)111 = [24; 013]. Their
weights are 1/48 and −1/48, respectively; multiply them by 8. Expanding the Jacobiator
and the Leibniz rule in the linear combination 16 (I f − Ih)111 yields the sum of Kontsevich
grap
1( hs:
6 [04; 31; 42] + [04; 12; 43] + [04; 23; 41] + [05; 31; 42] + [05; 12; 43] + [05; 23; 41]
− )[24; 01; 43] − [24; 13; 40] − [24; 30; 41] − [25; 01; 43] − [25; 13; 40] − [25; 30; 41] .
Two pairs of graphs cancel; namely [05; 31; 42] = [25; 30; 41] and [05; 23; 41] = [25; 13; 40].
The remaining eight terms equal A(3)111 as listed in [7, App. D].
5.2. The order ℏ4. Let us proceed with the term A(4) at ℏ4 in the associator Assoc(⋆)(P)(·, ·, ·)
mod ō(ℏ4). The numbers of Kontsevich oriented graphs in the star-product expansion
grow as fast as
⋆ = ℏ0 · (#graphs = 1) + ℏ1 · (# = 1) + ℏ2 · (# = 4) + ℏ3 · (# = 13) + ℏ4 · (# = 247)+
+ ℏ5 · (# = 2356) + ℏ6 · (# = 66041) + ō(ℏ6);
here we report the count of all nonzero-weight Kontsevich oriented graphs. Counting
them modulo automorphisms (which may also swap the sinks), Banks, Panzer, and Pym
10To collect and compare the Kontsevich orgraphs (built of wedges, i.e. ordered edge pairs issued
from internal vertices), we can bring every such graph to its normal form, that is, represent it using
the minimal base-(# sinks + # internal vertices) number, encoding the graph as the list of ordered pairs
of target vertices, by running over all the relabellings of internal vertices. (The labelling of ordered
sinks is always 0 ≺ 1 ≺ . . . ≺ m − 1.)
316 R. BURING AND A.V.KISELEV
obtain the numbers (ℏ0 : 1, ℏ1 : 1, ℏ2 : 3, ℏ3 : 8, ℏ4 : 133, ℏ5 : 1209, ℏ6 : 33268). This
shows that at orders ℏk⩾4, the use of graph-processing software is indispensible in the
task of verifying factorization (1) using weighted g(raph expan)sion (5) of the operator ^.Specifically, the number of Kontsevich oriented graphs at ℏk in the left-hand side ofthe factorization problem Assoc(⋆)(P)(·, ·, ·) = ^ P, [[P,P]] (·, ·, ·), and the number of
Leibniz graphs which assemble with nonzero coefficients to a solution ^ in the right-hand
side is presented in Table 2. At ℏ4, the expansion of Assoc(⋆)(P) mod ō(ℏ4) requires 241
Table 2. Number of graphs in either side of the factorization.
k 2 3 4 5 6 7
LHS: # K. orgraphs 3 (Jac) 39 740 12464 290305 ?
RHS: # L. orgraphs, 1 (Jac) 13 241
coeff , 0 ︸          ?   ︷︷             ︸? ?
Reference §4.3, [15] §5.1, [5] §5.2, [7] [3]
nonzero coefficients of Leibniz graphs on 3 sinks, 2 = n − 1 internal vertices for bi-
vectors P and one internal vertex for the tri-vector [[P,P]], and therefore, 2(n− 1)+ 3 =
2n + 3 − 2 = 7 oriented edges.
Remark 3. Again, this set of Leibniz graphs is well structured. Indeed, it is a disjoint
union of homogeneous differential operators arranged according to their differential
orders w.r.t. the sinks, e.g., (1, 1, 1), (2, 1, 1), (1, 2, 1), (1, 1, 2), etc., up to (3, 3, 1).
Example 8. The Leibniz graph L331 := [01; 01; 012] of differential orders (3, 3, 1) has
the weight 1/24(according to [3]. Multiplied by a universal (for all graphs at ℏ
4) factor
24 = 16 and the factor 1/(#Aut(L331)) = 1/2 due to this graph’s) symmetry (3 ⇄ 4),it expands to 13 [01; 01; 01; 52] + [01; 01; 12; 50] + [01; 01; 20; 51] by the definition of
Jacobi’s identity. This sum of three weighted Kontsevich orgraphs reproduces exactly
A(4)331, which is known from [7, Table 8 in App. D].
Example 9. The Leibniz graph L322 := [01; 02; 012] of differential orders (3, 2, 2) has
the weight 1/24 according to [3)]. Multiplied now by a universal (for(all graphs at ℏ
4)
factor 24 = 16 and the factor 1/(#Aut(L322)) = 1, it expands to 23 [01; 02; 01; 52] +
[01; 02; 12; 50] + [01; 02; 20; 51] . This sum reproduces A(4)322 (again, see [7, Table 8 in
App. D]).
Example 10. Consider at the differential order (1, 3, 2) at ℏ4 the three Leibniz graphs
L(1)132 := [12; 13; 012], L
(2)
132 := [12; 12; 014], and L
(3)
132 := [12; 01; 412]. They have no
symmetries, i.e. their automorphism groups are one-element, and their weights are
W(L(1) (2) (3)132) = 1/72, W(L132) = 1/48, and W(L132) = 1/48, respectively. Pre-multiplied by
their weights an(d universal factor 24 = 16, these Leibniz graphs e)xpand to2
9 [12; 1(3; 01; 52] + [12; 13; 12; 50] + [12; 13; 20; 51]
+ 13( )[12; 12; 01; 54] + [12; 12; 14; 50] + [12; 12; 40; 51]1 )+ 3 [12; 01; 41; 52] + [12; 01; 12; 54] + [12; 01; 24; 51] .
There is one cancellation, since [12; 01; 12; 54] = −[12; 12; 01; 54]. The remaining seven
terms reproduce exactly A(4)132; that component is known from [7, Table 8 in App. D].
FORMALITY MORPHISM AS THE MECHANISM OF ⋆-PRODUCT ASSOCIATIVITY 317
Actually, there was another Leibniz graph at this homogeneity order, L(4)132 := [12; 15; 012],
but its weight is zero and hence it does not contribute. (Indeed, we get an independent
verification of this by having already balanced the entire homogeneous component at
differential orders (1, 3, 2) in the associator.)
Intermediate conclusion. We have experimentally found the constants ck in Corollary 4
which balance the Kontsevich graph expansion of the ℏk-term A(k) in the associator
against an expansion of the respective term at ℏk in the r.-h.s. of (1) using the weighted
Leibniz graphs. Namely, we conjecture ck = k/6 in §4.3. The origin of these constants,
in particular how they arise from the sum over i < j in the L∞ condition (4) (perhaps,
in combination with different normalizations of the objects which we consider) still
remains to be explained, similar to the reasoning in [2, 18] where the signs are fixed.
Note that both in the associator, which is quadratic w.r.t. the weights of Kontsevich
graphs in ⋆, and in the operator ^, which is linear in the Kontsevich weights of Leibniz
graphs, the weight values are provided simultaneously, by using identical techniques
(for instance, from [3]). Indeed, the weights are provided by the integral formula which
is universal with respect to all the graphs under study [15].
Appendix A. Encodings of weighted Kontsevich-graph expansions for
(p, q, r)-homogeneous components (I f , . . . , S h)pqr
# 2/3 (S_f)_{221}
3 3 1 0 1 0 1 4 2 2/3
3 3 1 0 1 1 2 4 0 2/3
3 3 1 0 1 2 0 4 1 2/3
# 2/3 (S_g)_{122}
3 3 1 1 2 0 1 4 2 2/3
3 3 1 1 2 1 2 4 0 2/3
3 3 1 1 2 2 0 4 1 2/3
# -2/3 (S_h)_{212}
3 3 1 2 0 0 1 4 2 -2/3
3 3 1 2 0 1 2 4 0 -2/3
3 3 1 2 0 2 0 4 1 -2/3
# 1/6 (I_f)_{111}
3 3 1 0 4 3 1 4 2 1/6
3 3 1 0 4 1 2 4 3 1/6
3 3 1 0 4 2 3 4 1 1/6
3 3 1 0 5 3 1 4 2 1/6
3 3 1 0 5 1 2 4 3 1/6
3 3 1 0 5 2 3 4 1 1/6
# -1/6 (I_h)_{111}
3 3 1 2 4 0 1 4 3 -1/6
3 3 1 2 4 1 3 4 0 -1/6
3 3 1 2 4 3 0 4 1 -1/6
3 3 1 2 5 0 1 4 3 -1/6
3 3 1 2 5 1 3 4 0 -1/6
318 R. BURING AND A.V.KISELEV
3 3 1 2 5 3 0 4 1 -1/6
# 1/6 (I_f)_{112}
3 3 1 0 2 3 1 4 2 1/6
3 3 1 0 2 1 2 4 3 1/6
3 3 1 0 2 2 3 4 1 1/6
# 1/6 (I_g)_{112}
3 3 1 1 2 0 3 4 2 1/6
3 3 1 1 2 3 2 4 0 1/6
3 3 1 1 2 2 0 4 3 1/6
# -1/3 (S_h)_{112}
3 3 1 2 4 0 1 4 2 -1/3
3 3 1 2 4 1 2 4 0 -1/3
3 3 1 2 4 2 0 4 1 -1/3
3 3 1 2 5 0 1 4 2 -1/3
3 3 1 2 5 1 2 4 0 -1/3
3 3 1 2 5 2 0 4 1 -1/3
# 1/3 (I_f)_{121}
3 3 1 0 1 3 1 4 2 1/3
3 3 1 0 1 1 2 4 3 1/3
3 3 1 0 1 2 3 4 1 1/3
# -1/3 (I_h)_{121}
3 3 1 2 1 0 1 4 3 -1/3
3 3 1 2 1 1 3 4 0 -1/3
3 3 1 2 1 3 0 4 1 -1/3
# 1/3 (S_f)_{211}
3 3 1 0 4 0 1 4 2 1/3
3 3 1 0 4 1 2 4 0 1/3
3 3 1 0 4 2 0 4 1 1/3
3 3 1 0 5 0 1 4 2 1/3
3 3 1 0 5 1 2 4 0 1/3
3 3 1 0 5 2 0 4 1 1/3
# -1/6 (I_g)_{211}
3 3 1 1 0 0 3 4 2 -1/6
3 3 1 1 0 3 2 4 0 -1/6
3 3 1 1 0 2 0 4 3 -1/6
# -1/6 (I_h)_{211}
3 3 1 2 0 0 1 4 3 -1/6
3 3 1 2 0 1 3 4 0 -1/6
3 3 1 2 0 3 0 4 1 -1/6
Acknowledgements. The first author thanks the Organisers of international workshop
‘Symmetries & integrability of equations of Mathematical Physics’ (22–24 December
2018, IM NASU Kiev, Ukraine) for helpful discussions and warm atmosphere during
the meeting. A part of this research was done while RB was visiting at RUG and AVK
was visiting at JGU Mainz (supported by IM JGU via project 5020 and JBI RUG
FORMALITY MORPHISM AS THE MECHANISM OF ⋆-PRODUCT ASSOCIATIVITY 319
project 106552). The research of AVK is supported by the IHÉS (partially, by the
Nokia Fund).
References
[1] Alekseev A., Rossi C.A., Torossian C., Willwacher T. (2016) Logarithms and deforma-
tion quantization, Invent. Math. 206:1, 1–28. (Preprint arXiv:1401.3200 [q-alg]); Rossi
C.A., Willwacher T. (2014) P. Etingof’s conjecture about Drinfeld associators, Preprint
arXiv:1404.2047 [q-alg]
[2] Arnal D., Manchon D., Masmoudi M. (2002) Choix des signes pour la formalité de M. Kon-
tsevich. Pacific J. Math. 203:1, 23–66. (Preprint arXiv:q-alg/0003003)
[3] Banks P., Panzer E., Pym B. (2018) Multiple zeta values in deformation quantization,
71 p. (software available), Preprint arXiv:1812.11649 [q-alg]
[4] Bouisaghouane A., Buring R., Kiselev A. (2017) The Kontsevich tetrahedral flow revisited,
J. Geom. Phys. 119, 272–285. (Preprint arXiv:1608.01710 [q-alg])
[5] Buring R., Kiselev A.V. (2017) On the Kontsevich ⋆-product associativity mechanism,
PEPAN Letters 14:2 Supersymmetry and Quantum Symmetries’2015, 403–407. (Preprint
arXiv:1602.09036 [q-alg])
[6] Buring R., Kiselev A.V. (2019) The orientation morphism: from graph cocycles to de-
formations of Poisson structures, J. Phys.: Conf. Ser. 1194 Proc. 32nd Int. colloquium on
Group-theoretical methods in Physics: Group32 (9–13 July 2018, CVUT Prague, Czech
Republic), Paper 012017, 10 p. (Preprint arXiv:1811.07878 [math.CO])
[7] Buring R., Kiselev A.V. (2019) The expansion ⋆ mod ō(ℏ4) and computer-assisted proof
schemes in the Kontsevich deformation quantization, Experimental Math., 67 p. (revised).
(Preprint IHÉS/M/17/05, arXiv:1702.00681 [math.CO])
[8] Buring R., Kiselev A.V., Rutten N. J. (2018) Poisson brackets symmetry from the penta-
gon-wheel cocycle in the graph complex, Physics of Particles and Nuclei 49:5 Supersym-
metry and Quantum Symmetries’2017, 924–928. (Preprint arXiv:1712.05259 [math-ph])
[9] Cattaneo A. (2005) Formality and star products. (Lect. notes D. Indelicato) Poisson
geometry, deformation quantisation and group representations. London Math. Soc., Lect.
Note Ser. 323, 79–144 (Cambridge Univ. Press, Cambridge).
[10] Cattaneo A. S., Felder G. (2000) A path integral approach to the Kontsevich quantization
formula, Comm. Math. Phys. 212:3, 591–611. (Preprint arXiv:q-alg/9902090)
[11] Cattaneo A., Keller B., Torossian C., Bruguières A. (2005) Déformation, quantification,
théorie de Lie. Panoramas et Synthèses 20, Soc. Math. de France, Paris.
[12] Felder G., Willwacher T. (2010) On the (ir)rationality of Kontsevich weights, Int. Math.
Res. Not. IMRN 2010:4, 701–716. (Preprint arXiv:0808.2762 [q-alg])
[13] Kontsevich M. (1994) Feynman diagrams and low-dimensional topology, First Europ.
Congr. of Math. 2 (Paris, 1992), Progr. Math. 120, Birkhäuser, Basel, 97–121; Kontsevich
M. (1995) Homological algebra of mirror symmetry, Proc. Intern. Congr. Math. 1 (Zürich,
1994), Birkhäuser, Basel, 120–139.
[14] Kontsevich M. (1997) Formality conjecture. Deformation theory and symplectic geome-
try (Ascona 1996, D. Sternheimer, J. Rawnsley and S.Gutt, eds), Math. Phys. Stud. 20,
Kluwer Acad. Publ., Dordrecht, 139–156.
[15] Kontsevich M. (2003) Deformation quantization of Poisson manifolds, Lett. Math.
Phys. 66:3, 157–216. (Preprint arXiv:q-alg/9709040)
[16] Lada T., Stasheff J. (1993) Introduction to sh Lie algebras for physicists, Internat. J. The-
oret. Phys. 32:7, 1087–1103. (Preprint arXiv:hep-th/9209099)
320 R. BURING AND A.V.KISELEV
[17] Schlessinger M., Stasheff J. (1985) The Lie algebra structure of tangent cohomology and
deformation theory, J. Pure Appl. Alg. 38, 313–322.
[18] Willwacher T., Calaque D. (2012) Formality of cyclic cochains, Adv. Math. 231:2, 624–
650. (Preprint arXiv:0806.4095 [q-alg])
Chapter 13
The heptagon-wheel cocycle in the
Kontsevich graph complex
This chapter is based on the peer-reviewed journal publication R. Buring, A. V. Kiselev,
and N. J. Rutten, J. Nonlin. Math. Phys., 24: Suppl. 1 ‘Local & Nonlocal Symmetries
in Mathematical Physics’, 157–173, 2017. (Preprint arXiv:1710.00658 [math.CO] – 17
p.)
Commentary. In reference to Part I of the dissertation, the material of this chapter is
used in Chapters 4 and 5. The SageMath code (in Appendix B within this chapter) for
the graph insertion, bracket of graphs, and vertex-expanding differential served as the
beginning of the gcaops software. The encodings of undirected graph cocycles γ3, γ5, γ7,
and [γ3, γ5] are contained in Appendix E of the dissertation.
321
THE HEPTAGON-WHEEL COCYCLE
IN THE KONTSEVICH GRAPH COMPLEX
RICARDO BURING(a), ARTHEMY KISELEV(b,c), AND NINA RUTTEN(b)
Special Issue JNMP 2017 “Local & nonlocal symmetries in Mathematical Physics”
Abstract. The real vector space of non-oriented graphs is known to carry a dif-
ferential graded Lie algebra structure. Cocycles in the Kontsevich graph complex,
expressed using formal sums of graphs on n vertices and 2n− 2 edges, induce – under
the orientation mapping – infinitesimal symmetries of classical Poisson structures on
arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence
of a nontrivial cocycle that contains the (2ℓ + 1)-wheel graph with a nonzero coeffi-
cient at every ℓ ∈ N. We present detailed calculations of the differential of graphs;
for the tetrahedron and pentagon-wheel cocycles, consisting at ℓ = 1 and ℓ = 2 of one
and two graphs respectively, the cocycle condition d(γ) = 0 is verified by hand. For
the next, heptagon-wheel cocycle (known to exist at ℓ = 3), we provide an explicit
representative: it consists of 46 graphs on 8 vertices and 14 edges.
Introduction. The structure of differential graded Lie algebra on the space of non-
oriented graphs, as well as the cohomology groups of the graph complex, were introduced
by Kontsevich in the context of mirror symmetry [10, 11]. It can be shown that by
orienting a graph cocycle on n vertices and 2n− 2 edges (and by adding to every graph
in that cocycle two new edges going to two sink vertices) in all such ways that each of
the n old vertices is a tail of exactly two arrows, and by placing a copy of a given Poisson
bracket P in every such vertex, one obtains an infinitesimal symmetry of the space of
Poisson structures. This construction is universal with respect to all finite-dimensional
affine real manifolds (see [12] and [2]).1 Until recently two such differential-polynomial
symmetry flows were known (of nonlinearity degrees 4 and 6 respectively). Namely,
the tetrahedral graph flow Ṗ = Q1: 6 (P) was proposed in the seminal paper [12] (see
2
also [2, 3]). Consisting of 91 oriented bi-vector graphs on 5 + 1 = 6 vertices, the
Kontsevich–Willwacher pentagon-wheel flow will presently be described in [7].
Date: 24 November 2017.
2010 Mathematics Subject Classification. 13D10, 32G81, 53D17, 81S10, also 53D55, 58J10, 90C35.
Key words and phrases. Non-oriented graph complex, differential, cocycle, symmetry, Poisson
geometry.
(a)Address: Institut für Mathematik, Johannes Gutenberg–Universität, Staudingerweg 9, D-55128
Mainz, Germany.
(b)Address: Johann Bernoulli Institute for Mathematics and Computer Science, University of Gro-
ningen, P.O. Box 407, 9700 AK Groningen, The Netherlands. (c)E-mail: A.V.Kiselev@rug.nl.
1The dilation Ṗ = P, also universal with respect to all Poisson manifolds, is obtained by orienting
the graph • on one vertex and no edges, yet that graph is not a cocycle, d(•) = −•−• ̸= 0. The single-
edge graph •−• ∈ ker d on two vertices is a cocycle but its bi-grading differs from (n, 2n− 2). However,
by satisfying the zero-curvature equation d(•−•) + 12 [•−•, •−•] = 0 the graph •−• is a Maurer–Cartan
element in the graph complex.
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 323
The cohomology of the graph complex in degree 0 is known to be isomorphic to
the Grothendieck–Teichmüller Lie algebra grt (see [9] and [16]); under the isomor-
phism, the grt generators correspond to nontrivial cocycles. Using this correspondence,
Willwacher gave in [16, Proposition 9.1] the existence proof for an infinite sequence of
the Deligne–Drinfel’d nontrivial cocycles on n vertices and 2n − 2 edges. (Formulas
which describe these cocycles in terms of the grt Lie algebra generators are given in the
preprint [15].) To be specific, at each ℓ ∈ N every cocycle from that sequence contains
the (2ℓ+ 1)-wheel with nonzero coefficient (e.g., the tetrahedron alone making the co-
cycle γ3 at ℓ = 1), and possibly other graphs on 2ℓ + 2 vertices and 4ℓ + 2 edges. For
instance, at ℓ = 2 the pentagon-wheel cocycle γ5 consists of two graphs, see Fig. 1 on
p. 327 below.
In this paper we describe the next one, the heptagon-wheel cocycle γ7 from that
sequence(of∑solutions to the equation )
d (coefficient ∈ R) · (graph with an ordering of its edge set) = 0.
{graphs}
Our representative of the cocycle γ7 consists of 46 connected graphs on 8 vertices and
14 edges. (This number of nonzero coefficients can be increased by adding a cobound-
ary.) This solution has been obtained straightforwardly, that is, by solving the graph
equation d(γ7) = 0 directly. One could try reconstructing the cocycle γ7 from a set
of the grt Lie algebra generators, which are known in low degrees. Still an explicit
verification that γ7 ∈ ker d would be appropriate for that way of reasoning.
In this paper we also confirm that the three cocycles known so far – namely the tetra-
hedron and pentagon- and heptagon-wheel solutions – span the space of nontrivial coho-
mology classes which are built of connected graphs on n ⩽ 8 vertices and 2n− 2 edges.
At n = 9, there is a unique nontrivial cohomology class with graphs on nine vertices
and sixteen edges: namely, the Lie bracket [γ3,γ5] of the previously found cocycles.
(Brown showed in [4] that the elements σ2ℓ+1 in the Lie algebra grt which – under the
Willwacher isomorphism – correspond to the wheel cocycles γ2ℓ+1 generate a free Lie
algebra; hence it was expected that the cocycle [γ3,γ5] is non-trivial.) To verify that
the list of currently known d-cocycles is exhaustive – under all the assumptions which
were made about the graphs at our disposal – at every n ⩽ 9 we count the dimension
of the space of cocycles minus the dimension of the space of respective coboundaries.2
Our findings fully match the dimensions from [14, Table 1].
This text is structured as follows. Necessary definitions and some notation from the
graph complex theory are recalled in §1. These notions are illustrated in §2 where a step-
by-step calculation of the (vanishing) differentials d(γ3) and d(γ5) is explained. Our
main result is Theorem 7 with the heptagon-wheel solution of the equation d(γ7) =
0. Also in §3, in Proposition 8 we verify the count of number of cocycles modulo
coboundaries which are formed by all connected graphs on n vertices and 2n− 2 edges
(here 4 ⩽ n ⩽ 9). The graphs which constitute γ7 are drawn on pp. 334–340 in
Appendix A. The code in Sage programming language, allowing one to calculate the
differential for a given graph γ and ordering E(γ) on the set of its edges, is contained in
2The proof scheme is computer-assisted (cf. [2, 6]); it can be applied to the study of other cocycles:
either on higher number of vertices or built at arbitrary n ⩾ 2 from not necessarily connected graphs.
324 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
Appendix B; the same code can be run to calculate the dimension of graph cohomology
groups.
The main purpose of this paper is to provide a pedagogical introduction into the
subject.3 Besides, the formulas of the three cocycle representatives will be helpful
in the future search of an easy recipe to calculate all the wheel cocycles γ2ℓ+1. (No
general recipe is known yet, except for a longer reconstruction of those cohomology
group elements from the generators of Lie algebra grt.) Thirdly, our present knowledge
of both the cocycles γi and the respective flows Ṗ = Qi(P) on the spaces of Poisson
structures will be important for testing and verifying explicit formulas of the orientation
mapping O⃗r such that Qi = O⃗r(γi).
1. The non-oriented graph complex
We work with the real vector space generated by finite non-oriented graphs4 without
multiple edges nor tadpoles and endowed with a wedge ordering of edges: by definition,
an edge swap ei∧ ej = −ej ∧ ei implies the change of sign in front of the graph at hand.
Topologically equal graphs are equal as vector space elements if their edge orderings E
differ by an even permutation; otherwise, the graphs are opposite to each other (i.e.
they differ by the factor −1).
Definition 1. A graph which equals minus itself – under a symmetry that induces a
parity-odd permutation of edges – is called a zero graph. In particular (view •−•−•),
every graph possessing a symmetry which swaps an odd number of edge pairs is a zero
graph.
Notation. For a given labelling of vertices in a graph, we denote by ij (equivalently,
by ji) the edge connecting the vertices i and j. For instance, both 12 and 21 is the
notation for the edge between the vertices 1 and 2. (No multiple edges are allowed,
hence 12 is the edge. Indeed, by Definition 1 all graphs with multiple edges would be
zero graphs.) We also denote by N(v) the valency of a vertex v.
Example 1. The 4-wheel 12 ∧ 13 ∧ 14 ∧ 15 ∧ 23 ∧ 25 ∧ 34 ∧ 45 = I ∧ · · · ∧ V III or
likewise, the 2ℓ-wheel at any ℓ > 1 is a zero graph; here, the reflection symmetry is
I ⇄ III, V ⇄ V II, and V I ⇄ V III.
Note that every term in a sum of non-oriented graphs γ with real coefficients is fully
encoded by an ordering E on the set of adjacency relations for its vertices v (if N(v) > 0).
From now on, we assume N(v) ⩾ 3 unless stated otherwise explicitly.
Example 2. The tetrahedron (or 3-wheel) is the full graph on four vertices and six
edges (enumerated in the ascending order: 12 = I, . . ., 34 = V I), pp2γ3 = 12 ∧ 13 ∧ 14 ∧ 23 ∧ 24 ∧ 34 = I ∧ · · · ∧ V I = 1 p p3
This graph is nonzero. (The axis vertex is labelled 4 in this figure.)
3The first example of practical calculations of the graph cohomology –with respect to the edge
contracting differential – is found in [1]; a wide range of vertex-edge bi-degrees is considered there.
4The vector space of graphs under study is infinite dimensional; however, it is endowed with the
bi-grading (#vertices, #edges) so that all the homogeneous components are finite dimensional.
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 325
Example 3. The linear combination γ5 of two 6-vertex 10-edge graphs, namely, of the
pentagon wheel and triangular prism with one extra diagonal (here, 12 = I and so on),
γ5 = 12 ∧ 23 ∧ 34 ∧ 45 ∧ 51 ∧ 16 ∧ 26 ∧ 36 ∧ 46 ∧ 56
+ 5 · 12 ∧ 23 ∧ 34 ∧ 41 ∧ 45 ∧ 15 ∧ 56 ∧ 36 ∧ 26 ∧ 13
2
is drawn in Fig. 1 on p. 327 below (cf. [1]).
Let γ1 and γ2 be connected non-oriented graphs. The definition of insertion γ1 ◦iγ2 of
the entire graph γ1 into vertices of γ2 and the construction of Lie bracket [·, ·] of graphs
and differential d in the non-oriented graph complex, referring to a sign convention, are
as follows (cf. [12] and [8, 14, 16]); these definitions apply to sums of graphs by linearity.
Definition 2. The insertion γ1 ◦i γ2 of an n1-vertex graph γ1 with ord∑ered set of edgesE(γ1) into a graph γ2 with #E(γ2) edges on n2 vertices is a sum of graphs on n1+n2−1
vertices and #E(γ1)+#E(γ2) edges. Topologically, the sum γ1 ◦i γ2 = (γ1 → v in γ2)
consists of all the graphs in which a vertex v from γ2 is replaced by the entire graph
γ1 and the edges touching v in γ2 are re-attached to the vertices of γ1 in all possible
ways.5 By convention, in every new term the edge ordering is E(γ1) ∧ E(γ2).
To simplify sums of graphs, first eliminate the zero graphs. Now suppose that in a
sum, two non-oriented graphs, say α and β, are isomorphic (topologically, i.e. regardless
of the respective vertex labellings and edge orderings E(α) and E(β)). By using that
isomorphism, which establishes a 1–1 correspondence between the edges, extract the
sign from the equation E(α) = ±E(β). If “+”, then α = β; else α = −β. Collecting
similar terms is now elementary.
Lemma 1. The bi-linear graded skew-symmetric operation,
[γ , γ ] = γ ◦ γ − (−)#E(γ1)·#E(γ2)1 2 1 i 2 γ2 ◦i γ1,
is a Lie bracket on the vector space G of non-oriented graphs.6
Lemma 2. The operator d(graph) = [•−•, graph] is a differential: d2 = 0.
In effect, the mapping d blows up every vertex v in its argument in such a way that
whenever the number of adjacent vertices N(v) ⩾ 2 is sufficient, each end of the inserted
edge •−• is connected with the rest of the graph by at least one edge.
Theorem 3 ([12]). The real vector space G of non-oriented graphs is a differential
graded Lie algebra (dgLa) with Lie bracket [·, ·] and differential d = [•−•, ·]. The differ-
ential d is a graded derivation of the bracket [·, ·] (due to the Jacobi identity for this Lie
algebra structure).
5Let the enumeration of vertices in every such term in the sum start running over the enumerated
vertices in γ2 until v is reached. Now the enumeration counts the vertices in the graph γ1 and then it
resumes with the remaining vertices (if any) that go after v in γ2.
6The postulated precedence or antecedence of the wedge product of edges from γ1 with respect to
the edges from γ2 in every graph within γ1 ◦i γ2 produce the operations ◦i which coincide with or,
respectively, differ from Definition 2 by the sign factor (−)#E(γ1)·#E(γ2). The same applies to the Lie
bracket of graphs [γ1, γ2] if the operation γ1 ◦i γ2 is the insertion of γ2 into γ1 (as in [14]). Anyway, the
notion of d-cocycles which we presently recall is well defined and insensitive to such sign ambiguity.
326 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
The graphs γ3 and γ5 from Examples 2 and 3 are d-cocycles (this will be shown
in §2). Therefore, their commutator [γ3,γ5] is also in ker d. Neither γ3 nor γ5 is exact,
hence marking a nontrivial cohomology class in the non-oriented graph complex.
Theorem 4 ([8, Th. 5.5]). At every ℓ ∈ N in the connected graph complex there is
a nontrivial d-cocycle on 2ℓ + 1 vertices and 4ℓ + 2 edges. Such cocycle contains the
(2ℓ+1)-wheel in which, by definition, the axis vertex is connected with every other vertex
by a spoke so that each of those 2ℓ vertices is adjacent to the axis and two neighbours;
the cocycle marked by the (2ℓ+1)-wheel graph can contain other (2ℓ+1, 4ℓ+2)-graphs.
Example 4. For ℓ = 3 the heptagon wheel cocycle γ7, which we present in this paper,
consists of the heptagon-wheel graph on (2 · 3+1)+1 = 8 vertices and 2(2 · 3+1) = 14
edges and forty-five other graphs with equally many vertices and edges (hence of the
same number of generators of their homotopy groups, or basic loops: 7 = 14− (8− 1)),
and with real coefficients. All these weighted graphs are drawn in Appendix A (see
pp. 334–340). The chosen – lexicographic – ordering of edges in each term is read from
the encoding of every such graph (see also Table 1 on p. 331; each entry of that table
is a listing I ≺ · · · ≺ XIV of the ordered edge set, followed by the coefficient of that
graph). A verification of the cocycle condition d(γ7) = 0 for this solution is computer-
assisted; it has been performed by using the code (in Sage programming language)
which is contained in Appendix B.
2. Calculating the differential of graphs
Example 5 (dγ3 = 0). The tetrahedron γ3 is the full graph on n = 4 vertices; we are
free to choose any ordering of the six edges in it, so let it be lexicographic:
E(γ3) = 12 ∧ 13 ∧ 14 ∧ 23 ∧ 24 ∧ 34 = I ∧ II ∧ III ∧ IV ∧ V ∧ V I.
The differential of this graph is equal to
d(γ3) = [•−•,γ ] = •−• ◦ γ − (−)#E(•−•)·#E(γ3)3 i 3 γ3 ◦i •−• = •−• ◦iγ3 − γ3 ◦i •−•,
since#E(γ3) = 6. Note that every vertex of valency one appears twice in d(γ3): namely
in the minuend (where the edge ordering is E ∧ I ∧ · · · ∧ V I by definition of ◦i) and
subtrahend (where the edge ordering is I ∧ · · · ∧V I ∧E). Because these edge orderings
differ by a parity-even permutation, such graphs in •−•◦iγ3 and γ3 ◦i •−• carry the same
sign. Hence they cancel in the difference •−• ◦iγ3 − γ3 ◦i •−•, and no longer shall we
pay any attention to the leaves, absent in the diff(er)ential of any graph. It is readily
seen that the twenty-four graphs (24 = 4 vertices · 3 · 2 ends of •−•) we are left with
1
in d(γ3) are of the shape drawn rhere. A vertq ex is blown up to the new edge E = •−•edgre′′ vj
edge′ r r r = q qq q (see Remark 1)vi
whose ends are both attached to the rest of the graph along the old edges. This shape
can be obtained in two ways: by blowing up vi, so that edge′ is the newly inserted edge,
or by blowing up vj, so that edge′′ is the newly inserted edge. By Lemma 5 below we
conclude that d(γ3) = 0.
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 327
Remark 1. Incidentally, every graph which was obtained in d(γ3) itself is a zero graph.
Indeed, it is symmetric with respect to a flip over the vertical line and this symmetry
swaps three edge pairs (see Definition 1).
Lemma 5 (handshake). In the differential of any graph γ such that the valency of
all vertices in γ is strictly greater than two, the graphs in which one end of the newly
inserted edge •−• has valency two, all cancel.
Proof. Let v be such a vertex in d(γ), i.e. the vertex v is an end of the inserted edge •−•
and it has valency 2. Locally (near ′ ′ ′′ ′′v), we have either •E•Old • or •Old Ea v b a •v •b. In the two
respective graphs in d(γ) the rest, consisting only of old edges and vertices of valency⩾ 3
from γ, is the same. Yet the two graphs are topologically equal; furthermore, they have
the same ordering of edges except for E ′ = Old′′ and Old′ = E ′′. Recall that by
construction, the edge ordering of the first graph is E ′∧· · ·∧Old′∧· · · , whereas for the
second graph it is E ′′ ∧ · · · ∧Old′′ ∧ · · · ; the new edge always goes first. So effectively,
two edges are swapped. Therefore,
E ′′ ∧ · · · ∧Old′′ ∧ · · · = Old′ ∧ · · · ∧ E ′ ∧ · · · = −E ′ ∧ · · · ∧Old′ ∧ · · · .
Hence in every such pair in d(γ), the graphs occur with opposite signs. Moreover, the
initial hypothesis N(a) ⩾ 3 about the valency of all vertices a in the graph γ guarantees
that the cancelling pairs of graphs in d(γ) do not intersect,7 and thus all cancel. □
Corollary 6 (to Lemma 5). In the differential of any graph with vertices of valency > 2,
the blow up of a vertex of valency 3 produces only the handshakes, that is the graphs
which cancel out by Lemma 5 (cf. footnote 9 on p. 332 below).
Example 6 (dγ5 = 0). The pentagon-wheel cocycle is the sum of two graphs with
real coefficients which is drawnr in Fig. 1. The edges in every term are ordered by2
II I 4 r IIIr r
3
VII
3 VIIIVIII r r1 V6 5 r5 6
γ = VI + · rIV5 VII IIIII rIX r V 2X r VI IX r4 5 1 I 2
IV  X 
Figure 1. The Kontsevich–Willwacher pentagon-wheel cocycle γ5.
I ∧ · · · ∧ X. The differential of a sum of graphs is the sum of their differentials; this
is why we calculate them separately and then collect similar terms. By the above,
neither contains any leaves; likewise by the handshake Lemma 5, all the graphs – in
which a new vertex (of valency 2) appears as midpoint of the already existing edge –
cancel. By Corollary 6 it remains for us to consider the blow-ups of only the vertices
of valency ⩾ 4 (cf. [12]). Such are the axis vertex of the pentagon wheel and vertices
7This is why the assumption N(v) ⩾ 3 is important. Indeed, the disjoint-pair cancellation mecha-
nism does work only for chains with even numbers of valency-two vertices v in γ. Here is an example
(of one such vertex v between a and b) when it actually does not: in the differential of a graph that
contains a• I • IIv •b, we locally obtain • E• I •IIa a′ v •b +a• I • E II I II Ev •v′ •b +a• •v •b′•b, so that the middle
term can be cancelled against either the first or the last one but not with both of them simultaneously.
328 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
labelled 1 and 3 in the other graph (the prism). By blowing up the pentagon wheel
axis we shall obtain the (nonzero) ‘human’ and the (zero) ‘monkey’ graphs, presented
in what follows. Likewise from the prism graph in γ5 one obtains the ‘human’, the
‘monkey’, and the (zero) ‘stone’. Let us now discuss this in full detail.
From the pentagon wheel we obtain 2 · 5 Da Vinci’s ‘human’ graphs, two of which
are portrayed in Fig. 2. (The factor 2 occurs from the two distinct ways to attach three
versus two old edges in the wheelr to the loose ends of thre inserted edge •−•.) We claim
r II I II IVII VIIVIII r r r VIII r
III r VI r E rVIE =rIX X r V III r r VIX X
(a) IV (b) IV
Figure 2. Two of the fourteen Da Vinci’s ‘human’ graphs occurring with
weights in dγ5.
that all the five ‘human’ graphs (i.e. standing with their feet on the edges I, . . ., V in
the pentagon wheel) carry the same sign, providing the overall coefficient +10 = 2 ·(+5)
of such graph in the differential of the wheel. The graph (b) is topologically equal to
the graph (a); indeed, the matching of their edges is I(b) = V (a), II(b) = I(a), III(b) =
II(a), IV (b) = III(a), V (b) = IV (a), V I(b) = X(a), V II(b) = V I(a), V III(b) = V II(a),
IX(b) = V III(a), and X(b) = IX(a); also E(b) = E(a). Hence the postulated ordering of
edges in (b) is
E(b) ∧ I(b) ∧ · · · ∧X(b) = E(a) ∧ V (a) ∧ I(a) ∧ II(a) ∧ III(a) ∧ IV (a)∧
∧X(a) ∧ V I(a) ∧ V II(a) ∧ V III(a) ∧ IX(a) = +E(a) ∧ I(a) ∧ · · · ∧X(a), (1)
which equals the edge ordering of the graph (a). For the other three graphs of this
shape the equalities of wedge products are similar: a parity-even permutation of edges
works out the mapping of graphs, e.g., to the graph (a) which we take as the reference.
From the pentagon wheel we also obtain 2 · 5 ‘monkey’ graphs, a specimen of which
is shown in Fig. 3 below. Nrote that the ‘monkey’ graph is mirror-symmetric, see the
I
r II I r r rrVII 
Q AII
 VIQIQr VI AVVIII r VI r
 IX B
 E AE = VIII  B ArIV = 0III r r V
S  B 
IX X S III  BSSr BBrX
IV
Figure 3. The ‘monkey’ graph: animal touches earth with its palm; this
is an example of zero graph.
redrawing. This symmetry induces a permutation of edges which swaps 5 pairs, so
(since 5 is odd) the ‘monkey’ graph is equal to zero.
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 329
Now consider the graphs obtained by blowing up vertices 1 and 3 in the prism graph.
How are the four old neighbors distributed over the ends of the inserted edge? Whenever
those four old neighbours are distributed in proportion 4 = 3 + 1 (i.e. with valencies 4
and 2 for the two ends of the inserted edge), there is no contribution from the resulting
graphs to d(prism) by the handshake Lemma 5. So the graphs which could contibute
are only those with the 4 = 2+ 2 distribution (i.e. with valency 3 for either of the ends
of the inserted edge). For one fixed neighbour of one of the new edge’s ends there are
three ways to choose the second neighbour of that vertex. This is how the ‘human’,
‘monkey’, and ‘stone’ graphs are presently obtained.
Let us blow up vertex 1 in the prism in these three different ways. First we make
the end (now marked 1) of the inserted edge adjacent to 2 and 3, and the other end
(marked 1′) to vertices 4 and 5; the resulting graph is the ‘human’ graph shown in Fig. 4.
From the prism graph we obtain 2 · 2 = 4 such ‘human’ graphs. One of the factors 2 is
r2 r
r I IX II IrII VII1 X 3 r6 r VIII r r
r VIII = − VIIII EE r r VII III
r V
IV 4 V rIX X r
(z) 1′ 5VI (a) IV
Figure 4. One of the ‘human’ graphs obtained by blowing up –
according to a scenario discussed in the text – a vertex of valency four in
the prism graph from γ5.
obtained like before, namely by attaching a given set of old edges to one or the other
end of the inserted edge •−•, see p. 328; the other factor 2 comes by the rotational
symmetry of the prism graph. Indeed, the prism with one diagonal is symmetric under
the rotation by angle π that transposes the vertices 1 ⇄ 3, 2 ⇄ 4, and 5 ⇄ 6. This is
why the same ‘human’ graph is obtained when the vertex 3 is blown up according to a
similar scenario. We claim that the permutation of edges that relates the two graphs
is parity-even (similar to (1)), so they do not cancel but add up. Summarizing, the
overal coefficient of the ‘human’ graph – produced in d(prism) for the edge ordering
E ∧ I ∧ · · · ∧X shown in Fig. 4 – equals 2 · 2 = +4.
The count of an overall contribution 10 + 5 · (+4) · (−1 from edge ordering) = 0 to
2
the differential d(γ5) of the cocycle γ5 will be performed using Eq. (2); right now let
us inspect the vanishing of contributions from the other two types of graphs wich are
obtained by the two possible edge distribution scenarios (with respect to the ends of
the new edge •−• that replaces the blown-up vertex 1 or 3 in the prism). r2
The ‘monkey’ graph is obtained by blowing up the vertex 1 I IX
(or 3) in the prism and then attaching the new edge’s end, still 1 r II r6
marked 1, to the vertices 2 and 4. The other end, now marked 1′, IV r3 VIII
of the new edge becomes adjacent to the vertices 3 and 5. We rIIIE VII
keep in mind that every ‘monkey’ graph itself is equal to zero, X 4 V
hence no contribution to d(prism ′ r r) occurs. 1 5VI
330 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
So far, the new vertex 1 has always been a fixed neighbour of vertex 2, and it was
made adjacent to 3 in the ‘human’ and to 4 in the ‘monkey’ graphs, respectively. The
overall set of neigbours of the new edge 1–1′, apart from the fixed vertex 2, consists
of vertices 3, 4 and 5. So the third scenario to consider is the ‘stone’ graph in which
the new vertex 1 is adjacent to 1′, 2, and 5, whereas the new vertex 1′ neighbours 1,
3, and 4. This graph is mirror-symmetrric under the transposition of vertices 1′ ⇄ 25
@
III  rVI@I II
r X 1@ E@6 VIII r @ @
Z 2 1
′ rVI@r4
J 
 
= 0
Z IX V 
IVZJ 
ZZJJr

 I3
and 4 ⇄ 6, which induces the swaps in five edge pairs, namely, II ⇄ III, E ⇄ X,
V I ⇄ V III, V ⇄ IX, and I ⇄ IV . Arguing as before, we deduce that every such
‘stone’ graph (obtained by a blow up of either 1 or 3 in the prism) is zero.
Our final task in the calculation of d(γ5) is collecting the coefficients of the ‘hu-
man’ graphs from d(5-wheel) and d(prism), coming not only with coefficients 10 and 4
respectively, but also with the respective edge orderings. To discriminate edges be-
tween the two pictures, that is originating from the pentagon wheel and the prism, let
us use the superscripts (a) and (z), see Fig. 4. The edge matching is E(z) = III(a),
I(z) = II(a), II(z) = V II(a), III(z) = E(a), IV (z) = IX(a), V (z) = X(a), V I(z) = IV (a),
V II(z) = V (a), V III(z) = V I(a), IX(z) = I(a), and X(z) = V III(a). Consequently, for
the edge orderings we have
E(z) ∧ I(z) ∧ · · · ∧X(z) =
III(a) ∧ II(a) ∧ V II(a) ∧ E(a) ∧ IX(a) ∧X(a) ∧ IV (a) ∧ V (a) ∧ V I(a) ∧ I(a) ∧ V III(a)
= (−)23E(a) ∧ I(a) ∧ · · · ∧X(a). (2)
This argument shows that the graph differential of the linear combination (+1) ·
pentagon-wheel + 5 · prism, with either graph’s edge ordering specified as in Example 3,
2
vanishes. In other words, γ5 is a d-cocycle.
3. A representative of the heptagon-wheel cocycle γ7
It is already known that the heptagon-wheel cocycle γ7, the existence of which was
stated in Theorem 4, is unique modulo d-trivial terms in the respective cohomology
group of connected graphs on 8 vertices and 14 edges (hence with 7 basic loops), cf. [14].
Theorem 7. The encoding of every term in a representative of the cocycle γ7 is given
in Table 1, the format of lines in which is the lexicographic-ordered list of fourteen edges
I ∧ · · · ∧XIV followed by the nonzero real coefficient. The forty-six graphs that form
this representative of the d-cohomology class γ7 are shown on pages 334–340.
Proof scheme. This reasoning is computer-assisted. First, all connected graphs on 8 ver-
tices and 14 edges, and without multiple edges were generated. (There are 1579 such
graphs; note that arbitrary valency N(v) ⩾ 1 of vertices was allowed.) The coefficient
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 331
Table 1. The heptagon-wheel graph cocycle γ7.
Graph encoding Coeff. Graph encoding Coeff.
16 17 18 23 25 28 34 38 46 48 57 58 68 78 1 12 13 18 25 26 37 38 45 46 47 56 57 68 78 −7
12 14 18 23 27 35 37 46 48 57 58 67 68 78 −21/8 12 14 16 23 25 36 37 45 48 57 58 67 68 78 77/8
13 14 18 23 25 28 37 46 48 56 57 67 68 78 −77/4 13 16 17 24 25 26 35 37 45 48 58 67 68 78 −7
12 13 15 24 27 35 36 46 48 57 58 67 68 78 −35/8 14 15 17 23 26 28 37 38 46 48 56 57 68 78 49/4
12 13 18 24 26 37 38 46 47 56 57 58 68 78 49/8 12 16 18 27 28 34 36 38 46 47 56 57 58 78 −147/8
14 17 18 23 25 26 35 37 46 48 56 58 67 78 77/8 12 15 16 27 28 35 36 38 45 46 47 57 68 78 −21/8
12 13 18 26 27 35 38 45 46 47 56 57 68 78 −105/8 12 14 18 23 27 35 36 45 46 57 58 67 68 78 −35/8
12 14 18 23 27 36 38 46 48 56 57 58 67 78 7/8 14 15 16 23 26 28 37 38 46 48 57 58 67 78 −49/4
12 14 15 23 27 35 36 46 48 57 58 67 68 78 35/8 12 15 18 23 28 34 37 46 48 56 57 67 68 78 105/8
12 13 14 27 28 36 38 46 47 56 57 58 68 78 −49/8 12 14 17 23 26 37 38 46 48 56 57 58 68 78 −49/8
12 13 18 25 27 34 36 47 48 56 58 67 68 78 35/4 12 16 18 25 27 35 36 37 45 46 48 57 68 78 49/16
12 13 14 25 26 36 38 45 47 57 58 67 68 78 −119/16 12 13 18 25 27 35 36 46 47 48 56 57 68 78 7
12 13 15 24 28 36 38 47 48 56 57 67 68 78 49/8 12 14 18 25 28 34 36 38 47 57 58 67 68 78 −7
12 13 14 23 28 37 46 48 56 57 58 67 68 78 77/4 12 16 18 25 27 35 36 37 45 46 48 58 67 78 −77/16
12 15 17 25 26 35 36 38 45 47 48 67 68 78 −49/8 12 14 18 23 27 35 38 46 47 57 58 67 68 78 77/4
13 15 18 24 26 28 37 38 46 47 56 57 68 78 −49/4 12 14 15 23 27 36 38 46 48 57 58 67 68 78 35/2
13 14 18 25 26 28 36 38 47 48 56 57 67 78 −49/4 12 13 18 25 27 34 36 46 48 57 58 67 68 78 −105/8
12 14 18 23 28 35 37 46 48 56 57 67 68 78 −7 12 15 16 25 27 35 36 38 46 47 48 57 68 78 −7
12 14 18 23 28 36 38 46 47 56 57 58 67 78 −7 12 13 16 25 28 34 37 47 48 57 58 67 68 78 −147/16
12 15 16 25 27 35 36 38 46 47 48 58 67 78 49/8 12 13 17 25 26 35 37 45 46 48 58 67 68 78 −77/4
12 14 18 23 28 36 37 46 47 56 57 58 68 78 49/8 12 14 17 23 27 35 38 46 48 57 58 67 68 78 −49/8
12 13 15 26 27 35 36 45 47 48 58 67 68 78 −7 12 13 15 26 28 35 37 45 46 47 58 67 68 78 −7/4
12 13 18 24 28 35 38 46 47 57 58 67 68 78 7 12 14 18 23 26 36 38 47 48 56 57 58 67 78 −7
of the heptagon wheel was set equal to +1, all other coefficients still to be determined.
After calculating the differential of the sum of all these weighted graphs (we used a pro-
gram in Sage, see Appendix B), zero graphs were eliminated and the remaining terms
were collected (in the same way as is explained in §2). In the resulting sum of weighted
graphs on 9 vertices and 15 edges, we equated each coefficient to zero. We solved this
linear algebraic system w.r.t. the coefficients of graphs in γ7. There are Nim(7) = 35
free parameters in the general solution; such parameters count the coboundaries which
cannot modify the cohomology class marked by any particular representative (see Ta-
ble 2 on p. 332 below). Therefore the solution γ7 is unique modulo d-exact terms. All
those free parameters are now set to zero and the resulting nonzero values of the graph
coefficients are listed in Table 1. □
Proposition 8 (see [14, Table 1]). The space of nontrivial d-cocycles which are built
of connected graphs on n vertices and 2n − 2 edges at 1 ⩽ n ⩽ 9 is spanned by the
terahedron γ3, pentagon-wheel cocycle γ5 that consists of two graphs (see Example 3),
heptagon-wheel cocycle γ7 from Theorem 7, and the Lie bracket [γ3,γ5]. At the same
time, for either n = 5 or n = 7, the respective graph cohomology groups are trivial.8
Verification. The dimension Nker of the space of cocycles built of connected graphs γ on
n vertices and 2n− 2 edges is equal to the number of free parameters in the general so-
lution to the linear system d(sum of such graphs γ with undetermined coefficients) =
0. At the same time, to determine the dimension Nim of the subspace of cobound-
aries γ = d(δ), i.e. of those cocycles which are the differentials of connected graphs
on n − 1 vertices and 2n − 3 edges, we first count the number of Nδ of nonzero
8None of the results in Theorem 7 and Proposition 8 involves floating point operations in the way
how it is obtained; hence even if computer-assisted, both the claims are exact.
332 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
connected graphs δ in that vertex-edge bi-grading. Then we subtract from Nδ the
number N0 of free parameters in the general solution to the linear algebraic system
d(sums of such graphs δ with undetermined coefficients) = 0. This subtrahend counts
the number of relations between exact terms γ = d(δ); for n < 9 it is zero. The di-
mension of cohomology group H∗(n) in bi-grading (n, 2n − 2) is then Nker − Nim =
Nker − (Nδ −N0).
Our present count of the overall number of connected graphs (and of the zero graphs
among them) and the dimensions Nker, Nδ, N0 and Nim of the respective vector spaces
are summarized in Tables 2 and 3. □
Table 2. Dimensions of connected graph spaces and cohomology groups.
n #E #(graphs) #(= 0) #(̸= 0), Nδ Nker, N0 Nim dimH∗(n)
4 6 1 0 1 1 1
3 5 0 – – – – –
5 8 2 2 0 – 0
4 7 0 – – – – –
6 10 14 8 6 1 1
5 9 1 1 – 0 – –
7 12 126 78 48 1 0
6 11 9 8 – 1 0 1
8 14 1579 605 974 36 1
7 13 95 60 – 35 0 35
9 16 26631 7557 19074 883 1
8 15 1515 602 – 913 31 882
Remark 2. This reasoning covers all the connected graphs with specified number of
vertices and edges, meaning that the valency N(v) of every graph vertex v can be any
positive number (if n > 1). By Lemma 5 on p. 327 it is seen that for the subspaces V>2
of connected graphs restricted by N(v) > 2 for all v, the inclusion d(V>2) ⊆ V>2 holds.
Therefore, the dimensions of cohomology groups for graphs with such restriction on
valency cannot exceed the dimension of respective cohomology groups for all the graphs
under study (i.e. N(v) > 0).9 This means that trivial cohomology groups remain trivial
under the extra assumption N(v) > 2 on valency; yet we already know the generators
γ3, γ5, γ7, and [γ3,γ5] of all the nontrivial cohomology groups at n ⩽ 9. This is
confirmed in Table 3.
We finally note that the numbers of nonzero graphs with a specified number of vertices
and edges (and N(v) > 2), which we list in Table 3, all coincide with the respective
entries in Table II in the paper [17].
Remark 3. We expect that there are many d-cocycles on n vertices and 2n − 2 edges
other than the ones containing the (2ℓ + 1)-wheel graphs (which Theorem 4 provides)
or their iterated commutators. Namely, some terms in a weighted sum γ ∈ ker d can
9Indeed, we recall that these cohomology dimensions – in the count with versus without restriction
N(v) > 2 of the valency – are the same (e.g., see [16, Proposition 3.4] with a sketch of the proof).
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 333
Table 3. Dimensions of connected graph spaces with N(v) > 2 and
dimensions of cohomology groups in bi-degree (n, 2n− 2).
n #E #(graphs) #(= 0) #(̸= 0), Nδ Nker, N0 Nim dimH∗(n)
4 6 1 0 1 1 1
3 5 0 – – – – –
5 8 1 1 0 – 0
4 7 0 – – – – –
6 10 4 2 2 1 1
5 9 1 1 – 0 – –
7 12 18 12 6 1 0
6 11 5 4 – 1 0 1
8 14 136 61 75 11 1
7 13 30 20 – 10 0 10
9 16 1377 498 879 164 1
8 15 309 130 – 179 16 163
be disjoint graphs; moreover, the vertex-edge bi-grading of a connected component of
a given term can be other than (m, 2m − 2) for m∑∈ N. Indeed∑, for any tuple of d-
cocycles γi on n⊔i vertices and Ei edges satisfying i ni = n and i Ei = 2n − 2, one
has that γ := i γi ∈ ker d. The graphs γi can be restricted by a requirement that
each of them belongs to the domain of the orientation mapping O⃗r, so that O⃗r(γ) is a
Kontsevich bi-vector graph (see [12] and [2, 7]). In this way new classes of generators
of infinitesimal symmetries Ṗ = O⃗r(γ)(P) are obtained for Poisson structures P .
Acknowledgements. The authors are grateful to M. Kontsevich and T. Willwacher
for helpful discussion; the authors thank the referees for criticism and advice. This
research was supported in part by JBI RUG project 106552 (Groningen, The Nether-
lands). A part of this research was done while R. Buring and A.Kiselev were visiting at
the IHÉS (Bures-sur-Yvette, France) and A.Kiselev was visiting at the MPIM (Bonn)
and Johannes Gutenberg–Universität in Mainz, Germany.
References
[1] Bar-Natan D., McKay B.D. (2001) Graph cohomology — An overview and some computations,
13 p. (unpublished), http://www.math.toronto.edu/~drorbn/papers/GCOC/GCOC.ps
[2] Bouisaghouane A., Buring R., Kiselev A. (2017) The Kontsevich tetrahedral flow revisited,
J. Geom. Phys. 119, 272–285. (Preprint arXiv:1608.01710 [q-alg])
[3] Bouisaghouane A., Kiselev A.V. (2017) Do the Kontsevich tetrahedral flows preserve or destroy
the space of Poisson bi-vectors ? J. Phys.: Conf. Ser. 804 Proc. XXIV Int. conf. ‘Integrable
Systems and Quantum Symmetries’ (14–18 June 2016, ČVUT Prague, Czech Republic), Pa-
per 012008, 10 p. (Preprint arXiv:1609.06677 [q-alg])
[4] Brown F. (2012) Mixed Tate motives over Z, Ann. Math. (2) 175:2, 949–976.
[5] Buring R., Kiselev A.V. (2017) On the Kontsevich ⋆-product associativity mechanism, PEPAN
Letters 14:2, 403–407. (Preprint arXiv:1602.09036 [q-alg])
[6] Buring R., Kiselev A.V. (2017) The expansion ⋆ mod ō(ℏ4) and computer-assisted
proof schemes in the Kontsevich deformation quantization, Preprint IHÉS/M/17/05,
arXiv:1702.00681 [math.CO], 67 p.
334 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
[7] Buring R., Kiselev A.V., Rutten N. J. (2017) The Kontsevich–Willwacher pentagon-wheel sym-
metry of Poisson structures, SDSP IV (12–16 June 2017, ČVUT Děčín, Czech Republic).
[8] Dolgushev V. A., Rogers C. L., Willwacher T. H. (2015) Kontsevich’s graph complex, GRT, and
the deformation complex of the sheaf of polyvector fields, Ann. Math. 182:3, 855–943. (Preprint
arXiv:1211.4230 [math.KT])
[9] Drinfel’d V. G. (1990) On quasitriangular quasi-Hopf algebras and on a group that is closely
connected with Gal(Q/Q), Algebra i Analiz 2:4, 149–181 (in Russian); Eng. transl. in: Leningrad
Math. J. 2:4, 829–860 (1990).
[10] Kontsevich M. (1994) Feynman diagrams and low-dimensional topology, First Europ. Congr. of
Math. 2 (Paris, 1992), Progr. Math. 120, Birkhäuser, Basel, 97–121.
[11] Kontsevich M. (1995) Homological algebra of mirror symmetry, Proc. Intern. Congr. Math. 1
(Zürich, 1994), Birkhäuser, Basel, 120–139.
[12] Kontsevich M. (1997) Formality conjecture. Deformation theory and symplectic geometry (Ascona
1996, D. Sternheimer, J. Rawnsley and S.Gutt, eds), Math. Phys. Stud. 20, Kluwer Acad. Publ.,
Dordrecht, 139–156.
[13] Kontsevich M. (2017) Derived Grothendieck–Teichmüller group and graph complexes [after
T. Willwacher], Séminaire Bourbaki (69ème année, Janvier 2017), no. 1126, 26 p.
[14] Khoroshkin A., Willwacher T., Živković M. (2017) Differentials on graph complexes, Adv. Math.
307, 1184–1214. (Preprint arXiv:1411.2369 [q-alg])
[15] Rossi C.A., Willwacher T. (2014) P. Etingof’s conjecture about Drinfeld associators, Preprint
arXiv:1404.2047 [q-alg], 47 p.
[16] Willwacher T. (2015) M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie al-
gebra, Invent. Math. 200:3, 671–760. (Preprint arXiv:1009.1654 [q-alg])
[17] Willwacher T., Živković M. (2015) Multiple edges in M. Kontsevich’s graph complexes and
computations of the dimensions and Euler characteristics, Adv. Math. 272, 553–578. (Preprint
arXiv:1401.4974 [q-alg])
Appendix A. The heptagon-wheel cocycle γ7
In each term, the ordering of edges is lexicographic (cf. Table 1).
6 3
4
1 5
2
7
1 8 − 21γ7 = 3 8
7 8
1
6
2 5 4
5 2 3
6
7
3 5
8
− 77 6 8 −
35 1
4 8 7
1 4
4 2
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 335
5 3
7 2
4 5
6 7
49 77
+ 8 + 6
8 3 8
8
2
1
1 4
4 6 5
2
7
7
6
− 105 5 7+ 8 3
8 8 8
2
1
4
3 1
2 7 5
1
3 2 8
35 5 49
+ 7 −
8 8 4 6
8
1
4 6 3
5 2 2
1
8 1 3
6
35 − 119+ 6 84 7 16 5
4
4
3 7
336 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
5 1
6
7 3
2
3 4
49 77
+ 8 + 8
8 1 4
4 7
6
2 5
8 4
5 6
3
7 7 4 2
− 49 − 496
8 5 4
1 8
1
2 3
3 3
5
6 7
2 1 2
− 49 8 − 7
4 6 8
1
5
7 4 4
1 4
4 2 8
8 7
− 49 67 + 3
7 6 8
3
2
5
5 1
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 337
2 5 4
3
3
8
1
49 8
+ − 7 1 6
8 7 7
4
6 5 2
6 7
4 3
7 4
5 8
+ 7 5 8 − 7
2
1
6
3
1 2
2 1 1
7 3
4 6
77
+ 53 − 7
8 6 8
5
2
8
7 4
3 2
7
1
2 7 8
49 8
+ − 147
4 1 5 8 5
6
6
4
4 3
338 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
1 8
7
1
6 2
2
− 21
8
3 − 35 5
8 8
5 6
7 4
4 3
3 2
2 8 7 3 18
− 49 105+
4 5 8
6
4 7
5
4
1 6
5 7 6 1
8
8
4
− 49
3 49
4 +
8 1 16
3 7
2
6
2 5
4 5
8
6 7 2
+ 7 7 − 7 8
6
5 3 1
1
2 3 4
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 339
1
7 5 6
2
− 77 6 3 77 7+ 8
16 8 4 3 4
5
4 2 1
4 3 4
1 5
1 6
8
35 − 105
8
+ 6
2 7 8
2 2
7
3 5
3 6 4
3
4 7
8
1 147
+ 7 −
5 616
1 8
7 5
2 2
4 1
2
8 5 4
6
− 77
7
2 −
49
4 8
6
7 3
3
8
1 5
340 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
3 4
7
1
5
5 8
− 7
4 1
4 7
− 7 .
8
6
2
6 2 3
The sum of graphs γ7 is a d-cocycle because when the differential d(γ7) is constructed,
the images of many terms from γ7 overlap in d(γ7) (by graphs on 9 vertices and
15 edges). Finding out what the resulting adjacency table is for the forty-six graphs
in γ7 and –more generally – exploring whether such ‘meta-graphs’, the vertices of which
themselves are graphs that constitute d-cocycles modulo coboundaries, are in any sense
special, is an intriguing open problem. (We claim that for γ7, its meta-graph is con-
nected.)
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 341
Appendix B. Sage code for the graph differential
The following script, written in Sage version 7.2, can calculate the differential of an
arbitrary sum of non-oriented graphs with a specified ordering on the set of edges for
every term, and reduce sums of graphs modulo vertex and edge labelling.10 As an
illustration, it is shown how this can be used to find cocycles in the graph complex.
import itertools
def insert(user, victim, position):
result = []
victim = victim.relabel({k : k + position - 1 for k in victim.vertices()},
inplace=False)
victim = victim.copy(immutable=False)
for edge in victim.edges():
victim.set_edge_label(edge[0], edge[1], edge[2] + len(user.edges()))
user = user.relabel({k : k if k <= position else k + len(victim) - 1 for k in user.vertices()},
inplace=False)
for attachment in itertools.product(victim, repeat=len(user.edges_incident(position))):
new_graph = user.union(victim)
edges_in = user.edges_incident(position)
new_graph.delete_edges(edges_in)
new_edges = [(k if a == position else a, k if b == position else b, c)
for ((a,b,c), k) in zip(edges_in, attachment)]
new_graph.add_edges(new_edges)
result.append((1, new_graph))
return result
def graph_bracket(graph1, graph2):
result = []
for v in graph2:
result.extend(insert(graph2, graph1, v))
sign_factor = 1 if len(graph1.edges()) % 2 == 1 and len(graph2.edges()) % 2 == 1 else -1
for v in graph1:
result.extend([(sign_factor*c, g) for (c,g) in insert(graph1, graph2, v)])
return result
def graph_differential(graph):
edge = Graph([(1,2,1)])
return graph_bracket(edge, graph)
def differential(graph_sum):
result = []
for (c,g) in graph_sum:
result.extend([(c*d,h) for (d,h) in graph_differential(g)])
return result
def is_zero(graph):
for sigma in graph.automorphism_group():
edge_permutation = Permutation([graph.edge_label(sigma(i), sigma(j))
for (i,j,l) in sorted(graph.edges(), key=lambda (a,b,c): c)])
if edge_permutation.sign() == -1:
return True
return False
def reduce(graph_sum):
graph_table = {}
for (c,g) in graph_sum:
if is_zero(g): continue
10Another software package for numeric computation of the graph complex cohomology groups in
various degrees and loop orders is available from https://github.com/wilthoma/GHoL.
342 R.BURING, A.V.KISELEV, AND N. J.RUTTEN
# canonically label vertices:
g_canon, relabeling = g.canonical_label(certify=True)
# shift labeling up by one:
g_canon.relabel({k : k + 1 for k in g_canon.vertices()})
# canonically label edges (keeping track of the edge permutation):
count = 1
edges_seen = set([])
edge_relabeling = {}
for v in g_canon:
edges_in = sorted(g_canon.edges_incident(v), key = lambda (a,b,c): a if b == v else b)
for e in edges_in:
if frozenset([e[0], e[1]]) in edges_seen: continue
edge_relabeling[count] = e[2]
g_canon.set_edge_label(e[0], e[1], count)
edges_seen.add(frozenset([e[0], e[1]]))
count += 1
permutation = Permutation([edge_relabeling[i] for i in range(1, len(g.edges())+1)])
g_canon = g_canon.copy(immutable=True)
if g_canon in graph_table:
graph_table[g_canon] += permutation.sign()*c
else:
graph_table[g_canon] = permutation.sign()*c
return [(graph_table[g], g) for g in graph_table if not graph_table[g] == 0]
# Examples of graphs:
def wheel(n):
return Graph([(k, 1, k-1) for k in range(2, n+2)] + [(k, k+1 if k <= n else 2, n+k-1)
for k in range(2, n+2)])
tetrahedron = wheel(3)
fivewheel = wheel(5)
print "The differential of the tetrahedron is", reduce(graph_differential(tetrahedron))
# Finding all cocycles on 6 vertices and 10 edges:
n = 6
graph_list = list(filter(lambda G: G.is_connected() and len(G.edges()) == 2*n - 2, graphs(n)))
# shift labeling up by one
for g in graph_list:
g.relabel({k : k+1 for k in g.vertices()})
for (k, (i,j,_)) in enumerate(g.edges()):
g.set_edge_label(i, j, k+1)
# build an ansatz for a cocycle, with undetermined coefficients
nonzeros = filter(lambda g: not is_zero(g), graph_list)
coeffs = [var('c%d' % k) for k in range(0, len(nonzeros))]
cocycle = zip(coeffs, nonzeros)
# calculate its differential and reduce it
d_cocycle = []
for cocycle_term in cocycle:
d_cocycle.extend(reduce(differential([cocycle_term])))
d_cocycle = reduce(d_cocycle)
# set the coefficients of the graphs in the reduced sum to zero, and solve
linsys = []
for (c,g) in d_cocycle:
linsys.append(c==0)
print solve(linsys, coeffs)
THE HEPTAGON-WHEEL COCYCLE IN THE KONTSEVICH GRAPH COMPLEX 343
We finally recall that, to the best of our knowledge, the routines by McKay [1] for graph
automorphism computation are now used in SAGE (hence by the above program).

Chapter 14
Infinitesimal deformations of Poisson
bi-vectors using the Kontsevich
graph calculus
This chapter is based on the peer-reviewed conference proceedings R. Buring, A. V.
Kiselev, and N. J. Rutten, J. Phys.: Conf. Ser., 965: Proc. XXV Int.conf. ‘Integrable
Systems & Quantum Symmetries’ (6–10 June 2017, CVUT Prague, Czech Republic),
Paper 012010, 2018. (Preprint arXiv:1710.02405 [math.CO] – 12 p.)
Commentary. In reference to Part I of the dissertation, the material of this chapter is
used in §3.5 and Chapter 5. A way to encode Leibniz graphs was originally developed in
this chapter; we now refer to the Corrigendum in §3.3 which explains why the old method
could produce repetitions of Leibniz graphs.
345
Infinitesimal deformations of Poisson bi-vectors
using the Kontsevich graph calculus
Ricardo Buring1, Arthemy V Kiselev2 and Nina Rutten2
1 Institut für Mathematik, Johannes Gutenberg–Universität, Staudingerweg 9, D-55128
Mainz, Germany
2 Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen,
P.O. Box 407, 9700 AK Groningen, The Netherlands
E-mail: rburing@uni-mainz.de, A.V.Kiselev@rug.nl
Abstract. Let P be a Poisson structure on a finite-dimensional affine real manifold. Can
P be deformed in such a way that it stays Poisson ? The language of Kontsevich graphs
provides a universal approach –with respect to all affine Poisson manifolds – to finding a class of
solutions to this deformation problem. For that reasoning, several types of graphs are needed.
In this paper we outline the algorithms to generate those graphs. The graphs that encode
deformations are classified by the number of internal vertices k; for k ⩽ 4 we present all solutions
of the deformation problem. For k ⩾ 5, first reproducing the pentagon-wheel picture suggested
at k = 6 by Kontsevich and Willwacher, we construct the heptagon-wheel cocycle that yields a
new unique solution without 2-loops and tadpoles at k = 8.
Introduction. This paper contains a set of algorithms to generate the Kontsevich graphs
that encode polydifferential operators – in particular multi-vectors – on Poisson manifolds. We
report a result of implementing such algorithms in the problem of finding symmetries of Poisson
structures. Namely, continuing the line of reasoning from [1, 2], we find all the solutions of this
deformation problem that are expressed by the Kontsevich graphs with at most four internal
vertices. Next, we present one six-vertex solution (based on the previous work by Kontsevich [10]
and Willwacher [13]). Finally, we find a heptagon-wheel eight-vertex graph which, after the
orientation of its edges, gives a new universal Kontsevich flow. We refer to [8, 9] for motivations,
to [2, 4] for an exposition of basic theory, and to [6] and [5] for more details about the pentagon-
wheel (5+1)-vertex and heptagon-wheel (7+1)-vertex solutions respectively. Let us remark that
all the algorithms outlined here can be used without modification in the course of constructing
all k-vertex Kontsevich graph solutions with higher k ⩾ 5 in the deformation problem under
study.
Basic concept. We work with real vector spaces generated by finite graphs of the following
two types: (1) k-vertex non-oriented graphs, without multiple edges nor tadpoles, endowed with
a wedge ordering of edges, e.g., E = e1 ∧ · · · ∧ e2k−2; (2) oriented graphs on k internal vertices
and n sinks such that every internal vertex is a tail of two edges with a given ordering Left ≺
Right. Every connected component of a non-oriented graph γ is fully encoded by an ordering E
on the set of adjacency relations for its vertices.1 Every such oriented graph is given by the list
of ordered pairs of directed edges. An edge swap ei ∧ ej = −ej ∧ ei and the reversal Left ⇆
Right of those edges’ order in the tail vertex implies the change of sign in front of the graph at
hand.2
Example 1. The sum γ5 of two 6-vertex 10-edge graphs,
γ5 = 12
(I) ∧ 23(II) ∧ 34(III) ∧ 45(IV ) ∧ 51(V ) ∧ 16(V I) ∧ 26(V II) ∧ 36(V III) ∧ 46(IX) ∧ 56(X)
+5 · 12(I) ∧ 23(II)2 ∧ 34
(III) ∧ 41(IV ) ∧ 45(V ) ∧ 15(V I) ∧ 56(V II) ∧ 36(V III) ∧ 26(IX) ∧ 13(X),
is drawn in Theorem 7 on p. 355 below.
Example 2. The sum Q1: 6 of three oriented 8-edge graphs on k = 4 internal vertices and n = 2
2
sinks (enumerated using 0 and 1, see the notation on p. 349),
Q1: 6 = 2 4 1 0 1 2 4 2 5 2 3− 3(2 4 1 0 3 1 4 2 5 2 3+ 2 4 1 0 3 4 5 1 2 2 4)
2
is obtained from the non-oriented tetrahedron graph γ = 12(I) ∧ 13(II) ∧ 14(III) ∧ 23(IV )3 ∧
24(V ) ∧ 34(V I) on four vertices and six edges by taking all the admissible edge orientations (see
Theorem 4 and Remark 1).
I.1. Let γ1 and γ2 be connected non-oriented graphs. The definition of insertion γ1 ◦i γ2 of
the entire graph γ1 into vertices of γ2 and the construction of Lie bracket [·, ·] of graphs and
differential d in the non-oriented graph complex, referring to a sign convention, are as follows
(cf. [8] and [7, 11, 12]); these definitions apply to sums of graphs by linearity.
Definition 1. The insertion γ1 ◦i γ2 of a k1-vertex graph γ∑1 with ordered set of edges E(γ1)into a graph γ2 with #E(γ2) edges on k2 vertices is a sum of graphs on k1 + k2− 1 vertices and
#E(γ1) + #E(γ2) edges. Topologically, the sum γ1 ◦i γ2 = (γ1 → v in γ2) consists of all the
graphs in which a vertex v from γ2 is replaced by the entire graph γ1 and the edges touching
v in γ2 are re-attached to the vertices of γ1 in all possible ways.3 By convention, in every new
term the edge ordering is E(γ1) ∧ E(γ2).
To simplify sums of graphs, first eliminate the zero graphs. Now suppose that in a sum,
two non-oriented graphs, say α and β, are isomorphic (topologically, i.e. regardless of the
respective vertex labellings and edge orderings E(α) and E(β)). By using that isomorphism,
which establishes a 1–1 correspondence between the edges, extract the sign from the equation
E(α) = ±E(β). If “+”, then α = β; else α = −β. Collecting similar terms is now elementary.
Lemma 1. The bi-linear graded skew-symmetric operation,
[γ , γ ] = γ ◦ γ − (−)#E(γ1)·#E(γ2)1 2 1 i 2 γ2 ◦i γ1,
1 The edges are antipermutable so that a graph which equals minus itself – under a symmetry that induces a
parity-odd permutation of edges – is proclaimed to be equal to zero. In particular (view •−•−•), every graph
possessing a symmetry which swaps an odd number of edge pairs is a zero graph. For example, the 4-wheel
12 ∧ 13 ∧ 14 ∧ 15 ∧ 23 ∧ 25 ∧ 34 ∧ 45 = I ∧ · · · ∧ V III or the 2ℓ-wheel at any ℓ > 1 is such; here, the reflection
symmetry is I ⇄ III, V ⇄ V II, and V I ⇄ V III.
2 An oriented graph equals minus itself, hence it is a zero graph if there is a permutation of labels for its internal
vertices such that the adjacency tables for the two vertex labellings coincide but the two realisations of the same
graph differ by the ordering of outgoing edges at an odd number of internal vertices (see Example 3 below).
3 Let the enumeration of vertices in every such term in the sum start running over the enumerated vertices in
γ2 until v is reached. Now the enumeration counts the vertices in the graph γ1 and then it resumes with the
remaining vertices (if any) that go after v in γ2.
347
is a Lie bracket on the vector space G of non-oriented graphs.4
Lemma 2. The operator d(graph) = [•−•, graph] is a differential: d2 = 0.
In effect, the mapping d blows up every vertex v in its argument in such a way that whenever
the number of adjacent vertices N(v) ⩾ 2 is sufficient, each end of the inserted edge •−• is
connected with the rest of the graph by at least one edge.
Summarising, the real vector space G of non-oriented graphs is a differential graded Lie
algebra (dgLa) with Lie bracket [·, ·] and differential d = [•−•, ·]. The graphs γ5 and γ3 from
Examples 1 and 2 are d-cocycles. Neither is exact, hence marking a nontrivial cohomology class
in the non-oriented graph complex.
Theorem 3 ([7, Th. 5.5]). At every ` ∈ N in the connected graph complex there is a d-cocycle on
2`+1 vertices and 4`+2 edges. Such cocycle contains the (2`+1)-wheel in which, by definition,
the axis vertex is connected with every other vertex by a spoke so that each of those 2` vertices
is adjacent to the axis and two neighbours; the cocycle marked by the (2` + 1)-wheel graph can
contain other (2`+ 1, 4`+ 2)-graphs (see Example 1 and [5]).
I.2. The oriented graphs under study are built over n sinks from k wedges ←i−α • −j→α (here
←i−α ≺−j→α ) so that every edge is decorated with its own summation index which runs from 1 to
the dimension of a given affine Poisson manifold (N ,P). Each edge −→i encodes the derivation
∂/∂xi of the arrowhead object with respect to a local coordinate xi on N . By placing an αth
copy P iαjα(x) of the Poisson bi-vector P in the wedge top (1 ⩽ α ⩽ k), by taking the product
of contents of the n + k vertices (and evaluating all objects at a point x ∈ N ), and summing
over all indices, we realise a polydifferential operator in n arguments; the operator coefficients
are differential-polynomial in P. Totally skew-symmetric operators of differential order one in
each argument are well-defined n-vectors on the affine manifolds N .
The space of multi-vectors G encoded by oriented graphs is equipped with a graded Lie
algebra structure, namely the Schouten bracket [[·, ·]]. Its realisation in terms of oriented graphs
is shown in [2, Remark 4]. Recall that by definition the bi-vectors P at hand are Poisson by
satisfying the Jacobi identity [[P,P]] = 0. The Poisson differential ∂P = [[P, ·]] now endows
the space of multi-vectors on N with the differential graded Lie algebra (dgLa) structure. The
cohomology groups produced by the two dgLa structures introduced so far are correlated by the
edge orientation mapping O~r.
Theorem 4 ([8] and [12, App. K]). Let γ ∈ ker d be a cocycle on k vertices and 2k− 2 edges in
the non-oriented graph complex. Denote by {Γ} ⊂ G the subspace spanned by all those bi-vector
graphs Γ which are obtained from (each connected component in) γ by adding to it two edges to
the new sink vertices and then by taking the sum of graphs with all the admissible orientations of
the old 2k − 2 edges (so that a set of Kontsevich graphs built of k wedges is produced). Then in
that subspace {Γ} there is a sum of graphs that encodes a nonzero Poisson cocycle Q(P) ∈ ker ∂P .
Consequently, to find some cocycle Q(P) in the Poisson complex on any affine Poisson
manifold it suffices to find a cocycle in the non-oriented graph complex and then consider the sum
of graphs which are produced by the orientation mapping O~r. On the other hand, to list all the
∂P -cocycles Q(P) encoded by the bi-vector graphs made of k wedges ←•→, one must generate
all the relevant oriented graphs and solve the equation .∂P(Q) = 0 via [[P,P]] = 0, that is, solve
4 The postulated precedence or antecedence of the wedge product of edges from γ1 with respect to the edges
from γ2 in every graph within γ1 ◦i γ2 produce the operations ◦i which coincide with or, respectively, differ from
Definition 1 by the sign factor (−)#E(γ1)·#E(γ2). The same applies to the Lie bracket of graphs [γ1, γ2] if the
operation γ1 ◦i γ2 is the insertion of γ2 into γ1 (as in [11]). Anyway, the notion of d-cocycles which we presently
recall is well defined and insensitive to such sign ambiguity.
348
graphically the factorisation problem [[P, Q(P)]] = ♢(P, [[P,P]]) in which the cocycle condition
in the left-hand side holds by virtue of the Jacobi identity in the right. Such construction of
some and classification (at a fixed k > 0) of all universal infinitesimal symmetries of Poisson
brackets are the problems which we explore in this paper.
Remark 1. To the best of our knowledge [10], in a bi-vector graph Q(P) = O~r(γ), at every
internal vertex which is the tail of two oriented edges towards other internal vertices, the edge
ordering Left ≺ Right is inherited from a chosen wedge product E(γ) of edges in the non-
oriented graph γ. How are the new edges towards the sinks ordered, either between themselves
at a vertex or with respect to two other oriented edges, coming from γ and issued from different
vertices in Q(P) ? Our findings in [6] will help us to verify the order preservation claim and
assess answers to this question.
1. The Kontsevich graph calculus
Definition 2. Let us consider a class of oriented graphs on n+k vertices labelled 0, . . ., n+k−1
such that the consecutively ordered vertices 0, . . ., n−1 are sinks, and each of the internal vertices
n, . . ., n+ k− 1 is a source for two edges. For every internal vertex, the two outgoing edges are
ordered using L ≺ R: the preceding edge is labelled L (Left) and the other is R (Right). An
oriented graph on n sinks and k internal vertices is a Kontsevich graph of type (n, k).
For the purpose of defining a graph normal form, we now consider a Kontsevich graph Γ
together with a sign s ∈ {0,±1}, denoted by concatenation of the symbols: sΓ.
Notation (Encoding of the Kontsevich graphs). The format to store a signed graph sΓ for a
Kontsevich graph Γ is the integer number n > 0, the integer k ⩾ 0, the sign s, followed by the
(possibly empty, when k = 0) list of k ordered pairs of targets for edges issued from the internal
vertices n, . . ., n+ k − 1, respectively. The full format is then (n, k, s; list of ordered pairs).
Definition 3 (Normal form of a Kontsevich graph). The list of targets in the encoding of a
graph Γ can be considered as a 2k-digit integer written in base-(n + k) notation. By running
over the entire group S kk × (Z2) , and by this over all the different re-labellings of Γ, we obtain
many different integers written in base-(n + k). The absolute value |Γ| of Γ is the re-labelling
of Γ such that its list of targets is minimal as a nonnegative base-(n+ k) integer. For a signed
graph sΓ, the normal form is the signed graph t|Γ| which represents the same polydifferential
operator as sΓ. Here we let t = 0 if the graph is zero (see Example 3 below). 4 r R-r 3
Example 3 (Zero Kontsevich graph). Consider the graph with the encoding @ BL@  B
2 3 1 0 1 0 1 2 3. The swap of vertices 2 ⇄ 3 is a symmetry of this graph, Rr B
yet it also swaps the ordered edges (4→ 2) ≺ (4→ 3), producing a minus sign. r
	2@RBBN r
Equal to minus itself, this Kontsevich graph is zero. 0 1
Notation. Every Kontsevich graph Γ on n sinks (or every sum Γ of such graphs) yields the
sum Alt Γ of Kontsevich graphs which is totally skew-symmetric with respect to the n sinks
content s1, . . ., sn. Indeed, let ∑
(Alt Γ)(s1, . . . , sn) = (−)σ Γ(sσ(1), . . . , sσ(n)). (1)
σ∈Sn
Due to skew-symmetrisation, the sum of graphs Alt Γ can contain zero graphs or repetitions.
Example 4 (The Jacobiator). The left-hand side of the Jacobi identity is a skew sum of
Kontsevich graphs (e.g. it is obtained byrskew-symmetrizring the firstrterm)
• • r r L	r@ r	
Ri j k
 ?BBN := 	@R @Rr − riH kj 
r
 Hjr − 	r i j	r@Rr@kRr . (2)
1 2 3 1 2 3 1 2 3 1 2 3
The default ordering of edges is the one which34w9e see.
Definition 4 (Leibniz graph). A Leibniz graph is a graph whose vertices are either sinks, or
the sources for two arrows, or the Jacobiator (which is a source for three arrows). There must
be at least one Jacobiator vertex. The three arrows originating from a Jacobiator vertex must
land on three distinct vertices. Each edge falling on a Jacobiator works by the Leibniz rule on
the two internal vertices in it.
Example 5. The Jacobiator itself is a Leibniz graph (on one tri-valent internal vertex).
Definition 5 (Normal form of a Leibniz graph with one Jacobiator). Let Γ be a Leibniz graph
with one Jacobiator vertex Jac. From (2) we see that expansion of Jac into a sum of three
Kontsevich graphs means adding one new edge w → v (namely joining the internal vertices w
and v within the Jacobiator). Now, from Γ construct three Kontsevich graphs by expanding Jac
using (2) and letting the edges which fall on Jac in Γ be directed only to v in every new graph.
Next, for each Kontsevich graph find the relabelling τ which brings it to its normal form and
re-express the edge w → v using τ . Finally, out of the three normal forms of three graphs pick
the minimal one. By definition, the normal form of the Leibniz graph Γ is the pair: normal
form of Kontsevich graph, that edge τ(w)→ τ(v).
We say that a sum of Leibniz graphs is a skew Leibniz graph Alt Γ if it is produced from a
given Leibniz graph Γ by alternation using formula (1).
Definition 6 (Normal form of a skew Leibniz graph with one Jacobiator). Likewise, the normal
form of a skew Leibniz graph Alt Γ is the minimum of the normal forms of Leibniz graphs
(specifically, of the graph but not edge encodings) which are obtained from Γ by running over
the group of permutations of the sinks content.
Lemm(a 5 ([3]). )In order to show that a sum S of weighted skew-symmetric Kontsevich graphsvanishes for all Poisson structures P, it suffices to express S as a sum of skew Leibniz graphs:
S = ♢ P, Jac(P) .
1.1. Formulation of the problem
Let P →7 P + εQ(P) + ō(ε) be a deformation of bi-vectors that preserves their property to be
Poisson at least infinitesimally on all affine manifolds: [[P + εQ + ō(ε),P + εQ + ō(ε)]] = ō(ε).
Expanding and equating the first order terms, we obtain the equation P Q P .[[ , ( )]] = 0 via
[[P,P]] = 0. The language of( Konts)evich graphs allows one to convert this infinite analytic
problem within a given set-up N n,P in dimension n into a set of finite combinatorial problems
whose solutions are universal for all Poisson geometries in all dimensions n <∞.
Our first task in this paper is to find the space of flows Ṗ = Q(P) which are encoded by the
Kontsevich graphs on a fixed number of internal v(ertices k, )for 1 ⩽ k ⩽ 4. Specifically, we solvethe graph equation
[[P,Q(P)]] = ♢ P, Jac(P) (3)
for the Kontsevich bi-vector graph(s Q(P) and)Leibniz graphs ♢. We then factor out the Poisson-trivial and improper solutions, that is, we quotient out all bi-vector graphs that can be written
in the form Q(P) = [[P, X]] + ∇ P, Jac(P) , where X is a Kontsevich one-vector graph and
∇ is a Leibniz bi-vector graph. (The bi-vectors [[P, X]] make [[P,Q(P)]] vanish since [[P, ·]] is a
differential. The improper graphs ∇(P, Jac(P)) vanish identically at all Poisson bi-vectors P on
every affine manifold.
Before solving factorisation problem (3) with respect to the operator ♢, we must generate
– e.g., iteratively as described below – an ansatz for expansion of the right-hand side using skew
Leibniz graphs with undetermined coefficients.
350
1.2. How to generate Leibniz graphs iteratively
The first step is to construct a layer of skew Leibniz graphs, that is, all skew Leibniz graphs
which produce at least one graph in the input (in the course of expansion of skew Leibniz graphs
using formula (1) and then in the course of expansion of every Leibniz graph at hand to a sum
of Kontsevich graphs). For a given Kontsevich graph in the input S, one such Leibniz graph
can be constructed by contracting an edge between two internal vertices so that the new vertex
with three outgoing edges becomes the Jacobiator vertex. Note that these Leibniz graphs, which
are designed to reproduce S, may also produce extra Kontsevich graphs that are not given in
the input. Clearly, if the set of Kontsevich graphs in S coincides with the set of such graphs
obtained by expansion of all the Leibniz graphs in the ansatz at hand, then we are done: the
extra graphs, not present in S, are known to all cancel. Yet it could very well be that it is not
possible to express S using only the Leibniz graphs from the set accumulated so far. Then we
proceed by constructing the next layer of skew Leibniz graphs that reproduce at least one of the
extra Kontsevich graphs (which were not present in S but which are produced by the graphs in
the previously constructed layer(s) of Leibniz graphs). In this way we proceed iteratively until
no new Leibniz graphs are found; of course, the overall number of skew Leibniz graphs on a
fixed number of internal vertices and sinks is bounded from above so that the algorithm always
terminates. Note that the Leibniz graphs obtained in this way are the only ones that can in
principle be involved in the vanishing mechanism for S.
Notation. Let v be a graph vertex. Denote by N(v) the set of neighbours of v, by H(v) the
(possibly empty) set of arrowheads of oriented edges issued from the vertex v, and by T (v)
the (possibly empty) set of tails for oriented edges pointing at v. For example, #N(•) = 2,
#H(•) = 2, and T (•) = ∅ for the top • of the wedge graph ←•→.
Algorithm Consider a skew-symmetric sum S0 of oriented Kontsevich graphs with real
coefficients. Let Stotal := S0 and create an empty table L. We now describe the ith iteration of
the algorithm (i ⩾ 1).
Loop ⟳ Run over all Kontsevich graphs Γ in Si−1: for each internal vertex v in a graph Γ, run
over all vertices w ∈ T (v) in the set of tails of oriented edges pointing at v such that v ∈/ T (w) and
H(v)∩H(w) = ∅ for the sets o(f targets of oriented edges issued from v and w. Replace the edgew → v connecting w to v by Jacobiator )(2), that is, by a single vertex Jac with three outgoing
edges and such that T (Jac) = T (v) \ w ∪ T (w) and H(Jac) = H(v) ∪ (H(w) \ v) =: {a, b, c}.
Because we shall always expand the skew Leibniz graphs in what follows, we do not actually
contract the edge w → v (to obtain a Leibniz graph explicitly) in this algorithm but instead we
continue working with the original Kontsevich graphs containing the distinct vertices v and w.
For every edge that points at w, redirect it to v. Sum over the three cyclic permutations that
provide three possible ways to attach the three outgoing edges for v and w (excluding w → v)
– now seen as the outgoing edges of the Jacobiator – to the target vertices a, b, and c depending
on w and v. Skew-symmetrise5 each of these three graphs with respect to the sinks by applying
formula (1).
For every marked edge w → v indicating the internal edge in the Jacobiator vertex in a
graph, replace each sum of the Kontsevich graphs which is skew with respect to the sink content
by using the normal form of the respective skew Leibniz graph, see Definition 6. If this skew
Leibniz graph is not contained in L, apply the Leibniz rule(s) for all the derivations acting on the
Jacobiator vertex Jac. Otherwise speaking, sum over all possible ways to attach the incoming
edges of the target v in the marked edge w → v to its source w and target. To each Kontsevich
5 This algorithm can be modified so that it works for an input which is not skew, namely, by replacing skew
Leibniz graphs by ordinary Leibniz graph(s (that is,)by skipping the skew-symmetrisation). For example, thisstrategy has been used in [3, 4] to show that the Kontsevich star product ⋆ mod ō(ℏ4) is associative: although
the associator (f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) = ♢ P, Jac(P) 3is5n1ot skew, it does vanish for every Poisson structure P.
graph resulting from a skew Leibniz graph at hand assign the same undetermined coefficient,
and add all these weighted Kontsevich graphs to the sum Si. Further, add a row to the table L,
that new row containing the normal form of this skew Leibniz graph (with its coefficient that
has been made common to the Kontsevich graphs).
By now, the new sum of Kontsevich graphs Si is fully composed. Having thus finished
the current iteration over all graphs Γ in the set Si−1, redefine the algebraic sum of weighted
graphs Stotal by subtracting from it the newly formed sum Si. Collect similar terms in Stotal
and reduce this sum of Kontsevich graphs modulo their skew-symmetry under swaps L ⇄ R
of the edge ordering in every internal vertex, so that all zero graphs (see Example 3) also get
eliminated. ⟳ end loop
Increment i by 1 and repeat the iteration until the set of weighted (and skew) Leibniz
graphs L stabilizes. Finally, solve –with respect to the coefficients of skew Leibniz graphs – the
linear algebraic system obtained from the graph equation Stotal = 0 for the sum of Kontsevich
graphs which has been produced from its initial value S0 by running the iterations of the above
algorithm.
Example 6. For the skew sum of Kontsevich graphs in the right-hand side of (2), the algorithm
would produce just one skew Leibniz graph: namely, the Jacobiator itself.
Example 7 (The 3-wheel). For the Kontsevich tetrahedral flow Ṗ = Q1:6/2(P) on the spaces of
Poisson bi-vectors P, see [8, 9] and [1, 2], building a sufficient set of skew Leibniz graphs in the
r.-h. s. of factorisation problem (3) requires two iterations of the above algorithm: 11 Leibniz
graphs are produced at the first step and 50 more are added by the second step, making 61 in
total. One of the two known solutions of this factorisation problem [2] then consists of 8 skew
Leibni(z graphs (e)xpanding to 27 Leibniz graphs). In turn, as soon as all the Leibniz rules actingon the Jacobiators are processed and every Jacobiator vertex is expanded via (2), the right-hand
side ♢ P, Jac(P) equals the sum of 39 Kontsevich graphs which are assembled into the 9 totally
skew-symmetric terms in the left-hand side [[P,Q1:6/2]]. ( )
Example 8 (The 5-wheel). Consider the factorisation problem [[P,O~r(γ5)]] = ♢ P, Jac(P) for
the pentagon-wheel deformation Ṗ = O~r(γ5)(P) of Poisson bi-vectors P, see [10, 13] and [6].
The ninety skew Kontsevich graphs encoding the bi-vector O~r(γ5) are obtained by taking all the
admissible orientations of two (5+1)-vertex graphs γ5, one of which is the pentagon wheel with
five spokes, the other graph complementing the former to a cocycle in the non-oriented graph
complex. By running the iterations of the above algorithm for self-expanding construction of the
Leibniz tri-vector graphs in this factorisation problem, we achieve stabilisation of the number
of such graphs after the seventh iteration, see Table 1 below.
Table 1. The number of skew Leibniz graphs produced iteratively for [[P,O~r(γ5)]].
No. iteration i 1 2 3 4 5 6 7 8
# of graphs 1518 14846 41031 54188 56318 56503 56509 56509
of them new all +13328 +26185 +13157 +2130 +185 +6 none
2. Generating the Kontsevich multi-vector graphs
Let us return to problem (3): it is the ansatz for bi-vector Kontsevich graphs Q(P) with k
internal vertices, as well as the Kontsevich 1-vectors X with k − 1 internal vertices (to detect
trivial terms Q(P) = [[P,X]]) which must be generated at a given k. (At 1 ⩽ k ⩽ 4, one can still
352
expand with respect to all the Leibniz graphs in the r.-h.s. of (3), not employing the iterative
algorithm from §1.2. So, a generator of the Kontsevich (and Leibniz) tri-vectors will also be
described presently.)
The Kontsevich graphs corresponding to n-vectors are those graphs with n sinks (each
containing the respective argument of n-vector) in which exactly one arrow comes into each
sink, so that the order of the differential operator encoded by an n-vector graph equals one w.r.t.
each argument, and which are totally skew-symmetric in their n arguments. Let us explain how
one can economically obtain the set of one-vectors and skew-symmetric bi- and tri-vectors with
k internal vertices in three steps (including graphs with eyes • ⇄ • but excluding graphs with
tadpoles). This approach can easily be extended to the construction of n-vectors with any n ⩾ 1.
2.1. One-vectors
Each one-vector under study is encoded by a Kontsevich graph with one sink. Since the sink has
one incoming arrow, there is an internal vertex as the tail of this incoming arrow. The target of
another edge issued from this internal vertex can be any internal vertex other then itself.
Step 1. Generate all Kontsevich graphs on k − 1 internal vertices and one sink (i.e. graphs
including those with eyes yet excluding those with tadpoles, and not necessarily of differential
order one with respect to the sink content).
Step 2. For every such graph with k− 1 internal vertices, add the new sink and make it a target
of the old sink, which itself becomes the kth internal vertex. Now run over the k − 1 internal
vertices excluding the old sink and – via the Leibniz rule – make every such internal vertex the
second target of the old sink.
2.2. Bi-vectors
There are two cases in the construction of bi-vectors encoded by the Kontsevich graphs. At
all k ⩾ 1 the first variant is referred to those graphs with an internal vertex that has both sinks
as targets.
Variant 1: Step 1. Generate all k-vertex graphs on k − 1 internal vertices and one sink.
Variant 1: Step 2. For every such graph, add two new sinks and proclaim them as targets of
the old sink.
Note that the obtained graphs are skew-symmetric.
The second variant produces those graphs which contain two internal vertices such that one
has the first sink as target and the other has the second sink as target. The second target of
either such internal vertex can be any internal vertex other then itself. Note that for k = 1 only
the first variant applies.
Variant 2: Step 1. Generate all k-vertex Kontsevich graphs on k − 2 internal vertices and two
sinks. These sinks now become the (k − 1)th and kth internal vertices.
Variant 2: Step 2. For every such graph, add two new sinks, make the first new sink a target
of the first old sink and make the second new sink a target of the second old sink. Now run
over the k − 1 internal vertices excluding the first old sink, each time proclaiming an internal
vertex the second target of the first old sink. Simultaneously, run over the k−1 internal vertices
excluding the second old sink and likewise, declare an internal vertex to be the second target of
the second old sink.
Variant 2: Step 3. Skew-symmetrise each graph with respect to the content of two sinks using (1).
2.3. Tri-vectors
For k ⩾ 3, there exist two variants of tri-vectors. The first variant at all k ⩾ 2 yields those
Kontsevich graphs with two internal vertices such that one has two of the three sinks as its
targets while another internal vertex has the third sink as one of its targets. The second target
353
of this last vertex can be any internal vertex other then itself. The second variant contains
those graphs with three internal vertices such that the first one has the first sink as a target, the
second one has the second sink as a target, and the third one has the third sink as a target. For
each of these three internal vertices with a sink as target, the second target can be any internal
vertex other then itself.
Variant 1: Step 1. Generate all k-vertex Kontsevich graphs on k − 2 internal vertices and two
sinks.
Variant 1: Step 2. For every such graph, add three new sinks, make the first two new sinks
the targets of the first old sink and make the third new sink a target of the second old sink.
Now run over the k − 1 internal vertices excluding the second old sink and every time declare
an internal vertex the second target of the second old sink.
Variant 1: Step 3. Skew-symmetrise all graphs at hand by applying formula (1) to each of them.
Note that for k = 1 there are no tri-vectors encoded by Kontsevich graphs and also note that
for k = 2 only the first variant applies.
Variant 2: Step 1. Generate all Kontsevich graphs on k − 3 internal vertices and three sinks.
Variant 2: Step 2. For every such graph, add three new sinks, make the first new sink a target
of the first old sink, make the second new sink a target of the second old sink and make the
third new sink a target of the third old sink. Now run over the k− 1 internal vertices excluding
the first old sink and declare every such internal vertex the second target of the first old sink.
Independently, run over the k − 1 internal vertices excluding the second old sink and declare
each internal vertex to be the second target of the second old sink. Likewise, run over the k− 1
internal vertices excluding the third old sink and declare each internal vertex to be the second
target of the third old sink.
Variant 2: Step 3 Skew-symmetrise all the graphs at hand using (1).
2.4. Non-iterative generator of the Leibniz n-vector graphs
The following algorithm generates all Leibniz graphs with a prescribed number of internal
vertices and sinks. Note that not only multi-vectors, but also all graphs of arbitrary differential
order with respect to the sinks can be generated this way.
Step 1: Generate all Kontsevich graphs of prescribed type on k−1 internal vertices and n sinks,
e.g., all n-vectors.
Step 2: Run through the set of these Kontsevich graphs and in each of them, run through the
set of its internal vertices v. For every vertex v do the following: re-enumerate the internal
vertices so that this vertex is enumerated by k − 1. This vertex already targets two vertices, i
and j, where i < j < k − 1. Proclaim the last, (k − 1)th vertex to be the placeholder of the
Jacobiator (see (2)), so we must still add the third arrow. Let a new index ` run up to i − 1
starting at n if only the n-vectors are produced.6 For every admissible value of `, generate a
new graph where the `th vertex is proclaimed the third target of the Jacobiator vertex k − 1.
(Restricting ` by < i, we reduce the number of possible repetitions in the set of Leibniz graphs.
Indeed, for every triple ` < i < j, the same Leibniz graph in which the Jacobiator stands on
those three vertices would be produced from the three Kontsevich graphs: namely, those in
which the (k− 1)th vertex targets at the `th and ith, at the `th and jth, and at the ith and jth
vertices. In these three cases it is the jth, ith, and `th vertex, respectively, which would be
appointed by the algorithm as the Jacobiator’s third target.)
We use this algorithm to generate the Leibniz tri- and bi-vector graphs: to establish
Theorem 6, we list all possible terms in the right-hand side of factorisation problem (3) at k ⩽ 4
and then we filter out the improper bi-vectors in the found solutions Q(P).
6 If we want to generate not only n-vectors but all graphs of arbitrary differential orders, then we let ℓ start at 0
(so that the sinks are included).
354
Remark 2. There are at least 265,495 Leibniz graphs on 3 sinks and 6 internal vertices of which
one is the Jacobiator vertex. When compared with Table 1 on p. 352, this estimate suggests why
at large k ≳ 5, the breadth-first-search iterative algorithm from §1.2 generates a smaller number
of the Leibniz tri-vector graphs, namely, only the ones which can in principle be involved in the
factorisation under study.
3. Main result
Theorem 6 (k ⩽ 4). The few-vertex solutions of problem (3) are these (note that disconnected
Kontsevich graphs in Q(P) are allowed !):
• k = 1: The dilation Ṗ = P is a unique, nontrivial and proper solution.
• k = 2: No solutions exist (in particular, neither trivial nor improper).
• k = 3: There are no solutions: neither Poisson-trivial nor Leibniz bi-vectors.
• k = 4: A unique nontrivial and proper solution is the Kontsevich tetrahedral flow Q1: 6 (P)
2
from Example 2 (see [8, 9] and [1, 2]). There is a one-dimensional space of Poisson trivial (still
proper) solutions [[P, X]]; the Kontsevich 1-(vector X on three internal vertices is drawn in [2,App. F]. Intersecting with the former by {0}, there is a t)hree-dimensional space of improper (still
Poisson-nontrivial) solutions of the form ∇ P, Jac(P) .
None of the solutions Q(P) known so far contains any 2-cycles (or “eyes” •⇄ •).7
We now report a classification of Poisson bi-vector symmetries Ṗ = Q(P) which are given
by those Kontsevich graphs Q = O~r(γ) on k internal vertices that can be obtained at 5 ⩽ k ⩽ 9
by orienting k-vertex connected graphs γ without double edges. By construction, this extra
assumption keeps only those Kontsevich graphs which may not contain eyes.
We first find such graphs γ that satisfy d(γ) = 0, then we exclude the coboundaries γ = d(γ′)
for some graphs γ′ on k − 1 vertices and 2k − 3 edges.
Theorem 7 (5 ⩽ k ⩽ 8). Consider the vector space of non-oriented connected graphs on
k vertices and 2k − 2 edges, without tadpoles and without multiple edges. All nontrivial d-
cocycles for 5 ⩽ k ⩽ 8 are exhausted by the following ones: r  
• k = 5, 7: No solutions. r r
• k = 6: A unique solution8 is given by the Kontse- r r r 5γ = + r r
vi(ch–Willwac)her pentagon-wheel cocycle (see Exam-
5 2 r r
ple 1). The established factorisation r r[[P,O~r(γ5)]] =  
♢ P, Jac(P) will be addressed in a separate paper (see [6]).
• k = 8: The only solution γ7 consists of the heptagon-wheel and 45 other graphs (see Table 2,
in which the coefficient of heptagon graph is 1 in bold, and [5]).
Remark 3. The wheel graphs are built of triangles. The differential d cannot produce any
triangle since multiple edges are not allowed. Therefore, all wheel cocycles are nontrivial. Note
also that every wheel graph with 2` spokes is invariant under a mirror reflection with respect to a
diagonal consisting of two edges attached to the centre. Hence there exists an edge permutation
that swaps 2`− 1 pairs of edges. By footnote 1 such graph equals zero.
Appendix A. How the orientation mapping O~r is calculated
The algorithm lists all ways in which a given non-oriented graph can be oriented in such a way
that it becomes a Kontsevich graph on two sinks. It consists of two steps:
(i) choosing the source(s) of the two arrows pointing at the first and second sink, respectively;
7 Finding solutions Q(P) with tadpoles or extra sinks –with fixed arguments – is a separate problem.
8 There are only 12 admissible graphs to build cocycles from; of these 12, as many as 6 are zero graphs. This
count shows to what extent the number of graphs decreases if one restricts to only the flows Q = O⃗r(γ) obtained
from cocycles γ ∈ ker(d) in the non-oriented graph com35pl5ex.
Table 2. The heptagon-wheel graph cocycle γ7.
Graph encoding Coeff. Graph encoding Coeff.
16 17 18 23 25 28 34 38 46 48 57 58 68 78 1 12 13 18 25 26 37 38 45 46 47 56 57 68 78 −7
12 14 18 23 27 35 37 46 48 57 58 67 68 78 −21/8 12 14 16 23 25 36 37 45 48 57 58 67 68 78 77/8
13 14 18 23 25 28 37 46 48 56 57 67 68 78 −77/4 13 16 17 24 25 26 35 37 45 48 58 67 68 78 −7
12 13 15 24 27 35 36 46 48 57 58 67 68 78 −35/8 14 15 17 23 26 28 37 38 46 48 56 57 68 78 49/4
12 13 18 24 26 37 38 46 47 56 57 58 68 78 49/8 12 16 18 27 28 34 36 38 46 47 56 57 58 78 −147/8
14 17 18 23 25 26 35 37 46 48 56 58 67 78 77/8 12 15 16 27 28 35 36 38 45 46 47 57 68 78 −21/8
12 13 18 26 27 35 38 45 46 47 56 57 68 78 −105/8 12 14 18 23 27 35 36 45 46 57 58 67 68 78 −35/8
12 14 18 23 27 36 38 46 48 56 57 58 67 78 7/8 14 15 16 23 26 28 37 38 46 48 57 58 67 78 −49/4
12 14 15 23 27 35 36 46 48 57 58 67 68 78 35/8 12 15 18 23 28 34 37 46 48 56 57 67 68 78 105/8
12 13 14 27 28 36 38 46 47 56 57 58 68 78 −49/8 12 14 17 23 26 37 38 46 48 56 57 58 68 78 −49/8
12 13 18 25 27 34 36 47 48 56 58 67 68 78 35/4 12 16 18 25 27 35 36 37 45 46 48 57 68 78 49/16
12 13 14 25 26 36 38 45 47 57 58 67 68 78 −119/16 12 13 18 25 27 35 36 46 47 48 56 57 68 78 7
12 13 15 24 28 36 38 47 48 56 57 67 68 78 49/8 12 14 18 25 28 34 36 38 47 57 58 67 68 78 −7
12 13 14 23 28 37 46 48 56 57 58 67 68 78 77/4 12 16 18 25 27 35 36 37 45 46 48 58 67 78 −77/16
12 15 17 25 26 35 36 38 45 47 48 67 68 78 −49/8 12 14 18 23 27 35 38 46 47 57 58 67 68 78 77/4
13 15 18 24 26 28 37 38 46 47 56 57 68 78 −49/4 12 14 15 23 27 36 38 46 48 57 58 67 68 78 35/2
13 14 18 25 26 28 36 38 47 48 56 57 67 78 −49/4 12 13 18 25 27 34 36 46 48 57 58 67 68 78 −105/8
12 14 18 23 28 35 37 46 48 56 57 67 68 78 −7 12 15 16 25 27 35 36 38 46 47 48 57 68 78 −7
12 14 18 23 28 36 38 46 47 56 57 58 67 78 −7 12 13 16 25 28 34 37 47 48 57 58 67 68 78 −147/16
12 15 16 25 27 35 36 38 46 47 48 58 67 78 49/8 12 13 17 25 26 35 37 45 46 48 58 67 68 78 −77/4
12 14 18 23 28 36 37 46 47 56 57 58 68 78 49/8 12 14 17 23 27 35 38 46 48 57 58 67 68 78 −49/8
12 13 15 26 27 35 36 45 47 48 58 67 68 78 −7 12 13 15 26 28 35 37 45 46 47 58 67 68 78 −7/4
12 13 18 24 28 35 38 46 47 57 58 67 68 78 7 12 14 18 23 26 36 38 47 48 56 57 58 67 78 −7
(ii) orienting the edges between the internal vertices in all admissible ways, so that only
Kontsevich graphs are obtained.
Step 1. Enumerate the k vertices of a given non-oriented, connected graph using 2, . . . , k + 1.
They become the internal vertices of the oriented graph. Now add the two sinks to the non-
oriented graph, the sinks enumerated using 0 and 1. Let a and b be a non-strictly ordered
(a ⩽ b) pair of internal vertices in the graph. Extend the graph by oriented edges a → 0 and
b→ 1 from vertices a and b to the sinks 0 and 1, respectively.
Remark 4. The choice of such a base pair, that is, the vertex or vertices from which two arrows
are issued to the sinks, is an external input in the orientation procedure. Let us agree that if,
at any step of the algorithm, a contradiction is achieved so that a graph at hand cannot be of
Kontsevich type, the oriented graph draft is discarded; one proceeds with the next options in
that loop, or if the former loop is finished, with the next level-up loops, or – having returned to
the choice of base vertices – with the next base. In other words, we do not exclude in principle
a possibility to have no admissible orientations for a particular choice of the base for a given
non-oriented graph.
Notation. Let v be an internal vertex. Recalling from p. 351 the notation for the set N(v) of
neighbours of v, the (initially empty) set H(v) of arrowheads of oriented edges issued from the
vertex v, and the (initially empty) set T (v) of tails for oriented edges pointing at v, we now put
by definition F (v) := N(v) \ (H(v) ∪ T (v)). In other words, F (v) is the subset of neighbours
connected with v by a non-oriented edge.
Step 2.1. Inambiguous orientation of (some) edges. Here we use that every internal vertex
of a Kontsevich graph should be the tail of exactly two outgoing arrows. We run over the
set of all internal vertices v. For every vertex such that the number of elements #H(v) = 2,
356
proclaim T (v) := N(v)\H(v), whence F (v) = ∅. If for a vertex v we have that #H(v) = 1 and
#F (v) = 1, then include F (v) ↪→ H(v), that is, convert a unique non-oriented edge touching v
into an outgoing edge issued from this vertex. If #H(v) = 0 and #F (v) = 2, also include
F (v) ↪→ H(v), effectively making both non-oriented edges outgoing from v.
Repeat the three parts of Step 2.1 while any of the sets F (v), T (v), or S(v) is modified for
at least one internal vertex v unless a contradiction is revealed. Summarising, Step 2.1 amounts
to finding the edge orientations which are implied by the choice of the base pair a, b and by all
the orientations of edges fixed earlier.
Step 2.2. Fixing the orientation of (some) remaining edges. Choose an internal vertex v such
that H(v) < 2 and such that H(v) 6= ∅ or T (v) 6= ∅, that is, choose a vertex that is not yet
equipped with two outgoing edges and that is attached to an oriented edge. If #H(v) = 1, then
run over the non-empty set F (v): for every vertex w in F (v), include {w} ↪→ H(v) and start
over at Step 2.1. Otherwise, i.e. if H(v) = ∅, run over all ordered pairs (u, v) of vertices in the
set F (v): for every such pair, make H(v) := {u,w} and start over at Step 2.1.
By realising Steps 1 and 2 we accumulate the sum of fully oriented Kontsevich graphs.
Acknowledgments
A.V.Kiselev thanks the Organising committee of the international conference ISQS’25 on
integrable systems and quantum symmetries (6–10 June 2017 in ČVUT Prague, Czech Republic)
for a warm atmosphere during the meeting. The authors are grateful to M. Kontsevich and
T. Willwacher for helpful discussion. We also thank Center for Information Technology of the
University of Groningen for providing access to Peregrine high performance computing cluster.
This research was supported in part by JBI RUG project 106552 (Groningen, The Netherlands).
A part of this research was done while R.Buring and A.V.Kiselev were visiting at the IHÉS
(Bures-sur-Yvette, France) and A.V.Kiselev was visiting at the MPIM (Bonn, Germany).
References
[1] Bouisaghouane A., Kiselev A.V. (2017) Do the Kontsevich tetrahedral flows preserve or destroy the
space of Poisson bi-vectors ? J. Phys.: Conf. Ser. 804 Proc. XXIV Int. conf. ‘Integrable Systems and
Quantum Symmetries’ (14–18 June 2016, ČVUT Prague, Czech Republic), Paper 012008, 10 p. (Preprint
arXiv:1609.06677 [q-alg])
[2] Bouisaghouane A., Buring R., Kiselev A. (2017) The Kontsevich tetrahedral flow revisited, J. Geom. Phys.
119, 272–285. (Preprint arXiv:1608.01710 [q-alg])
[3] Buring R., Kiselev A.V. (2017) On the Kontsevich ⋆-product associativity mechanism, PEPAN Letters 14:2,
403–407. (Preprint arXiv:1602.09036 [q-alg])
[4] Buring R., Kiselev A.V. (2017) The expansion ⋆ mod ō(ℏ4) and computer-assisted proof schemes in the
Kontsevich deformation quantization, Preprint arXiv:1702.00681 [math.CO]
[5] Buring R., Kiselev A.V., Rutten N. J. (2017) The heptagon-wheel cocycle in the Kontsevich graph complex,
J. Nonlin. Math. Phys. 24 Suppl. 1 ‘Local & Nonlocal Symmetries in Mathematical Physics’, 157–173.
(Preprint arXiv:1710.00658 [math.CO])
[6] Buring R., Kiselev A.V., Rutten N. J. (2017) Poisson brackets symmetry from the pentagon-wheel cocycle
in the graph complex, Preprint arXiv:1712.05259 [math-ph]
[7] Dolgushev V. A., Rogers C. L., Willwacher T. H. (2015) Kontsevich’s graph complex, GRT, and
the deformation complex of the sheaf of polyvector fields, Ann. Math. 182:3, 855–943. (Preprint
arXiv:1211.4230 [math.KT])
[8] Kontsevich M. (1997) Formality conjecture. Deformation theory and symplectic geometry (Ascona 1996,
D. Sternheimer, J.Rawnsley and S.Gutt, eds), Math. Phys. Stud. 20, Kluwer Acad. Publ., Dordrecht,
139–156.
[9] Kontsevich M. (2017) Derived Grothendieck–Teichmüller group and graph complexes [after T. Willwacher],
Séminaire Bourbaki (69ème année, Janvier 2017), no. 1126, 26 p.
[10] Kontsevich M. (2017) Private communication.
[11] Khoroshkin A., Willwacher T., Živković M. (2017) Differentials on graph complexes, Adv. Math. 307, 1184–
1214. (Preprint arXiv:1411.2369 [q-alg])
357
[12] Willwacher T. (2015) M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra, Invent.
Math. 200:3, 671–760. (Preprint arXiv:1009.1654 [q-alg])
[13] Willwacher T. (2017) Private communication.
358
Chapter 15
The Kontsevich tetrahedral flow
revisited
This chapter is based on the peer-reviewed journal publication A. Bouisaghouane, R. Bur-
ing, and A.V. Kiselev, J. Geom. Phys., 119, 272–285, 2017. (Preprint arXiv:1608.01710
[q-alg] – 29 p.)
Commentary. In reference to Part I of the dissertation, the material of this chapter is
used in Chapter 2, §5.1, §6.1, Chapter 7, Chapter 8, and Chapter 9. With this chaper we
amend the formula for the tetrahedral flow (originally proposed by Kontsevich in Ascona
’96). The Poisson cocycle factorization problem which we study here is a particular
example of the general construction, which we address in Chapter 17. In the meantime,
we discover multiple solutions to that factorization problem (see §3.5.2 of Part I).
359
THE KONTSEVICH TETRAHEDRAL FLOW REVISITED
A. BOUISAGHOUANE, R. BURING, AND A. KISELEV∗,§
Abstract. We prove that the Kontsevich tetrahedral flow Ṗ = Qa:b(P), the right-
hand side of which is a linear combination of two differential monomials of degree four
in a bi-vector P on an affine real Poisson manifold Nn, does infinitesimally preserve
the space of Poisson bi-vectors on Nn if and only if the two monomials in Qa:b(P) are
balanced by the ratio a : b = 1 : 6. The proof is explicit; it is written in the language
of Kontsevich graphs.
Introduction. The main question which we address in this paper is how Poisson struc-
tures can be deformed in such a way that they stay Poisson. We reveal one such method
that works for all Poisson structures on affine real manifolds; the construction of that
flow on the space of bi-vectors was proposed in [11]: the formula is derived from two
differently oriented tetrahedral graphs on four vertices. The flow is a linear combina-
tion of two terms, each quartic-nonlinear in the Poisson structure. By using several
examples of Poisson brackets with high polynomial degree coefficients, the first and last
authors demonstrated in [1] that the ratio 1 : 6 is the only possible balance at which
the tetrahedral flow can preserve the property of the Cauchy datum to be Poisson. But
does the Kontsevich tetrahedral flow Ṗ = Q1:6(P) with ratio 1 : 6 actually preserve the
space of all Poisson bi-vectors?
We prove the infinitesimal version of this claim: namely, we show that [[P ,Q1:6(P)]] =
0 for every bi-vector P satisfying the master-equation [[P ,P ]] = 0 for Poisson structures.
The proof is graphical: to prove that equation (2) holds, we find an operator ♢, encoded
by using the Kontsevich graphs, that solves equation (10). We also show that there is
no universal mechanism (that would involve the language of Kontsevich graphs) for the
tetrahedral flow to be trivial in the respective Poisson cohomology.
The text is structured as follows. In section 1 we recall how oriented graphs can be
used to encode differential operators acting on the space of multivectors. In particular,
differential polynomials in a given Poisson structure are obtained as soon as a copy of
that Poisson bi-vector is placed in every internal vertex of a graph. Specifically, the
right-hand side Qa:b = a · Γ1 + b · Γ2 of the Kontsevich (tetrahe)dral flow Ṗ = Qa:b(P) on
the space of bi-vectors on an affine Poisson manifold Nn,P is a linear combination
Date: 24 November 2016, revised 12 March 2017.
2010 Mathematics Subject Classification. 53D55, 58E30, 81S10; secondary 53D17, 58Z05, 70S20.
Key words and phrases. Poisson bracket, affine manifold, graph complex, tetrahedral flow, Poisson
cohomology.
Address: Johann Bernoulli Institute for Mathematics and Computer Science, University of Gronin-
gen, P.O. Box 407, 9700 AK Groningen, The Netherlands. ∗E-mail: A.V.Kiselev@rug.nl
∗ Institut des Hautes Études Scientifiques, 35 route de Chartres, Bures-sur-Yvette, F-91440 France.
§ Present address: Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany.
360
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 361
of two differential monomials, Γ1(P) and Γ2(P), of degree four in the bi-vector P that
evolves.
We determine at which balance a : b the Kontsevich tetrahedral flow Ṗ = Qa:b(P)
infinitesimally preserves the space of Poisson bi-vectors, that is, the bi-vector P +
εQa:b(P) + ō(ε) satisfies the equation
[[P + εQ .a:b(P) + ō(ε),P + εQa:b(P) + ō(ε)]] = ō(ε) via [[P ,P ]] = 0; (1)
here we denote by [[·, ·]] the Schouten bracket (see formula (5) on page 364). Expanding,
we obtain the cocycle condition,
.
[[P ,Qa:b(P)]] = 0 via [[P ,P ]] = 0, (2)
with respect to the Poisson differential ∂P = [[P , ·]]. Viewed as an equation with respect
to the ratio a : b, condition (2) is the main object of our study.
Recent counterexamples [1] show that the bi-vector P + εQa:b(P) + ō(ε) can stay
Poisson only if the balance a : b in Qa:b is equal to 1 : 6. We now prove the infinitesimal
part of sufficiency: the deformation P+εQ1:6(P)+ō(ε) is always infinitesimally Poisson,
whence the balance a : b = 1 : 6 in the Kontsevich tetrahedral flow is universal for all
Poisson bi-vectors P on all affine manifolds Nn. The proof is explicit: in section 2 we
reveal the mechanism of factorization – via the Jacobi identity – in (2) at a : b = 1 : 6.
On the left-hand side of factorization problem (2) we expand the Poisson differential of
the Kontsevich tetrahedral flow at the balance 1 : 6 into the sum of 39 graphs (see Fig-
ure 3 on page 367 and Table 2 in Appendix A). On the other side of that factorization,
we take the sum that runs with undetermined coefficients over all those fragments of
differential consequences of the Jacobi identity [[P ,P ]] = 0 which are known to vanish
independently. We then find a linear polydifferential operator ♢(P , ·) that acts on the
filtered components of the Jacobiator Jac (P) := [[P ,P ]] fo(r the bi-vector P); the oper-
ator ♢ provides the factorization [[P ,Q1:6(P)]](f, g, h) = ♢ P , Jac (P)(·, ·, ·) (f, g, h) of
the ∂P-cocycle condition, see (2), through the Jacobi identity Jac (P) = 0. To describe
the differential operators that produce such consequences of the Jacobi identity, we use
the pictorial language of graphs: every internal vertex contains a copy of the bi-vector P
and the operators are reduced by using its skew-symmetry. There remain 7, 025 graphs,
the coefficients of which are linear in the unknowns. We now solve the arising inho-
mogeneous linear algebraic system. Its solution yields a polydifferential operator ♢,
encoded using Leibniz graphs (see p. 371), that provides the sought-for factorization
[[P ,Q1:6]] = ♢(P , Jac(P)). It is readily seen from formula (11) that this operator ♢ is
completely determined by only 8 nonzero coefficients (out of 1132 total).1 Therefore,
although finding an operator ♢ was hard, verifying that it does solve the factorization
problem has become almost immediate, as we show in the proof of Theorem 3. We
thereby establish the main result (namely, Corollary 4 on page 369). The paper con-
cludes with the formulation of an open problem about the integration of tetrahedral
flow in (1) to higher order expansions in ε, see (13) on p. 373.
In Appendix B we outline a different method to tackle the factorization problem,
namely, by making the Jacobi identity visible in (2) by perturbing the original struc-
ture P 7→ P̃ in such a way that P̃ is not Poisson and Q1:6(P̃) 6= 0. Hence P̃ contributes
1The maximally detailed description of that solution ♢ is contained in Appendix A.
362 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
to the right-hand side of (2) such that the respectively perturbed bi-vector Q1:6(P̃)
stops being compatible with the perturbed Poisson structure P̃ . The first-order bal-
ance of both sides of perturbed equation (2) then suggests the coefficients of those
differential consequences of the Jacobiatior which are actually involved in the factoriza-
tion mechanism. The coefficients of operators realized by graphs which were found by
following this scheme are reproduced in the full run-through that gave us the solution ♢
in section 2.
1. The main problem: From graphs to multivectors
1.1. The language of graphs. Let us formalise a way to encode polydifferential op-
erators – in particular multivectors – using oriented graphs [9, 10]. In an affine real
manifold Nn (here 2 ⩽ n <∞), take a chart Uα ↪→ Rn and denote the Cartesian coor-
dinates by x = (x1, . . . , xn). We now consider only the oriented graphs whose vertices
are either tails for an ordered pair of arrows, each decorated with its own index, or sinks
(with no issued edges) like the vertices 1, 2 in (1 ←i) − • −→j (2). The arrowtail vertices
are called internal. Every internal vertex • carries a copy of a given Poisson bi-vector
P = P ij(x) ∂i ∧ ∂j with its own pair of indices. For each internal vertex •, the pair of
out-going edges is ordered L ≺ R. The ordering L ≺ R of decorated out-going edges
coincides with the ordering “first ≺ second” of the indexes in the coefficients of P .
Namely, the left edge (L) carries the first index and the other edge (R) carries the
second index. By definition, the decorated edge • −→i • denotes at once the deriv∑ation
∂/∂xi ≡ ∂i (that acts on the content of the arrowhead vertex) and the summation ni=1
(over the index i in the object which is contained within the arrowtail vertex). As it
has been explained in [7, 12], the operator which every graph encodes is equal to the
sum (running over all the indexes) of products (running over all the vertices) of those
vertices content (differentiated by the in-coming arrows, if any). Moreover, we let the
sinks be ordered (like 1 ≺ 2 above), so that every such graph defines a polydifferential
operator: its arguments are thrown into the respective sinks.
Examp∑le 1. The wedge graph 1 ←i( ) − P ij j(x) −→ (2) encodes the bi-differential op-
erator n ←− ij −→
L R
i,j=1(1)∂i ·P (x) · ∂j (2). Such graph specifies a Poisson bracket (on every
chart Uα ⊆ Nn) if it satisfies the Jacobi identity, see (4) below.
Remark 1. In principle, we allow the presence of both the tadpoles and cycles over two
vertices (or “eyes”), see Fig. 1. However, in hindsight there will be neither tadpoles nor
eyes in the solution which we shall have found in section 2 below.

?r - r RrI
Figure 1. A tadpole and an “eye”.
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 363
Remark 2. Under the above assumptions, there exist inhabited graphs that encode zero
differential operators. Namely, consider the graph with a double edge:
r i r j j iL R L R flip La z =a r R rz = −a r R rz = −a r R rz.
R  R  L  R 
j i i j
By first relabelling the summation indic∑es and then swapping∑L ⇄ R (and redrawing)
we evaluate the operator acting at z to n iji,j=1 a ∂i∂j(z) = −
n
i,j=1 a
ij∂i∂j(z); whence
the operator is zero. In the same way, any graph containing a double edge encodes a
zero operator. Graphs can also encode zero differential operators in a more subtle way.
For example consider the wedge on two wedges:
3 r R-r2
@ B
L@  B
Rr = 0.r  B r (3)
	1@RBBN
f g
Swapping the labels 1 ⇄ 2 of the lower wedges does not change the operator. On the
other hand, doing the same in a different way, namely, by swapping ‘left’ and ‘right’ in
the top wedge introduces a minus sign. Hence the graph encodes a differential operator
equal to minus itself, i.e. zero. Proving that a graph which contains the left-hand side
of (3) as a subgraph equals zero is an elementary exercise (cf. Example 3 on p. 376).
Besides the trivial vanishing mechanism in Remark 2, there is the Jacobi identity
together with its differential consequences, which will play a key role in what follows.
For any three arguments 1 , 2 , 3 ∈ C∞(Nn), the Jacobi identity JacP(1 , 2 , 3 ) = 0 is
realized2 by the graph
r r r r• • L	@ 	
R @RrH ri j k i k i j k

 2?
BBN := 	r@Rr @Rr − r j 
r
 Hjr − 	r 	r@Rr = 0. (4)
1 3 1 2 3 1 2 3 1 2 3
In our notation this identity’s left-hand side encodes a sum over all (i, j, k); instead
restricting to fixed (i, j, k) corresponds to taking a coefficient of the differential oper-
ator (cf. Lemma 1 below), which yields the respective component JacijkP of the Jaco-
biator Jac(P). Clearly, the Jacobiator is totally skew-symmetric with respect to its
arguments 1, 2, 3.
In fact, the Jacobiator Jac(P) is the Schouten bracket of a given Poisson bi-vector P
with itself: Jac (P) = [[P ,P ]] (depending on conventions, times a constant which is here
omitted, cf. [8]). The bracket [[·, ·]] is a unique extension of the commutator [·, ·] on the
space of vector fields X1(Nn) to the space X∗(Nn) of multivector fields. Let us recall
its inductive definition in the finite-dimensional set-up.
Definition 1. The Schouten bracket [[·, ·]] : X∗(Nn)×X∗(Nn)→ X∗(Nn) coincides with
the commutator [·, ·] when evaluated on 1-vectors; when evaluated at a p-vector X,
2The notation JacP(1, 2, 3) is synonymic to Jac(P)(1 ⊗ 2 ⊗ 3 ).
364 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
q-vector Y, and r-vector Z for p, q, r ⩾ 1, the Schouten bracket is shifted-graded skew-
symmetric, [[X,Y]] = −(−1)(p−1)(q−1)[[Y,X]], and it works over each argument via the
graded Leibniz rule: [[X,Y ∧ Z]] = [[X,Y]]∧Z+(−1)(p−1)qY∧ [[X,Z]]. The bracket is then
extended by linearity from homogeneous components to the entire space of multivector
fields on Nn.
Remark 3. The construction of Schouten bracket also reads as follows. Denote by ξi the
parity-odd canonical conjugate of the variable xi for every i = 1, . . ., n. For instance,
every bi-vector is realised in terms of local coordinates xi and ξ ∗ ni on ΠT N by using
P = 1〈ξ ijiP (x) ξj〉. The Schouten bracket [[·, ·]] is the parity-odd Poisson bracket which2
is locally determined on ΠT ∗Nn by the canonical symplectic structure dx ∧ dξ. Our
working formula is3
←− −→ ←− −→
P Q P ∂ · ∂ ∂ ∂[[ , ]] = ( ) (Q)− (P) · (Q). (5)
∂xi ∂ξ ∂ξ ∂xii i
It is now readily seen that the Schouten bracket of homogeneous arguments satisfies its
own, shifted-graded Jacobi identity,
[[X, [[Y, · ]]]](Z)− (−)(|X|−1)·(|Y|−1)[[Y, [[X, · ]]]](Z) = [[[[X,Y]], · ]](Z).
Hence for a bi-vector P such that [[P ,P ]] = 0, the map ∂P = [[P , ·]] : Xℓ(Nn) →
Xℓ+1(Nn) is a differential.
Remark 4. The graphical calculation of the Schouten bracket [[·, ·]] of two arguments
amounts to the action – via the Leibniz rule – of every out-going edge in an argument
on all the internal vertices in the other argument. For the Schouten bracket of a k-vector
with an `-vector, the rule of signs is this. For the sake of definition, enumerate the sinks
in the first and second arguments by using 0, . . ., k − 1 and 0, . . ., ` − 1, respectively.
Then the arrow into the jth sink in the second argument acts on the internal vertices of
the first argument, acquiring the sign factor (−)j; here 0 ⩽ j < `. On the other hand,
the arrow to the ith sink in the first argument acts on the second argument’s internal
vertices with the sign factor −(−)(k−1)−i for 0 ⩽ i ⩽ k − 1.
The rule of signs, as it has been phrased above, is valid — provided that, for a k-
vector X and `-vector Y, the numbers 0, . . . of the k (or k − 1) sinks originating in
the (k + ` − 1)-vector [[X,Y]] from the first argument X precede the numbers of ` − 1
(resp., `) sinks originating from Y in the overall enumeration of those k + `− 1 sinks.4
For example, it is this ordeqringq of sinks usq ing(1 ≺q2 whichq is)shown in (6),A 
C
[[ 
A
 AU , ?]] = + q?−  AUq − 
q C ; (6)
1 2 1 
A
 AU  ? ? CCW
1 2 1 2 2 1
here k = 2, ` = 1 and the enumeration of arguments begins at 1 .
3In the set-up of infinite jet spaces J∞(π) (see [13] and [5, 6, 7]) the four partial derivatives in
formula (5) for [[·, ·]] become the variational derivatives with respect to the same variables, which now
parametrise the fibres in the Whitney sum π ×Mm Ππ̂ of (super-)bundles over the m-dimensional
base Mm.
4Such is the default convention which formula (5) suggests for the product of parity-odd variables ξi ,α
where 1 ⩽ α ⩽ k + ℓ− 1.
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 365
Still let us note that in its realization via Kontsevich graphs, the calculation of the
Schouten bracket [[·, ·]] effectively amounts to a consecutive plugging of one of its argu-
ments into each of the other argument’s sinks (see (6) again). Therefore, it would be
more natural to start enumerating the sinks of the graph that acts (on the new content
in one of its sinks, possibly the first), but when that new argument is reached, to inter-
rupt and now enumerate the argument’s own sinks, then continuing the enumeration
of sinks (if there still remain any to be counted) in the graph that acts. This change
of enumeration strategy comes at a price of having extra sign factors in front of the
graphs. Namely, the arrow into the jth sink in the second argument acquires the extra
sign factor (−)j·k. Similarly, the arrow to the ith sink in the first argument of [[·, ·]] must
now be multiplied by (−)ℓ·(k−1−i); we recall that 0 ⩽ i < k and 0 ⩽ j < `. We note that
for k and ` even (e.g., k = 2 and ` = 2 in formula (9)) no extra sign factors appear at
all from the re-ordering at a price of (−)j·k and (−)ℓ·(k−1−i). For example, such is the
final ordering of the 3 = 2 + 1 = 1 + 2 sinks which is shown in Fig. 3 on p. 367.
Summarizing, to be Poisson a bi-vector P must satisfy the master-equation,
[[P ,P ]] = 0, (7)
of which formula (4) is the component expansion with respect to the indices (i, j, k) in
the tri-vector [[P ,P ]](x, ξ).
Definition 2. Let P be a Poisson bi-vector on the manifold Nn at hand and consider its
deformation P 7→ P + εQ(P) + ō(ε). We say that after such deformation the bi-vector
stays infinitesimally Poisson if
[[P + εQ(P) + ō(ε),P + εQ(P) + ō(ε)]] = ō(ε), (1′)
that is, the master-equation is still satisfied up to ō(ε) for a given solution P of (7).
Remark 5. Nowhere above should one expect that the leading deformation term Q
in P + εQ + ō(ε) itself would be a Poisson bi-vector. This may happen for Q only
incidentally.
Expanding the left-hand side of equation (1) and using the shifted-graded skew-
symmetry of the Schouten bracket [[·, ·]], we extract the deformation equation
P Q .[[ , ]] = 0 via [[P ,P ]] = 0. (2)
Let us consider a class of its solutions Q = Q(P) which are universal with respect to
all finite-dimensional affine Poisson manifolds (Nn,P).
1.2. The Kontsevich tetrahedral flow. In the paper [11], Kontsevich proposed a
particular construction of infinitesimal deformations P 7→ P+εQ(P)+ ō(ε) for Poisson
structures on affine real manifolds. One such flow Ṗ = Q(P) on the space of Poisson
bi-vectors P is associated with the complete graph on four vertices, that is, the tetra-
hedron. Up to symmetry, there are two essentially different ways, resulting in Γ1 and
Γ′2, to orient its edges, provided that every vertex is a source for two arrows and, as an
elementary count suggests, there are two arrows leaving the tetrahedron and acting on
the arguments of the bi-differential operator which the tetrahedral graph encodes. The
two oriented tetrahedral graphs are shown in Fig. 2. Unlike the operator encoded by Γ1,
that of Γ′2 is generally speaking not skew-symmetric with respect to its arguments. By
366 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
′

}RCb 
b
m
b C}b
R
 C > k′ 
 C 

 R C  

ℓ′  CL L ℓ j
Γ = P


 C  Γ′ = P
=

 C 
1 P
L PqPCCW
 2 Pk PqPCCW
m

A ? i
L
 AR ?

 AU
Figure 2. The Kontsevich tetrahedral graphs encode two bi-linear bi-
differential operators on the product C∞(Nn)× C∞(Nn).
definition, put Γ2 := 1(Γ′2(1, 2) − Γ′2(2, 1)) to extract the antisymmetric part, that is,2
the bi-vector encoded by Γ′2. Explicitly, the quartic-nonlinear differential polynomials
Γ1(P) and Γ2(P),∑dep(endinn ∑g on a Poisson bi-vector P , are given b)y the formulaen ∂3P ij ∂Pkk′ ∂Pℓℓ′ ∂Pmm′ ∂ ∂
Γ1(P) = ∧ (8a)
∂xk∂xℓ∂xm ∂xℓ′ ∂xm′ ∂xk′ ∂xi ∂xj
i,j=1 k,ℓ,m,k′,ℓ′,m′=1
and ∑n ( ∑n )∂2P ij ∂2Pkm ∂Pk′ℓ ′ ′P ∂Pm ℓ ∂ ∂Γ2( ) = ′ ′ ′ ∧ , (8b)∂xk∂xℓ ∂xk ∂xℓ ∂xm ∂xj ∂xi ∂xm
i,m=1 j,k,ℓ,k′,ℓ′,m′=1
respectively. To construct a class of flows on the space of bi-vectors, Kontsevich sug-
gested to consider linear combinations, balanced by using the ratio a : b, of the bi-
vectors Γ1 and Γ2. We recall from section 1.1 that every internal vertex of each graph
is inhabited by a copy of a given Poisson bi-vector P , so that the linear combination of
two graphs encodes the bi-vector Qa:b(P) = a · Γ1(P) + b · Γ2(P), quartic in P and bal-
anced using a : b. We now inspect at which ratio a : b the bi-vector P + εQa:b(P)+ ō(ε)
stays infinitesimally Poisson, that is,
[[P + εQa:b(P) + ō(ε),P + εQa:b(P) + ō(ε)]] = ō(ε). (1)
The left-hand side of the deformation equation,
[[P ,Q .a:b(P)]] = 0 via [[P ,P ]] = 0, (2)
can be seen in terms of graphst: ( )|

C}b 
}Cb }b
P · > b 

C
[[ , a Γ1 + b · Γ2]] = 
A , a · 



 C + · 
=
C
 − =
C
 . (9)

 AU PPC 2 PPC P


PC
1 2 qCW qCW qCW


AUA ? ? =~
1 2 1 2 1 2
Let a : b = 1 : 6 (specifically, a = 1 and b = 3). Then the left-hand side of (2)
4 2
takes the shape depicted in Fig. 3. After the expansion of Leibniz rules and skew-
symmetrization, the sum in Fig. 3 simplifies to 39 graphs; they are listed in Table 2
on p. 374 below. Collecting, we conclude that the left-hand side of (2) is the sum of
9 manifestly skew-symmetric expressions, see Fig. 4 (and Table 3 in Appendix A). For
example, when outlining a proof of our main theorem (see p. 371), we shall explain
how the coefficient −1 of the first and second graphs in Fig. 4 is accumulated from the
2
terms in the right-hand side of (10). Simultaneously, we shall track how the coefficients
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 367

}Cb 
}Cb 
C}b 
}Cb 
C}b 
}b
1 
>
C
− C

− 1 
> 3    

  

 


 C  + 
=
C  − 3 =

 3 
  3 
 
P C P C P C 

C + 
=C − 
=C
4 PqWC 4 PqCW - 4 ?PqWC 4 PPqCCW
 
- 4 PPqCCW 4 PjPqCCW



AAU AUA 

 

AAU 

AAU ? ? 

AUA 

AAU ~ 

AAU
# # # # 

A 
}bAU C 

}Cb 
A 
A 
}Cb 
C}b 
A
− 1 
 "

> > 
 AU   


 

C
4 PPqCCW


 !
1"
C !A − 3 
 3 + P
 C A 
=
C 
 + 
=C APqCW 4 A 4 
 "PPqCWC!4"PPqCWC!A

AUA 

AAU 
 ? ? ? ? A
 AU 
 AU
# # 

AAU 

}Cb 

}Cb
3 


A
− 

 3 

 
P=

C
C + 
P=

C A
4 PqCW PqCCW
 A

 "!4"!

 =~ =~
A
AU
Figure 3. Incoming arrows act on th∑e content of boxes via the Leibnizrule; to obtain the tri-vector, the entire picture must be skew-symmetrized
over the content of three sinks using σσ∈S (−) . Expanding and skew-3
symmetrizing, one obtains 39 graphs in the left-hand side of (2).
∑ { 7 7
}Cb 6 
}Cb 6 }b }b }b1 > 1 > 
C> 
C 
C 3− 3   3  ( )σ − 

 C − 
 
 
  
 C 
A C C  3 
C 
2 5PPqCCW 2 5PPqC

A + AU
 

P C + 
P= C + 
=∈ 3 4 CW
4 2 PqCW 2 PqCW- 2 

PPqCWC
σ S3


AUA A
A
AU 

AAU AU 

 

AAU ? 

AAU 

AAU ?
0 1 2 0 1 2

AAU

C}b }b 


A
 
C 

 C
}b
 
C
}b
 }
  A
− 3 
AAU
=
C 
+ 3 
=C
 
  
 

 P C P C
A + 3 
 

=C  − 3 
=CPqCW PqCW 
A .

 P  P 
 PqCCW PqCCW A


 ? ? ? ?AUA 
 =~ =~ AUA
Figure 4. This sum of graphs is the skew-symmetrized content of Fig. 3.
In what follows, we realize these 9 terms in the left-hand side of (2) by
using an operator ♢ acting, in the right-hand side of (10) below, on the
Jacobiator (4).
cancel out for the two other graphs which are produced by expanding the same Leibniz
rules (that gave the above two graphs).
1.3. Main result. The reason why we are particularly concerned with the ratio a :
b = 1 : 6 is that this condition is necessary for equation (2) to hold. This has been
proved in [1] by producing examples of Poisson bi-vector P such that [[P ,Qa:b(P)]] = 0
only when a : b = 1 : 6. Let us now inspect whether this condition is also sufficient.
The task is to factorize the content of Fig. 4 through the Jacobi identity in (4).
368 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
We first examine the mechanism for the tri-vector [[P ,Q1:6(P)]] in (2) to vanish by
virtue of the Jacobi identity Jac(P) = 0 for a given Poisson bi-vector P on an affine
manifold Nn of any dimension. We claim that the Jacobiator JacP (·, ·, ·) is not neces-
sarily (indeed, far not always! ) evaluated at the three arguments f, g, h of the tri-vector
[[P ,Q1:6(P)]]. A sample graph that can actually appear in such factorizing operators ♢
is drawn in Fig. 5 below. ∑
Lemma∑1 ([2]). A tri-differential operator C = |I|,|J |,|K|⩾0 c
IJK ∂I ⊗ ∂J ⊗ ∂K with
coefficients cIJK ∈ C∞(Nn) vanishes identically iff all its homogeneous components
C = IJKijk |I|=i,|J |=j,|K|=k c ∂I ⊗ ∂J ⊗ ∂K vanish for all differential orders (i, j, k) of
the respective multi-indices (I, J,K); here ∂ = ∂α1L 1 ◦ · · · ◦ ∂αnn for a multi-index L =
(α1, . . . , αn).
In practice, Lemma 1 states that for every arrow falling on the Jacobiator JacP(1 , 2 , 3 )
– for which, in turn, a triple of arguments 1 , 2 , 3 is specified – the expansion of the Leib-
n'iz ruleryields$four fragmr ents which vranishseparatelry: e.g., we harve that@R@ ? r
r r	r@ r  r ?r	r@ r r r	r@ r  r r	@ r	@ r	@	@&R @R = 	@R @R + 	@% R @R + 	?@Rr @Rr + 	r@Rr?@Rr + 	r@Rr @Rr?.1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Namely, there is the fragment such that the derivation acts on the content P of the
Jacobiator’s two internal vertices, and there are three fragments such that the arrow
falls on the first, second, or third argument of the Jacobiator. Now it is readily seen
that the action of a derivative ∂i on an arg(ument of the)Jacobiator amounts to an
app(ropriate) redefinition of t(hat argument: ∂i J︸ ︷︷ ︸ ︸ ︷︷ )︸ ︸ (
acP(1 , 2 , 3 )) =
∂i JacP (1 , 2 , 3 )+ JacP ∂i(1 ), 2 , 3 +JacP 1 ,︷∂︷i(2 ), 3 ︸ ( )+J︸acP 1 ,︷2︷ , ∂i(3 )︸ = 0.
=0 =0 =0 =0
Let us introduce a name for the (class of) graphs which make the first term – out of
four – in the expansion of Leibniz rule in the above formula.
Definition 3. A Leibniz graph is a graph whose vertices are either sinks, or the sources
for two arrows, or the Jacobiator (which is a source for three arrows). There must be
at least one Jacobiator vertex. The three arrows originating from a Jacobiator vertex
must land on three distinct vertices. Each edge falling on a Jacobiator works by the
Leibniz rule on the two internal vertices in it.
An example of a Leibniz graph is given in Fig. 5. Every Leibniz graph can be
expanded to a sum of Kontsevich graphs, by expanding both the Leibniz rule(s) and
all copies of the Jacobiator; e.g. see (12). In this way Leibniz graphs also encode
(poly)differential operators, depending on the bi-vector P and the tri-vector Jac(P).
Proposition 2. For every Poisson bi-vector P the value – at the Jacobiator Jac(P) –
of every (poly)differential operator encoded by the Leibniz graph(s) is zero.
Theorem 3. There exists a( po∧lydifferential op∧erator2 3 ∧3 )
♢ ∈ PolyDiff Γ( TNn)× Γ( TNn)→ Γ( TNn)
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 369
? • There is a cycle,
• • • there is a loop,r • there are no tadpoles in this
6r@ r 
 B graph,@ 
 B R
 • an arrow falls back on Jac(P),B
 B • and Jac(P) does not stand on? ? BN
( ) ( ) ( ) all of the three sinks.
Figure 5. This is an example of Leibniz graph of which the factorizing
operators can consist.
which solves the factorization problem ( )
[[P ,Q1:6(P)]](f, g, h) = ♢ P , JacP (·, ·, ·) (f, g, h). (10)
The polydifferential operator ♢ is realised using Leibniz graphs in formula (11), see
p. 371 below.
Corollary 4 (Main result). Whenever a bi-vector P on an affine real manifold Nn is
Poisson, the deformation P + εQ1:6(P) + ō(ε) using the Kontsevich tetrahedral flow is
infinitesimally Poisson.
Remark 6. It is readily seen that the Kontsevich tetrahedral flow Ṗ = Q1:6(P) is well
defined on the space of Poisson ∣bi-vectors on a given affine manifold N
n. Indeed, it does
not depend on a choice of coor∣dinates up to their arbitrary affine reparametrisations.In other words, the velocity Ṗ ∈ n does not depend on the choice of a chart U 3 uu N
from an atlas in which only affine changes of variables are allowed. (Let us remember
that affine manifolds can of course be topologically nontrivial.)
Suppose however that a given affine structure on the manifold Nn is extended to a
larger atlas on it; for the sake of definition let that atlas be a smooth one. Assume that
the smooth structure is now reduced – by discarding a number of charts – to another
affine structure on the same manifo∣ld. The tetrahedral flow Ṗ = Q1:6(P) which oneinitially had can be contrasted with ∣the tetrah∣edral flow P˙̃ = Q1:6(P̃) which one finallyobtains for the Poisson bi-vector P̃ = P∣ in the course of a nonlinear change of
ũ(u) u
coordinates on Nn. Indeed, the respective velocities Ṗ and P˙̃ can be different whenever
they are expressed by using essentially different parametrisations of a neighbourhood of
a point u in Nn. For example, the tetrahedral flow vanishes identically when expressed
in the Darboux canonical variables on a chart in a symplectic manifold. But after a
nonlinear transformation, the right-hand side Q1:6(P̃) can become nonzero at the same
points of that Darboux chart.
This shows that an affine structure on the manifold Nn is a necessary part of the
input data for construction of the Kontsevich tetrahedral flows Ṗ = Q1:6(P).
2. Solution of the factorization problem
Expanding the Leibniz rules in [[P ,Q1:6(P)]], we obtain the sum of 39 graphs with
5 internal vertices and 3 sinks (so that from Figure 3 we produce Table 2, see page 374
370 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
∧
below). By construction, the Schouten bracket [[P ,Q 3 n1:6(P)]] ∈ Γ( TN ) is a tri-
vector on the underlying manifold Nn, that is, it is a totally antisymmetric tri-linear
polyderivation C∞(Nn)× C∞(Nn)× C∞(Nn)→ C∞(Nn). At the same time, we seek
to recognize the tri-vector [[P ,Q1:6(P)]] as the result of application of a (poly)differential
operator ♢ (see (10) in Theorem 3) to the Jacobiator Jac(P) (see (4) on p. 363).
We now explain how the operator ♢ is found.5 The ansatz for ♢ is the sum – with
undetermined coefficients – of all (separately vanishing) Leibniz graphs containing one
Jacobiator and three wedges, and having differential order (1, 1, 1) with respect to the
sinks (see Fig. 6). We thus have 28, 202 unknowns introduced (counted with possi-
ble repetitions of graphs which they refer to). Expanding all the Leibniz rules and
Jacobiators, wes obtasin a ssum of Kontsesvich gsraphss with 5 internsal vesrticesson 3 sinks.“3”: “2”: “1”(1):
• • • • • •

 A A A


 ? AAU ^ ? AUA ^ R AAU
s ( )s ( )s ( ) s ( ) s ( ) s ( ) s( ) s( ) s ( )
“1”(2): “0”(1): C “0”(2):
• •  C C • • • •
  C
  CU U   WC R U U U
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Figure 6. This is the list of all different types of Leibniz graphs which
are linear in the Jacobiator and which have differential order (1, 1, 1) with
respect to the sinks. The list is ordered by the number of ground vertices
on which the Jacobiator stands.
As soon as we take into account the order L ≺ R and the antisymmetry of graphs
under the reversion of that ordering at an internal vertex, the graphs that encode zero
differential operators are eliminated.6 There remain 7, 025 admissible graphs with 5
internal vertices on 3 sinks; the coefficient of every such graph is a linear combination
of the undetermined coefficients of the Leibniz graphs. In conclusion, we view (10)
as the system of 7, 025 linear inhomogeneous equations for the coefficients of Leibniz
graphs in the operator ♢. Solving this linear system is a way towards a proof of our
main result (which is expressed in Corollary 4). The process of finding a solution ♢ itself
does not constitute that proof. Therefore, the justification of the claim in Theorem 3
will be performed separately. In the meantime, using software tools, we solve the
linear algebraic system at hand. The duplications of graph labellings are conveniently
5Another method for solving the factorization problem is outlined in Appendix B.
6The relevant algebra of sums of graphs modulo skew-symmetry and the Jacobi identity has been
realized in software by the second author. An implementation of those tools in the problem of high-
order expansion of the Kontsevich ⋆-product is explained in a separate paper, see [3].
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 371
eliminated by our request for the program to find a solution with a minimal number
of nonzero components. Totally antisymmetric in tri-vector’s arguments, the solution
consists of 27 Leibniz graphs, which are assimilated into the sum of 8 manifestly skew-
symmetric terrms as rfollows:
@ 
@Rr • • • •
♢ ? ? ?
∑ 
 ?r B ∑  r? B= + 3 (−)τ B + 3  B
• • @ @τ∈S r   @B ⟳  @B2 
 R@BBNr r	 @RB

 J H

 BNr

 ? Ĵ ? ? ? ?HHj ?
( ) ( ) ( ) [( ) ( )] ( ) ( ) ( ) ( )
{ ? ? ?∑ • • ? ? • • }
+ 3 • • r AAU r + r + 
 @⟳  * 
 @Rr 	@@Rr  ?r  YH r 
 r?@ @H HY
 ? @R 
 ? @R H? 
 = H
 ? H?r
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
{ ? ?∑ • • • • }
+ 3 (−)σ 
 @R@r + 
 @ .
σ∈ @RrS3 


 r? 
 *

 
HHjH?r 
 r?CHY
 H
  
 CW H?r? ?
( ) ( ) ( ) ( ) ( ) ( ) (11)
To display the L ≺ R ordering at every internal vertex and to make possible the
arithmetic and algebra of graphs, we use the notation which is explained in Appendix A.
Remark 7. We remember that the set {1, 2, 3} of three arguments of the Jacobiator need
not coincide with the set {f, g, h} of the arguments of the tri-vector ♢(P , Jac(P)). Of
course, the two sets can intersect; this provides a natural filtration for the components
of solution (11). Namely, the number of elements in the intersection runs from three
for the first term to zero in the second or third graph.
In fact, Remark 7 reveals a highly nontrivial role of the operator ♢ in (10). Some of
the three internal vertices of its graphs can be arguments of Jac(P) whereas some of the
other such vertices (if any) can be tails for the arrows falling on Jac(P). In retrospect,
the two subsets of such vertices of ♢ do not intersect; every vertex in the intersection,
if it were nonempty, would produce a two-cycle, but there are no “eyes” in (11).
Proof of Theorem 3. So far, we have constructed operator (11); it involves a reason-
ably small number of Leibniz graphs so that the factorization in (10) can be verified
by a straightforward calculation. The sums in (11) contain 27 Leibniz graphs. Now
expand all the Leibniz rules; this yields the sum of 201 Kontsevich graphs with 3 sinks
372 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
and 5 internal vertices: together with their coefficients, they are listed in Table 5 in
Appendix A, see page 374. We claim that by collecting similar terms, one obtains the
39 graphs from the left-hand side of (10), see Fig. 4 and the encoding of those graphs
in Table 3 on page 375. Because we are free to enumerate the five internal vertices in
every graph in a way we like, and because the ordering of every pair of outgoing edges
is also under our control, at once do we bring all the graphs to their normal form.7
It is readily seen that there are many repetitions in Table 5. We must inspect what
vanishes and what stays. Let us do a sample reasoning first. Namely, let us inspect the
contribution toq the qleft-hand side of (10) from the first term of (11). We have that
@@Rq ∑{ 7 7 p p
C}b 6? ? ? > 
}Cb 6 Y p Y p }  > p 1 p 14
• • = 5P

 C 

 + 
 C + 3 p? + 3 p	? . (12)
⟳ PqCCW


A 5P C
3 3 PqWC
 Rp)? p?)
A 4

 ? Ĵ 

AAU AU 

AAU AAU 	 R ^ 	 R ^
0 1 2 0 1 2
The right-hand side of (12) expands into the sum of 12 different graphs. They are
marked in the first twenty-four lines of Table 5 by ♦i,♥i,♣i and ♠i for 1 ⩽ i ⩽ 3,
respectively; by definition, a suit with different values of its subscript i denotes the
ith cyclic permutation of the ground vertices for the same graph.8 For example, the
symbols ♦1,♦2,♦3 mark the three cyclic permutations of arguments in the first term in
the right-hand side of (12). The sum of the first two terms in the right-hand side of (12)
– marked by ♦i and ♥i, respectively – equals the sum of the first two terms in Fig. 4.9
At the same time, the sum of the last two terms – whose encodings with coefficients
±1 are marked by ♣i and ♠i, respectively – cancels against the contributions from the
fourth and sixth terms in solution (11) – with coefficients ±3, also marked by ♣i and ♠i
in the rest of Table 5. In Table 1 we calculate the coefficient of each graph marked by
the respective indexed symbol.
Now, in the same way all other similar terms are collected. There remain only 39
terms with nonzero coefficients. One verifies that those 39 terms are none other than
the entries of Table 2, that is, realizations of the 39 graphs in the left-hand side of (10).
This shows that equation (10) holds for the operator ♢ contained in (11). □
7The normal form of a graph is obtained by running over the group S5×(Z2)5 of all the relabellings of
internal vertices and swaps L ⇄ R of orderings at each vertex. (We recall that every swap negates the
coefficient of a graph; the permutations from S5 are responsible for encoding a given topological profile
in seemingly “different” ways.) By definition, the normal form of a graph is the minimal sequence of
five ordered pairs of target vertices viewed as 10-digit base-(3+5) numbers. (By convention, the three
ordered sinks are enumerated 0, 1, 2 and the internal vertices are the octonary digits 3, . . . , 7.)
8By taking a graph, placing it consecutively over three cyclic permutations of its sinks’ content,
and bringing the three graph encodings to their normal form, see above, one can obtain an extra sign
factor in front of some of these graphs. This is due to a convention about “minimal” graph encoding,
not signalling any mismatch in the arithmetic. For example, after the normalization such is the case
with the columns in Table 1: each column refers to a cyclic permutation of three arguments and the
coefficients in every line would coincide if one encoded the graphs for the last column not using the
respective minimal 10-digit octonary numbers. To make all the three coefficients in each line coinciding,
it is enough to swap L ⇄ R in one internal vertex in every graph from the third column.
9We inspect further that no other graphs in Table 5 make any contribution to the coefficients of
these two graphs.
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 373
Table 1. The coefficients of graphs marked by the four suits.
♦1 : −1 ♦2 : −1 ♦3 : +1
♥1 : −1 ♥2 : −1 ♥3 : +1
♣1 : −1− 1− 1 + 3 = 0 ♣2 : −1− 1− 1 + 3 = 0 ♣3 : +1 + 1 + 1− 3 = 0
♠1 : −1− 1− 1 + 3 = 0 ♠2 : −1− 1− 1 + 3 = 0 ♠3 : +1 + 1 + 1− 3 = 0
Remark 8. Operator (11) is not a unique solution of factorization problem (10). We
claim that apart from this sum of 27 Leibniz graphs, there is another solution which
consists of 102 Leibniz graphs; it is also linear with respect to the Jacobiator (that is,
its realization in the form ♢(P , Jac(P), Jac(P)) is not possible).
Discussion
Non-triviality. A flow specified on the space of Poisson bi-vectors by using the Kon-
tsevich graphs can be Poisson cohomology trivial modulo a sum of Leibniz graphs that
would vanish identically at any Poisson structure. However, this is not the case of the
Kontsevich tetrahedral flow Ṗ = Q1:6(P).
Proposition 5. There is no 1-vector field X encoded over Nn by the Kontsevich graphs
and there is no operator ∇ encoded using the Leibniz graphs such that
Q1:6(P) = [[P ,X]] +∇(P , Jac(P)).
The claim is established by a run-through over all Kontsevich graphs with three internal
vertices and one sink (making an ansatz for X) and all Leibniz graphs (in the operator∇)
with two copies of P and one Jacobiator in the internal vertices; all such graphs of both
types are taken with undetermined coefficients. The resulting inhomogeneous linear
algebraic system has no solution.
Integrability. By using the technique of Kontsevich graphs one can proceed with a
higher order expansion of the tetrahedral deformation,
P 7→ P + εQ1:6(P) + εR(P) + · · ·+ ō(εd), d ⩾ 2,
for Poisson structures P . Assuming that the master-equation holds up to ō(εd),
[[P + εQ1:6(P
.
) + εR(P) + · · ·+ ō(εd),P + εQ1:6(P) + εR(P) + · · ·+ ō(εd)]] = ō(εd)
via [[P ,P ]] = 0, (13)
we obtain a chain of linear equations for the higher order expansion terms, namely,
.
2[[P ,R(P)]] + [[Q1:6(P),Q1:6(P)]] = 0 via [[P ,P ]] = 0, etc. (14)
A solution consisting of R(P) and consecutive terms at higher powers of the deforma-
tion parameter10 can be sought using the same factorization techniques and computer-
assisted proof schemes [3] which have been implemented in this paper — whenever such
solution actually exists. It is clear that there can be Poisson cohomological obstructions
10In every graph at εk the number of internal vertices is 3k + 1.
374 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
to resolvability of cocycle conditions (14). Hence the integrability issue for the Kontse-
vich tetrahedral flow may be Poisson model-dependent, unlike the universal nature of
such deformation’s infinitesimal part.
Appendix A. Encoding of the solution
Let Γ be a labelled Kontsevich graph with n internal and m external vertices. We
assume the ground vertices of Γ are labelled [0, . . ., m − 1] and the internal vertices
are labelled [m, . . ., m + n − 1]. We define the encoding of Γ to be the prefix (n,m),
followed by a list of targets. The list of targets consists of ordered pairs where the
kth pair (k ⩾ 0) contains the two targets of the internal vertex number m+ k.
The expansion of the Schouten bracket [[P ,Qa:b]] for the ratio a : b = 1 : 6 depicted
in Figure 3 simplifies to a sum of 39 graphs with coefficients ±1 , ±3 . The encodings of
4 4
these graphs, followed by their respective coefficients, are listed in Table 2. The graphs
Table 2. Machine-readable encoding of Fig. 3 on p. 367.
1.1 3 5 4 2 0 1 4 6 4 7 4 5 1/4 7.1 3 5 6 2 7 0 1 4 4 5 5 6 3/4
1.2 3 5 4 0 1 2 4 6 4 7 4 5 1/4 7.2 3 5 6 0 7 1 2 4 4 5 5 6 3/4
1.3 3 5 4 1 2 0 4 6 4 7 4 5 1/4 7.3 3 5 6 1 7 2 0 4 4 5 5 6 3/4
2.1 3 5 7 0 3 5 3 6 3 4 1 2 1/4 8.1 3 5 7 2 7 0 1 4 4 5 5 6 3/4
2.2 3 5 7 1 3 5 3 6 3 4 2 0 1/4 8.2 3 5 7 0 7 1 2 4 4 5 5 6 3/4
2.3 3 5 7 2 3 5 3 6 3 4 0 1 1/4 8.3 3 5 7 1 7 2 0 4 4 5 5 6 3/4
3.1 3 5 5 2 0 1 4 6 4 7 4 5 3/4 9.1 3 5 4 2 7 1 0 4 4 5 5 6 −3/4
3.2 3 5 5 0 1 2 4 6 4 7 4 5 3/4 9.2 3 5 4 0 7 2 1 4 4 5 5 6 −3/4
3.3 3 5 5 1 2 0 4 6 4 7 4 5 3/4 9.3 3 5 4 1 7 0 2 4 4 5 5 6 −3/4
4.1 3 5 6 7 0 3 3 4 4 5 1 2 3/4 10.1 3 5 5 2 7 1 0 4 4 5 5 6 −3/4
4.2 3 5 6 7 1 3 3 4 4 5 2 0 3/4 10.2 3 5 5 0 7 2 1 4 4 5 5 6 −3/4
4.3 3 5 6 7 2 3 3 4 4 5 0 1 3/4 10.3 3 5 5 1 7 0 2 4 4 5 5 6 −3/4
5.1 3 5 4 2 7 0 1 4 4 5 5 6 3/4 11.1 3 5 6 2 7 1 0 4 4 5 5 6 −3/4
5.2 3 5 4 0 7 1 2 4 4 5 5 6 3/4 11.2 3 5 6 0 7 2 1 4 4 5 5 6 −3/4
5.3 3 5 4 1 7 2 0 4 4 5 5 6 3/4 11.3 3 5 6 1 7 0 2 4 4 5 5 6 −3/4
6.1 3 5 5 2 7 0 1 4 4 5 5 6 3/4 12.1 3 5 7 2 7 1 0 4 4 5 5 6 −3/4
6.2 3 5 5 0 7 1 2 4 4 5 5 6 3/4 12.2 3 5 7 0 7 2 1 4 4 5 5 6 −3/4
6.3 3 5 5 1 7 2 0 4 4 5 5 6 3/4 12.3 3 5 7 1 7 0 2 4 4 5 5 6 −3/4
13.1 3 5 6 0 7 3 3 4 4 5 1 2 −3/4
13.2 3 5 6 1 7 3 3 4 4 5 2 0 −3/4
13.3 3 5 6 2 7 3 3 4 4 5 0 1 −3/4
are collected into groups of three, consisting of the skew-symmetrization – by a sum
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 375
over cyclic permutations – of a single graph. Within the encodings in the groups of
three, the lists of targets only differ by a cyclic permutation of the target vertices 0, 1, 2.
Table 3. Machine-readable encoding of Fig. 4 on p. 367.
3 5 0 1 2 3 4 6 4 7 4 5 −1/2
3 5 0 4 1 2 4 6 4 7 4 5 −1/2
3 5 0 4 5 6 1 2 5 7 4 5 3/2
3 5 0 1 2 5 6 7 3 4 4 6 3/2
3 5 0 4 5 6 1 2 3 7 3 4 3/2
3 5 0 4 5 6 1 6 2 7 4 5 −3
3 5 0 4 5 6 1 7 5 7 2 4 3
3 5 0 4 1 5 2 6 4 7 4 5 3
3 5 0 4 2 5 6 7 1 4 4 6 −3
Consisting of 8 skew-symmetric terms, the solution (see (11) on p. 371) is encoded
in Table 4: the sought-for values of coefficients are written after the encoding of the
respective 27 Leibniz graphs. Here the sums over permutations of the ground vertices
Table 4. Machine-readable encoding of solution (11) on p. 371.
1.1 3 5 4 6 5 6 3 6 0 1 6 2 −1 6.1 3 5 1 2 3 5 3 6 0 3 6 4 3
6.2 3 5 0 2 3 5 3 6 1 3 6 4 −3
2.1 3 5 0 4 1 5 2 3 3 4 6 5 −3 6.3 3 5 4 6 0 1 3 4 2 4 6 5 −3
2.2 3 5 0 4 2 5 1 3 3 4 6 5 3
7.1 3 5 1 5 3 5 2 6 0 3 6 4 −3
3.1 3 5 0 4 1 2 3 4 3 4 6 5 −3 7.2 3 5 1 5 3 5 0 6 2 3 6 4 3
3.2 3 5 0 1 2 3 3 4 3 4 6 5 −3 7.3 3 5 0 5 3 5 2 6 1 3 6 4 3
3.3 3 5 0 2 1 3 3 4 3 4 6 5 3 7.4 3 5 2 5 3 5 1 6 0 3 6 4 3
7.5 3 5 2 5 3 5 0 6 1 3 6 4 −3
4.1 3 5 4 5 1 6 4 6 0 2 6 3 −3 7.6 3 5 0 5 3 5 1 6 2 3 6 4 −3
4.2 3 5 4 5 0 6 4 6 1 2 6 3 3
4.3 3 5 5 6 3 5 2 6 0 1 6 4 −3 8.1 3 5 1 4 2 5 3 6 0 3 6 4 −3
8.2 3 5 1 5 2 3 4 6 0 3 6 4 −3
5.1 3 5 1 4 5 6 3 6 0 2 6 3 3 8.3 3 5 0 4 2 5 3 6 1 3 6 4 3
5.2 3 5 0 4 5 6 3 6 1 2 6 3 −3 8.4 3 5 0 5 2 3 4 6 1 3 6 4 3
5.3 3 5 5 6 2 3 4 6 0 1 6 4 −3 8.5 3 5 4 6 0 5 1 3 2 4 6 5 −3
8.6 3 5 4 6 1 5 0 3 2 4 6 5 3
are expanded (thus making the 27 Leibniz graphs out of the 8 skew-symmetric groups).
In every entry of Table 4, the sum of three graphs in Jacobiator (4) is represented by
its first term. For all the in-coming arrows, the vertex 6 is the placeholder for the
Jacobiator (again, see (4) on p. 363); in earnest, the Jacobiator contains the internal
vertices 6 and 7. This convention is helpful: for every set of derivations acting on the
Jacobiator with internal vertices 6 and 7, only the first term is listed, namely the one
where each edge lands on 6.
376 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
Example 2. The first entry of Table 4 encodes a graph containing a three-cycle over
internal vertices 3, 4, 5. Issued from each of these three, the other edge lands on the
vertex 6: the placeholder for the Jacobiator. This entry is the first term in (11) on
p. 371.
Example 3. The entry 3.1 is one of three terms produced by the third graph in so-
lution (11); the Jacobiator in this entry is expanded using formula (4), resulting in
three terms (by definition). It is easy to see that the first term contains picture (3)
from Remark 2 as a subgraph. Hence the polydifferential operator encoded by this
graph vanishes due to skew-symmetry. However, the other two terms produced in the
entry 3.1 by formula (4) do not vanish by skew-symmetry. Likewise, there is one term
vanishing by the same mechanism in the entry 3.2 and in 3.3.
The proof of Theorem 3 amounts to expanding the Leibniz rules on Jacobiators in
Table 4 according to the rules above (resulting in Table 5 on p. 377, where the prefix
“3 5” of each graph has been omitted for brevity), simplifying by collecting terms,
and seeing that one obtains Table 3.
Appendix B. Perturbation method
In section 2 above, the run-through method gave all the terms at once in the operator ♢
that establishes the factorization [[P ,Q1:6]] = ♢(P , Jac(P)). At the same time, there
is another method to find ♢; the operator ♢ is then constructed gradually, term after
term in (11), by starting with a zero initial approximation for ♢. This is the perturba-
tion scheme which we now outline. (In fact, the perturbation method was tried first,
revealing the typical graph patterns and their topological complexity.)
The difficulty is that because the condition [[P ,Q1:6]] = 0 and the Jacobi identity
[[P ,P ]] = 0 are valid, it is impossible to factorize one through the other; both are
invisible. So, we first make both expressions visible by perturbing the Poisson bi-vector
P 7→ Pϵ = P + ∆ in such a way that the tri-vector [[Pϵ,Q1:6(Pϵ)]] and the Jacobiator
[[Pϵ,Pϵ]] stop vanishing identically:
[[Pϵ,Q1:6(Pϵ)]] 6= 0 and [[Pϵ,Pϵ]] =6 0.
To begin with, put ♢ := 0. Now consider a class of Poisson brackets on R3 (cf. [4]) by
using the pre-factor f(x, y, z) and arbitrary f(unction g(x), y, z) in the formula
{ } · ∂(g, u, v)u, v P = f det ;
∂(x, y, z)
it is helpful to start with some very degenerate dependencies of f and g of their ar-
guments (see [1] and [14]). The next step is to perturb the coefficients of the Poisson
bracket {·, ·}P at hand; in a similar way, one starts with degenerate dependency of the
perturbation ∆. The idea is to take perturbations which destroy the validity of Jacobi
identity for Pϵ in the linear approximation in the deformation parameter . It is readily
seen that the expansion of (10) in  yields the equality
[[Pϵ,Q1:6]]() = (♢+ ō(1)) ([[Pϵ,Pϵ]]) = 2 ·(♢+ ō(1)) ([[P ,∆]])+(♢+ ō(1)) ([[P ,P ]])+ ō().
Knowing the left-hand side at first order in  and taking into account that [[P ,P ]] ≡ 0
for the Poisson bi-vector P which we perturb by ∆, we reconstruct the operator ♢ that
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 377
Table 5. Expansion of Leibniz rules on Jacobiators in Table 4.
♢1 0 1 2 3 3 6 3 7 3 5 −1 0 4 2 5 6 7 1 3 3 6 −3 0 4 2 5 3 6 3 4 1 6 −3
♣1 0 1 2 3 3 6 3 7 4 5 −1 0 4 5 6 1 3 3 5 2 3 3 0 4 2 5 3 6 1 7 3 4 −3
♣1 0 1 2 3 3 6 3 7 4 5 −1 0 4 5 6 1 3 5 7 2 3 −3 0 4 2 5 3 6 1 4 3 6 3
♠1 0 1 2 3 3 6 4 7 4 5 −1 0 4 5 6 1 7 3 5 2 3 3 0 4 2 5 3 6 3 7 1 4 3
♣1 0 1 2 3 3 6 3 7 4 5 −1 0 4 5 6 1 7 5 7 2 3 −3 0 4 5 6 1 3 2 3 5 6 −3
♠1 0 1 2 3 3 6 4 7 4 5 −1 0 4 1 2 3 4 3 7 4 5 −3 0 4 5 6 2 3 5 7 1 3 −3
♠1 0 1 2 3 3 6 4 7 4 5 −1 ♣2 0 4 1 2 3 6 4 7 4 5 3 0 4 5 6 2 3 3 5 1 6 −3
♡1 0 1 2 3 4 6 4 7 4 5 −1 0 4 5 6 1 2 5 7 4 5 3 0 4 5 6 1 7 2 3 3 6 3
♢2 0 4 1 2 4 6 4 7 4 5 −1 0 4 5 6 1 2 5 7 3 5 3 0 4 5 6 1 6 2 3 3 5 −3
♣2 0 4 1 2 3 6 4 7 4 5 −1 0 4 5 6 1 2 3 5 3 5 3 0 4 5 6 2 3 3 7 1 5 3
♣2 0 4 1 2 3 6 4 7 4 5 −1 0 4 5 6 1 2 5 7 3 5 −3 0 4 2 5 1 3 3 4 5 6 3
♠2 0 4 1 2 3 6 3 7 4 5 −1 0 4 1 5 2 6 3 4 3 5 3 0 4 2 5 6 7 1 3 3 4 3
♣2 0 4 1 2 3 6 4 7 4 5 −1 0 4 1 5 2 6 4 7 3 5 −3 0 4 2 5 3 6 3 4 1 5 −3
♠2 0 4 1 2 3 6 3 7 4 5 −1 0 4 5 6 1 6 2 7 4 5 −3 0 4 2 5 1 6 3 7 3 4 −3
♠2 0 4 1 2 3 6 3 7 4 5 −1 0 4 5 6 1 6 2 7 3 5 −3 0 4 2 5 1 6 3 4 3 5 3
♡2 0 4 1 2 3 6 3 7 3 5 −1 0 4 2 5 3 6 1 4 3 6 −3 0 4 2 5 3 6 1 7 3 4 3
♢3 0 2 1 3 3 6 3 7 3 5 1 0 4 2 5 6 7 1 4 3 6 3 0 4 1 5 2 3 3 5 4 6 3
♣3 0 2 1 3 3 6 3 7 4 5 1 0 4 1 5 3 6 2 4 3 6 3 0 4 5 6 1 7 3 7 2 3 3
♣3 0 2 1 3 3 6 3 7 4 5 1 0 4 1 5 6 7 2 4 3 6 −3 0 4 5 6 2 3 3 5 1 4 −3
♠3 0 2 1 3 3 6 4 7 4 5 1 0 4 5 6 1 7 2 5 4 6 −3 0 4 1 5 6 7 2 3 3 6 −3
♣3 0 2 1 3 3 6 3 7 4 5 1 0 4 5 6 1 7 2 5 3 6 −3 0 4 1 5 3 6 2 3 4 6 −3
♠3 0 2 1 3 3 6 4 7 4 5 1 0 4 2 5 1 6 3 4 3 5 −3 0 4 5 6 1 7 2 3 3 6 −3
♠3 0 2 1 3 3 6 4 7 4 5 1 0 4 2 5 1 6 4 7 3 5 3 0 4 1 5 2 3 3 4 5 6 −3
♡3 0 2 1 3 4 6 4 7 4 5 1 0 4 1 5 2 3 3 7 4 5 −3 0 4 1 5 6 7 2 3 3 4 −3
0 1 2 5 3 6 3 4 3 4 −3 0 4 1 5 2 6 3 7 4 5 −3 0 4 1 5 3 6 3 4 2 5 3
0 1 2 5 3 6 4 7 3 4 3 0 4 5 6 1 7 5 7 2 4 3 0 4 1 5 2 6 3 7 3 4 3
0 1 2 5 6 7 3 4 3 4 −3 0 4 5 6 1 7 5 7 2 3 3 0 4 1 5 2 6 3 4 3 5 −3
0 1 2 5 6 7 3 4 4 6 3 0 4 5 6 1 6 2 3 3 5 3 0 4 1 5 3 6 2 7 3 4 −3
0 4 1 5 2 6 4 7 4 5 3 0 4 5 6 1 6 2 7 3 5 3 0 4 2 5 1 3 3 5 4 6 −3
0 4 1 5 2 6 4 7 3 5 3 0 4 2 5 1 3 3 7 4 5 3 0 4 5 6 2 7 3 7 1 3 −3
0 4 1 5 2 6 3 7 4 5 3 0 4 2 5 1 6 3 7 4 5 3 0 4 5 6 1 3 3 5 2 4 3
0 4 1 5 2 6 3 7 3 5 3 0 4 5 6 2 7 5 7 1 4 −3 0 4 2 5 6 7 1 3 3 6 3
0 4 2 5 3 6 3 4 1 3 −3 0 4 5 6 2 7 5 7 1 3 −3 0 4 2 5 3 6 1 3 4 6 3
0 4 2 5 3 6 4 7 1 3 3 0 4 5 6 1 3 2 5 3 6 3 0 4 5 6 1 3 2 7 3 5 −3
0 4 2 5 6 7 1 3 3 4 −3 0 4 5 6 1 7 2 5 3 6 3 0 1 2 3 3 4 3 5 4 6 3
0 4 2 5 6 7 1 3 4 6 3 0 4 5 6 1 2 3 5 3 5 −3 ♣1 0 1 2 3 3 6 3 7 4 5 3
0 1 2 3 3 4 3 7 4 5 −3 0 4 5 6 1 2 3 7 3 5 −3 0 1 2 5 3 6 3 7 3 5 −3
0 1 2 5 3 6 4 7 3 4 −3 0 4 5 6 3 7 3 7 1 2 −3 0 1 2 5 3 6 3 7 3 4 −3
♠1 0 1 2 3 3 6 4 7 4 5 3 0 4 5 6 3 6 3 7 1 2 3 0 1 2 5 3 6 3 4 3 4 3
0 1 2 5 3 6 4 7 4 5 3 0 4 5 6 2 3 3 5 1 5 −3 0 1 2 5 3 6 3 7 3 4 3
0 4 1 5 6 7 2 4 4 6 3 0 4 5 6 2 3 3 7 1 5 −3 0 4 1 5 3 6 2 3 4 6 3
0 4 1 5 6 7 2 3 4 6 3 0 4 5 6 1 7 3 7 2 3 −3 0 4 1 5 3 6 4 7 2 3 3
0 4 1 5 6 7 2 4 3 6 3 0 4 5 6 1 7 3 5 2 3 −3 0 4 1 5 3 6 3 4 2 6 3
0 4 1 5 6 7 2 3 3 6 3 0 4 5 6 1 3 3 5 2 5 3 0 4 1 5 3 6 2 7 3 4 3
0 4 5 6 2 3 3 5 1 3 −3 0 4 5 6 1 3 3 7 2 5 3 0 4 1 5 3 6 2 4 3 6 −3
0 4 5 6 2 7 3 5 1 3 −3 0 4 5 6 2 7 3 7 1 3 3 0 4 1 5 3 6 3 7 2 4 −3
0 4 5 6 2 3 5 7 1 3 3 0 4 5 6 2 7 3 5 1 3 3 0 4 5 6 1 3 2 5 3 6 −3
0 4 5 6 2 7 5 7 1 3 3 0 4 1 2 3 4 3 5 4 6 3 0 4 5 6 1 3 3 7 2 5 −3
0 2 1 5 3 6 3 4 3 4 3 ♠2 0 4 1 2 3 6 3 7 4 5 3 0 4 5 6 1 3 3 5 2 6 3
0 2 1 5 3 6 4 7 3 4 −3 0 4 5 6 1 2 3 7 3 5 3 0 4 5 6 1 3 2 7 3 5 3
0 2 1 5 6 7 3 4 3 4 3 0 4 5 6 1 2 3 7 3 4 3 0 4 5 6 1 3 2 3 5 6 3
0 2 1 5 6 7 3 4 4 6 −3 0 4 2 5 3 6 3 4 1 4 −3 0 4 5 6 1 3 5 7 2 3 3
0 4 2 5 1 6 4 7 4 5 −3 0 4 2 5 3 6 3 7 1 4 −3 0 1 2 3 3 4 3 4 5 6 −3
0 4 2 5 1 6 4 7 3 5 −3 0 4 1 5 2 6 3 7 3 5 −3 0 1 2 3 3 4 3 7 4 5 3
0 4 2 5 1 6 3 7 4 5 −3 0 4 1 5 2 6 3 7 3 4 −3 0 1 2 3 3 4 3 5 4 6 −3
0 4 2 5 1 6 3 7 3 5 −3 0 4 1 5 3 6 3 4 2 4 3 0 2 1 3 3 4 3 4 5 6 3
0 4 1 5 3 6 3 4 2 3 3 0 4 1 5 3 6 3 7 2 4 3 0 2 1 3 3 4 3 7 4 5 −3
0 4 1 5 3 6 4 7 2 3 −3 0 4 2 5 1 6 3 7 3 5 3 0 2 1 3 3 4 3 5 4 6 3
0 4 1 5 6 7 2 3 3 4 3 0 4 2 5 1 6 3 7 3 4 3 0 4 1 2 3 4 3 4 5 6 −3
0 4 1 5 6 7 2 3 4 6 −3 0 2 1 3 3 4 3 5 4 6 −3 0 4 1 2 3 4 3 7 4 5 3
0 2 1 3 3 4 3 7 4 5 3 ♣3 0 2 1 3 3 6 3 7 4 5 −3 0 4 1 2 3 4 3 5 4 6 −3
♠3 0 2 1 3 3 6 4 7 4 5 −3 0 2 1 5 3 6 3 7 3 5 3 0 4 1 5 2 3 3 4 5 6 3
0 2 1 5 3 6 4 7 3 4 3 0 2 1 5 3 6 3 7 3 4 3 0 4 1 5 2 3 3 7 4 5 3
0 2 1 5 3 6 4 7 4 5 −3 0 2 1 5 3 6 3 4 3 4 −3 0 4 1 5 2 3 3 5 4 6 −3
0 4 2 5 6 7 1 4 4 6 −3 0 2 1 5 3 6 3 7 3 4 −3 0 4 2 5 1 3 3 4 5 6 −3
0 4 2 5 6 7 1 4 3 6 −3 0 4 2 5 3 6 1 3 4 6 −3 0 4 2 5 1 3 3 7 4 5 −3
0 4 2 5 6 7 1 3 4 6 −3 0 4 2 5 3 6 4 7 1 3 −3 0 4 2 5 1 3 3 5 4 6 3
378 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
now acts on the known tri-vector 2[[P ,∆]]. In this sense, the Jacobiator [[P ,P ]] shows
up through the term [[P ,∆]].
For each pair (P ,∆), the above balance at 1 contains sums over indexes that mark
the derivatives falling on the Jacobiator. By taking those formulae, we guess the candi-
dates for graphs that form the next, yet unknown, part of the operator ♢. Specifically,
we inspect which differential operator(s), acting on the Jacobi identity, become visible
and we list the graphs that provide such differential operators via the Leibniz rule(s).
For a while we keep every such candidate with an undetermined coefficient. By repeat-
ing the iteration, now for a different Poisson bi-vector P or its new, less degenerate
perturbation ∆, we obtain linear constraints for the already introduced undetermined
coefficients. Simultaneously, we continue listing the new candidates and introducing
new coefficients for them.
Remark 9. By translating formulae into graphs, we convert the dimension-dependent
expressions into the dimension-independent operators which are encoded by the graphs.
An obvious drawback of the method which is outlined here is that, presumably, some
parts of the operator ♢ could always stay invisible for all Poisson structures over R3 if
they show up only in the higher dimensions. Secondly, the number of variants to con-
sider and in practice, the number of irrelevant terms, each having its own undetermined
coefficient, grows exponentially at the initial stage of the reasoning.
By following the loops of iterations of this algorithm, we managed to find two non-
zero coefficients and five zero coefficients in solution (11). Namely, we identified the
coefficient ±1 for the tripod, which is the first term in (11), and we also recognized the
coefficient ±3 of the sum of ‘elephant’ graphs, which is the second to last term in (11).
Remark 10. Because of the known skew-symmetry of the tri-vector [[P ,Q1:6]] with re-
spect to its arguments f, g, h, finding one term in a sum within formula (11) for ♢
means that the entire such sum is reconstructed. Indeed, one then takes the sum over
a subgroup of S3 acting on f, g, h, depending on the actual skew-symmetry of the term
which has been found.
For instance, the first term in (11), itself making a sum running over {id} ≺ S3,
is obviously totally antisymmetric with respect to its arguments. The other graph
which we found by using the perturbation method (see the last graph in the second
line of formula (11) on p. 371) is skew-symmetric with respect to its second and third
arguments but it is not yet totally skew-symmetric with respect to the full set of its
arguments. This shows that is suffices to take the sum over the group ⟳ = A3 ≺ S3 of
cyclic permutations of f, g, h, thus reconstructing the sixth term in solution (11).
Acknowledgements. A.K. thanks M.Kontsevich for posing the problem; the authors
are grateful to P.Vanhaecke and A.G. Sergeev for stimulating discussions. The authors
are profoundly grateful to the referee for constructive criticism and advice.
This research was supported in part by JBI RUG project 106552 (Groningen) and
by the IHÉS and MPIM (Bonn), to which A.K. is grateful for warm hospitality. A. B.
and R.B. thank the organizers of the 8th international workshop GADEIS VIII on
Group Analysis of Differential Equations and Integrable Systems (12–16 June 2016,
Larnaca, Cyprus) for partial financial support and warm hospitality. A. B. and R.B. are
also grateful to the Graduate School of Science (Faculty of Mathematics and Natural
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 379
Sciences, University of Groningen) for financial support. We thank the Center for
Information Technology of the University of Groningen for providing access to the
Peregrine high performance computing cluster.
References
[1] Bouisaghouane A., Kiselev A.V. (2017) Do the Kontsevich tetrahedral flows pre-
serve or destroy the space of Poisson bi-vectors ? J. Phys.: Conf. Ser. Proc.
XXIV Int. conf. ‘Integrable Systems and Quantum Symmetries’ (14–18 June 2016,
CVUT Prague, Czech Republic), to appear, 10 p. (Preprint arXiv:1609.06677
[q-alg])
[2] Buring R., Kiselev A.V. (2017) On the Kontsevich ?-product associativity mecha-
nism, PEPAN Letters 14:2, 403–407. (Preprint arXiv:1602.09036 [q-alg])
[3] Buring R., Kiselev A.V. (2017) Software modules and computer-assisted proof
schemes in the Kontsevich deformation quantization, Preprint arXiv:1702.00681
[math.CO], 44 + xvi p.
[4] Grabowski J., Marmo G., Perelomov A.M. (1993) Poisson structures: towards a
classification, Mod. Phys. Lett. A8:18, 1719–1733.
[5] Kiselev A. V. (2013) The geometry of variations in Batalin–Vilkovisky formalism,
J. Phys.: Conf. Ser. 474, Paper 012024, 1–51. (Preprint 1312.1262 [math-ph])
[6] Kiselev A.V. (2016) The calculus of multivectors on noncommutative jet spaces.
Preprint arXiv:1210.0726 (v4) [math.DG], 53 p.
[7] Kiselev A.V. (2015) Deformation approach to quantisation of field models, Preprint
IHÉS/M/15/13 (Bures-sur-Yvette, France), 37 p.
[8] Kiselev A.V., Ringers S. (2013) A comparison of definitions for the Schouten
bracket on jet spaces, Proc. 6th Int. workshop ‘Group analysis of differential
equations and integrable systems’ (18–20 June 2012, Protaras, Cyprus), 127–141.
(Preprint arXiv:1208.6196 [math.DG])
[9] Kontsevich M. (1994) Feynman diagrams and low-dimensional topology. First Eu-
rop. Congr. of Math. 2 (Paris, 1992), Progr. Math. 120, Birkhäuser, Basel, 97–121.
[10] Kontsevich M. (1995) Homological algebra of mirror symmetry. Proc. Intern. Congr.
Math. 1 (Zürich, 1994), Birkhäuser, Basel, 120–139.
[11] Kontsevich M. (1997) Formality conjecture. Deformation theory and symplectic
geometry (Ascona 1996, D. Sternheimer, J. Rawnsley and S.Gutt, eds), Math. Phys.
Stud. 20, Kluwer Acad. Publ., Dordrecht, 139–156.
[12] Kontsevich M. (2003) Deformation quantization of Poisson manifolds, Lett. Math.
Phys. 66:3, 157–216. (Preprint q-alg/9709040)
[13] Olver P. J. (1993) Applications of Lie groups to differential equations, Grad. Texts
in Math. 107 (2nd ed.), Springer–Verlag, NY.
[14] Vanhaecke P. (1996) Integrable systems in the realm of algebraic geometry, Lect.
Notes Math. 1638, Springer–Verlag, Berlin.
380 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
Extra references
[15] Bouisaghouane A. (2017) The Kontsevich tetrahedral flow in 2D: a toy model,
Preprint arXiv:1702.06044 [math.DG], 6 p.
[16] Dubrovin B. (2005) Bihamiltonian structures of PDE’s and Frobenius mani-
folds, Summer School ICTP, http://indico.ictp.it/event/a04198/session/
47/contribution/26/material/0/0.pdf.
[17] Kiselev A.V. (2012) The twelve lectures in the (non)commutative geometry of
differential equations, Preprint IHÉS/M/12/13 (Bures-sur-Yvette, France), 140 pp.
[18] Kontsevich M. (1993) Formal (non)commutative symplectic geometry, The
Gel’fand Mathematical Seminars, 1990-1992 (L. Corwin, I. Gelfand, and J. Lepow-
sky, eds), Birkhäuser, Boston MA, 173–187.
[19] Laurent–Gengoux C., Picherau A., Vanhaecke P. (2013) Poisson structures. Gründ-
lehren der mathematischen Wissenschaften 347, Springer–Verlag, Berlin.
[20] Manin Yu. I. (1999) Frobenius manifolds, quantum cohomology, and moduli spaces.
AMS Colloquium publications 47, Providence RI.
[21] Merkulov S.A. (2010) Exotic automorphisms of the Schouten algebra of polyvector
fields. Preprint arXiv:0809.2385 (v6) [q-alg], 37 p.
[22] Olver P. J., Sokolov V.V. (1998) Integrable evolution equations on associative
algebras, Comm. Math. Phys. 193:2, 245–268;
Olver P. J., Sokolov V.V. (1998) Non-abelian integrable systems of the derivative
nonlinear Schrödinger type, Inverse Prob. 14:6, L5–L8.
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 381
Appendix C. The condition a : b = 1 : 6 is necessary (and maybe
sufficient ?)
Proposition 6 ([1]). The tetrahedral flow Ṗ = Qa:b(P) preserves the property of
P + εQa:b(P) + ō(ε) to be (at least infinitesimally) Poisson for all Poisson bi-vectors P
on all affine real manifolds Nn only if the ratio is a : b = 1 : 6.
Our proof amounts to producing at least one counterexample when any ratio other than
1 : 6 violates equation (2) for a given Poisson bi-vector P .
Proof. Let x, y, z(be the Cartesia/n coordin)ates on R3. Consider the Poisson bracket
{u, v}P = x · det ∂(xyz + y, u, v)(∂(x, y, z) given by the Jacobian, so that the coeffi-cient matrix is )
0 x2y −x(xz+1)
P ij = −x2y 0 xyz .
−x(xz+1) −xyz 0
The coefficient m(atrices of both bi-vecto)rs are
0 −x5y −x4 ( )(xz+1) 0 x5y x4(xz+2)
Γ1(P) = 6 · x5y 0 −x3y , Γ2(P) = −x5y 0 −2x3y .
x4(xz+1) x3y 0 −x4(xz+2) 2x3y 0
It is readily seen that no non-trivial linear combination a · Γ1(P) + b · Γ2(P) of the two
flows vanishes everywhere on R3 3 (x, y, z) for this example. Acting on the bi-vectors Γ1
and Γ2 by the Poisson differential [[P , ·]], we obtain two tri-vectors which are completely
determined by one component each. Namely, we have that
[[P ,Γ1(P)]]123 = 36x6yz + 48x5y, [[P ,Γ2(P)]]123 = −6x6yz − 8x5y.
Clearly, the balance a : b = 1 : 6 is the only ratio at which the non-trivial linear
combination Qa:b(P) = a · Γ1(P) + b · Γ2(P) solves the equation [[P ,Qa:b(P)]] ≡ 0. □
In fact, more is known — this time, about the sufficiency of the condition a : b = 1 : 6.
First, let us recall from [4] that on R3 with coordinates x, y, and z there is a class of
Poisson brackets that admit first in(tegrals at l)east locally:11
{ } · ∂(g, u, v)u, v = f det for u, v ∈ C∞ 3P (R ), (15)
∂(x, y, z)
where the free(parameter)g is a function and th∣ e parameter f(is a density s)o that
· ∂(g, u, v) ∂(g, u, v)f(x, y, z) det dxdydz = f(x, y, z)∣∣∣ · det dx′dy′dz′∣ .∂(x, y, z) x=x(x′,y′,z′) ∂(x′, y′, z′)y=y(x′,y′,z′)
z=z(x′,y′,z′)
In any given coordinate system the parameter f can be chosen freely; then it is recal-
culated as shown above.
11The referee points out that not all the Poisson brackets are given by the Jacobian determinants.
Indeed, the function g in (15) is always a Casimir of such bracket, but there are real Poisson structures
on R3 which do not have (smooth) Casimirs near all of its points: some point(s) can be singular so
that in no neighbourhood of it would a Casimir exist. In fact, no exhaustive description is known for
Poisson brackets on R3.
382 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
Proposition 7 (R3,{·, ·}P). The tetrahedral flow Ṗ = Q1:6(P) does preserve the prop-
erty of P + εQa:b(P) + ō(ε) to be infinitesimally Poisson for all Poisson structures (15)
on R3.
We used Proposition 7 as an heuristic motivation to our main Theorem 3 in which the
claim from Proposition 7 is extended to all Poisson structures on all finite-dimensional
affine real manifolds. Therefore, in hindsight, Proposition 7 above has been proven
rigorously as soon as Theorem 3 was established.
To verify the claim in Proposition 7 by direct calculation, it would take years for man
still only a few seconds for a computer.12 A computer-assisted proof of Proposition 7 is
realized through running the script in Maple (see below). (All computations are done
with the coefficient matrices of bi-vectors at hand. The bi-vectors are computed by
using working formulas (8a) and (8b).) For the balanced flow we have:
FlowQ := proc (P, y, a, b)
description "Eval flow Q_a:b of q-dim bi-vector P.";
local i, j, q, A, F, G, B, T, C;
q := op(P)[1];
F := proc (i, j, k, l, m, n, p, r) options operator, arrow;
a*(diff(P[i, j], y[k], y[l], y[m]))*(diff(P[k, n], y[p]))
*(diff(P[l, p], y[r]))*(diff(P[m, r], y[n])) end proc;
G := proc (i, j, k, l, m, n, p, r) options operator, arrow;
b*(diff(P[i, j], y[k], y[l]))*(diff(P[k, m], y[n], y[p]))
*(diff(P[n, l], y[r]))*(diff(P[r, p], y[j])) end proc;
B := Array(1 .. q, 1 .. q);
T := combinat:-cartprod([seq([seq(1 .. q)], i = 1 .. 8)]);
while not T[finished] do
C := op(T[nextvalue]());
B[C[1], C[2]] := B[C[1], C[2]]+F(C);
B[C[1], C[5]] := B[C[1], C[5]]+G(C);
end do;
A := Array(1 .. q, 1 .. q);
for i from 1 to q do
for j from 1 to q do
A[i, j] := simplify((1/2)*B[i, j]-(1/2)*B[j, i]);
end do;
end do;
Matrix(A);
end proc:
To implement the Schouten bracket of two bi-vectors A and B, we use a component
expansion (cf. [16]∑): n
[[A,B]]ijk = AskBij +Bsks A
ij
s + A
sjBki +Bsjs A
ki + Asis B
jk
s +B
siAjks ,
s=1
where superscripts and subscripts denote the bi-vector components and partial deriva-
tives with respect to the coordinates ys, respectively.
12Running the script below took us approximately 5 seconds.
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 383
SchoutenBracket := proc (A, B, y)
description "Evaluate the Schouten-bracket of A and B.";
local T, t, F, n, res, cnt;
n := op(A)[1];
F := proc (i, j, k) options operator, arrow;
A[s, k]*(diff(B[i, j], y[s]))+B[s, k]*(diff(A[i, j], y[s]))+
A[s, j]*(diff(B[k, i], y[s]))+B[s, j]*(diff(A[k, i], y[s]))+
A[s, i]*(diff(B[j, k], y[s]))+B[s, i]*(diff(A[j, k], y[s])) end proc;
T := combinat:-choose(n, 3);
for t in T do
print([[t[1], t[2], t[3]],simplify(add(F(t[1], t[2], t[3]), s = 1 .. n))]);
end do;
end proc:
Finally, the following script provides a computer-assisted proof of Proposision 7.
# All 3-dimensional Poisson bi-vectors are of the following form.
> P:=<<0,-f(x,y,z)*(diff(g(x,y,z),z)),f(x,y,z)*(diff(g(x,y,z),y))>|
<f(x,y,z)*(diff(g(x,y,z),z)),0,-f(x,y,z)*(diff(g(x,y,z),x))>|
<-f(x,y,z)*(diff(g(x,y,z),y)),f(x,y,z)*(diff(g(x,y,z),x)),0>>:
# We evaluate the balanced flow Q_{1:6} on the above bi-vector.
> Q:=FlowQ(P,{x,y,z},1,6)
[Length of output exceeds limit of 1000000]
# If so, let us inspect whether the flow Q_{1:6} vanishes.
> LinearAlgebra:-Equal(Q,Matrix(1..3,1..3,0))
false
# Still, let us act on this Q_{1:6} by the Poisson differential.
> SchoutenBracket(P,Q,{x,y,z})
[[1,2,3], 0]
This reasoning hints us that the condition a : b = 1 : 6 could be sufficient for equation (2)
to hold for all Poisson structures on all finite dimensional affine real manifolds. A
rigorous proof of the respective claim in Theorem 3 is provided in section 2.
Appendix D. The count of Leibniz graphs in Fig. 6
We count all possible differential consequences of the Jacobi identity, that is, we consider
the differential operators acting on the Jacobiator. We do this by constructing all
possible graphs that encode trivector-valued differential consequences (see Lemma 1 on
p. 368). The graphs that encode such differential consequences have 3 ground vertices.
The Schouten bracket [[P ,Q1:6(P)]] consists of graphs with 5 internal vertices. Since
two of these internal vertices are accounted for by the Jacobi identity, there remain
3 spare internal vertices.
First, let the Jacobiator stand, with all its three edges, on the 3 ground vertices. The
only freedom that remains is how the 3 free internal vertices act on each other and on
the Jacobiator. With its first edge, every free internal vertex can act on itself, on its 2
neighbouring free vertices, or on the Jacobiator; there are 4 possible targets. No second
edge can meet the first edge at the same target (as this would yield no contribution due
384 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
to the anti-symmetry, which is explained in Remark 2). Hence there are only 3 possible
targets for this second edge. Finally, again due to anti-symmetry, every possibility is
constructed exactly twice this way. Swapping the targets of the first and second edge
only contributes to the si(gn o)f the graph. The total number of this type of differential
consequence is therefore 4·3 3 = 216 graphs. This type of graph is drawn first from
2
the top-left in Figure 6.
Now let the Jacobiator stand on only 2 of the ground vertices. The remaining edge
of the Jacobiator has only 3 possible targets, as the third edge cannot fall back onto the
Jacobiator itself. One of the free internal vertices acts with an edge on the remaining
ground vertex. The other edge has 4 candidates as its target, namely the vertex itself,
the neighbouring 2 free i(nter)nal vertices, and the Jacobiator. The 2 internal vertices notfalling on a ground vertex have each 4·3 possible targets. The total number of graphs is2
therefore equal to 3 · 4 · 4·3 2 = 432. This type of graph is the second from the top-left
2
in Figure 6.
Next, let the Jacobiator stand on only 1 ground vertex. We distinguish between
two cases: namely, the case where 1 free internal vertex stands on both the remaining
ground vertices and the case where two different internal vertices act by one edge each
on the remaining two ground vertices. These are the third and fourth graphs from the
top-left in Figure 6, respectively.
• In the first case, the remaining 2 internal vertices each have 4·3 possible targets.
2
Th(e Ja)cobiator must act with its two remaining free edges on two different targets outof the 3 available, yielding 3 possibilities. The number of graphs in the first case is
3 · 4·3 2 = 108.
2
• For the second case, two internal vertices can each act on themselves, on the neigh-
bouring 2 internal vertices, or on the Jacobiator. With two of its edges, the Jacobiator
can act in 3 different ways on the 3 internal vertices. The third internal vertex has
4·3 possible targets. This brings the total number of graphs for the second case to
2
4 · 4 · 4·3 · 3 = 288.
2
The last case to consider is where the Jacobiator does not act on any of the ground
vertices. Again, since the outgoing edges of the Jacobiator must have different targets,
it is clear that the Jacobiator acts in a unique way on all 3 internal vertices. We now
distinguish two cases: namely, the case where 1 free internal vertex stands on 2 ground
vertices, 1 free internal vertex acts on 1 ground vertex, and 1 free internal vertex falling
on no ground vertex, and the second case where each internal vertex acts with one edge
on one ground vertex. These two cases are represented by the last 2 graphs in Figure 6,
respectively.
• In the first case, there is a free internal vertex with one free edge, which has 4
possible targets. The remaining free internal vertex with two free edges has 4·3 possible
2
targets. The total number of graphs for this case is 4 · 4·3 = 24.
2
• In the second case, each internal vertex can act on itself, on its 2 neighbouring
internal vertices, and on the Jacobiator. This results in a total of 43 = 64 graphs.
Summarizing, the total number of all trivector-valued Leibniz graphs, linear in the
Jacobiator and containing five internal vertices, is 1132.
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 385
Appendix E. Properties of the found solution
Remark 11. Let us recall that equation (2) yields the linear system of 7,025 inhomoge-
neous equations for the coefficients of 1132 patterns from Fig. 6. This shows that the
algebraic system at hand is extremely overdetermined. Moreover, out of those 1132 ad-
missible totally antisymmetric graphs, solution (11) involves only 8 of them. In this
sense, the factorising operator ♢ in (2) is special; for it expands via (11) over a very
low dimensional affine subspace in the affine space of unknowns in that inhomogeneous
linear algebraic system.
Property 1. The relevant Leibniz graphs, with respect to which the solution ♢(P , · )
expands, do not contain tadpoles nor two-cycles (or “eyes”, see Fig. 1 on p. 362).
• None of the arrows that act back on the Jacobiator is issued from any of its argu-
ments.
• In all the graphs the source vertices (if any), on which no arrows fall after all the
Leibniz rules are expanded, belong to the Jacobiator (cf. (4) on p. 363).
Property 2. The found solution ♢ does contain the graphs in which two or three
arrows fall on the Jacobiator.13
It has been explained in [5, 7] that the existence of two or more such arrows falling
on the equation [[P ,P ]] = 0 is an obstruction to an extension of the main claim,
.
[[P ,Q1:6(P)]] = 0 via [[P ,P ]] = 0, (2)
to the infinite-dimensional geometry of jet spaces J∞(π) for affine bundles over a man-
ifold Mm or jet spaces J∞(Mm → Nn) of maps from Mm, and of variational Poisson
brackets { , }P for functionals on such jet spaces (see [13, 17] and [6, 7]). Namely, it
can then be that
[[P ,Q1:6(P)]] ≇ 0 although [[P ,P ]] ∼= 0. (16)
We denote here by [[ , ]] the variational Schouten bracket; the variational bi-vector Q1:6
is constructed from the variational Poisson bi-vector P by using techniques from the
geometry of iterated variations of functionals (see [5, 6, 7]). An explicit counterexample
of (16) is known from [1] for the variational Poisson structure of the Harry Dym partial
differential equation.
The reason why the obstruction arises is that in the variational setting, the second
and higher order variations of a trivial integral functional Jac(P) ∼= 0 in the horizontal
cohomology can still be nonzero (although its first variation would of course vanish).14
Remark 12. The eight graphs in (11) represent a linear differential operator with re-
spect to the Jacobiator Jac(P). However, a quadratic nonlinearity with respect to the
two-vertex argument Jac(P) could be hidden in the five-vertex graphs in formula (11),
so that it would in fact encode a bi-differential operator ♢(P , · , · ). If this be the
13For instance, the first term in ♢ is the tripod standing on Jac(P).
14The same effect has been foreseen for a variational lift of deformation quantisation [12]: it has
been argued in [7] why the associativity of noncommutative star-product ⋆ = ×+ ℏ{ · , · }P + ō(ℏ) can
leak and it has been shown in [2] that if it actually does at O(ℏk), the order k at which this leak of
associativity can occur is high: k ⩾ 4.
386 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
case, expansion of one or the other copy of the Jacobiator using (4) in such a polyd-
ifferential operator ♢(P , · , · ) would produce two seemingly distinct linear differential
operators ♢(P , · ).
The scenarios to build the bi-linear, bi-differential terms in the operator ♢ are drawn
in Fig. 7 below.sWe consider – in fact, without ansy loss of generality – only those eight“3”: • • • • “2”(1): • • • •
 
 

 
 

	 
 ? ? 
 ?
( ) ( ) ( ) ( ) ( ) ( )
s s
“2”(2): • • • • “1”(1): • • • •
 

 

U 	 
 U ? ?
( ) ( ) ( ) ( ) ( ) ( )
s
“1”(2): • • • •


U R 	
( ) ( ) ( )
Figure 7. The Leibniz graphs by using which a quadratic –with respect
to the Jacobiator – part ♢(P , ·, ·) of the factorizing operator could be
sought for in (10); such quadratic part (if any) itself is necessarily totally
skew-symmetric with respect to the three sinks.
Leibniz graphs in which
• the three arguments of each copy of Jacobiator (4) are different; in particular,
• neither of the Jacobiators acts on the other copy by two or three arrows (so that
only none or one such arrow is possible).
We recall that known solution (11) is the sum of 39 graphs from which a linear depen-
dence on the Jacobiator Jac(P) is retrieved by using the 27 Leibniz graphs (see Table 4
on p. 375). Let us inspect whether any solution of equation (10) can be nonlinear
in Jac(P); in particular, let us check whether there is (or is not) a bi-linear dependence
in Jac(P) hidden in (11).
Proposition 8. There is no quadratic part in all the solutions of equation (10).
This claim is supported by a computer-assisted run-through over all Leibniz graphs with
linear and with quadratic dependence on the Jacobiator, combined with a requirement
that at least one coefficient of those quadratic (in Jac(P)) Leibniz graphs be nonzero.
There is no solution.
UNIVERSAL INFINITESIMAL DEFORMATION OF POISSON STRUCTURES 387
Appendix F. O(pen probl)ems
F.1(. For th)e factorisation [[P ,Q1:6(P)]] = ♢ P , Jac(P) to guarantee that the equality
∂P Q1:6(P) = 0 holds if Jac(P) = 0, its mechanism is nontrivial. Relying on Lemma 1
(see [2]), it tells us how the differential consequences of Jacobi identity are split into
separately vanishing expressions. This mechanism works not only in the construction
of flows that satisfy (2) but also in the associativity,
Assoc .P(f, g, h) := (f ? g) ? h− f ? (g ? h) = 0 via [[P ,P ]] = 0,
of the non-commutative unital star-product ? = ×+ ℏ{ · , · }P + o(ℏ). The formula for
?-products was given in [12], establishing the deformation quantisation × 7→ ? of the
usual product × in the algebra C∞(Nn) 3 f, g, h on a finite-dimentional affine Poisson
manifold (Nn,P), see also [2, 7]. In fact, the construction of graph complex and the
pictorial language of graphs [11, 12] that encode polydifferential operators is common
to all these deformation procedures (cf. [3], also [21]).
Open problem 1. Consider the Kontsevich star-product ? = ×+ℏ{ · , · }P+o(ℏ) in the
algebra C∞(Nn)[[ℏ]] on a finite-dimensional affine Poisson manifold (Nn,P). Given by
the tetrahedra Γ and Γ′1 2 (see Fig. 2 on p. 366), the infinitesimal deformation P 7→ P +
εQ1:6(P)+o(ε) induces the infinitesimal deformation ? 7→ ?+ℏε [[[[Q1:6(P), · ]], · ]]+o(ε)
of the star-product. What are the properties of this infinitesimally deformed ?(ε)-
product ? In particular, is the condition that Q1:6(P) be ∂P-trivial necessary for the
?(ε)-product to be gauge-equivalent to the unperturbed ?-product at ε = 0 ?
We recall that the theory of (infinitesimal) deformations of associative algebra struc-
tures is very well studied in the broadest context (e.g., of the Yang–Baxter equation,
Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation, Frobenius manifolds and F-
structures, etc.), see [16, 20]. We expect that in that theory’s part which is specific
to the deformation of associative structures on finite-dimensional affine Poisson mani-
folds Nn, there must be a dictionary between the construction of Kontsevich flows for
spaces of Poisson bi-vectors and other instruments to deform the associative product in
the algebra C∞(Nn).
F.2. The Kontsevich tetrahedral flow Ṗ = Q1:6(P) is a universal procedure to deform
a given Poisson bi-vector P on any finite-dimensional affine real manifold Nn (i. e. not
necessarily topologically trivial). For consistency, let us recall that generally speak-
ing, not every infinitesimal deformation P →7 P + εQ + ō(ε) of a Poisson bi-vector P
can be compl}et/ed{to a Poisson deforma(t∧ion P )7→} P + Q(ε) at all o{rders in(∧ε. Th)eobstructions are contained in the third ∂P-cohomology group H3P = T ∈ Γ 3 TN
| ∂P(T) = 0 T = ∂P(R), R ∈ Γ 2 TN . Indeed, cast the master-equation
[[P +Q(ε),P +Q(ε)]] = 0 for the Poisson deformation to the coboundary statement
[[Q(ε),Q(ε)]] = ∂P(−P − 2Q(ε)), whence ∂P([[Q(ε),Q(ε)]] ≡ 0 by ∂2P = 0. Therefore,
the vanishing of the third ∂P-cohomology group guarantees the existence of a power
series solution Q(ε) to the cocycle-coboundary equation [[Q(ε),Q(ε)]] = −2∂P(Q(ε)):
known to be a cocycle, the left-hand side has been proven to be a coboundary as well.
(In other words, an infinitesimal deformation P 7→ P + εQ1:6(P) + o(ε) can be com-
pleted to the construction of Poisson bi-vector P(ε) such that P(ε = 0) = P and
388 A. BOUISAGHOUANE, R. BURING, AND A. V. KISELEV
∣
d ∣ P(ε) = Q1:6(P) if the third Poisson cohomology group H3P(Nn) with respect todε ε=0
the Poisson differential ∂P = [[P , · ]] vanishes for the manifold Nn.)
In the symplectic case, i. e. for n even and bracket { · , · }P nondegenerate, the Poisson
complex is known to be isomorphic to the de Rham complex for Nn (see [19]). We
are not yet aware of any way to constrain the Poisson cohomology groups Hk nP(N )
for degenerate Poisson brackets { · , · }P on real manifolds Nn of not necessarily even
dimension n < ∞. (E.g., the algorithm for construction of cubic Poisson brackets on
the basis of a class of R-matrices, which is explained in [19], yields a rank-six bracket
on N9 ⊂ R9.)
F.3. The second Poisson cohomology group H2 n nP(N ) of the manifold N , if nonzero,
provides room for the ∂P-nontrivial deformations of P usingQ1:6(P) such thatQ1:6(P) 6=
[[P ,X]] for all globally defined 1-vectors X on Nn. In particular, this implies that there
are no ∂P-nontrivial tetrahedral graph flows on even-dimensional star-shaped domains
equipped with nondegenerate Poisson brackets.
A possibility for the right-hand side Q1:6(P) of the tetrahedral flow to be ∂P-trivial
is thus a global, topological effect; it cannot always be seen within a single chart in Nn.
Moreover, it is not universal with respect to the calculus of graphs.
Remark 13. Kontsevich notes [11] that if n = 2 so that every bi-vector P on N2 is
Poisson and every flow Ṗ = Qa:b(P) preserves this property, the tetrahedron Γ1 (or,
equivalently, the velocity Q1:0(P)) is always ∂P-exact. The required 1-vector field X(P)
in the coboundary statement Q1:0(P) = [[P ,Xp ]] cpan be expressed in terms of the bi-vector P , e.g., by the Leibniz-rule graph X =R p ?	I. (This is a particular, not general?
solution.) We recall that after the dimension n is fixed (here n = 2), a given differential
polynomial in P can be encoded by the Kontsevich graphs in non-unique way (cf. [15]
for details).
Open problem 2. The formalism developed in [11] suggests that there are, most
likely, infinitely many Kontsevich graph flows on the spaces of Poisson bi-vectors on
finite-dimensional affine Poisson manifolds. Forming an example Q1:6(P) of such a
cocycle in the graph complex, the tetrahedra Γ1 and Γ′2 in Fig. 2 are built over four
internal vertices. What is or are the next – with respect to the ordering of natural
numbers – Poisson cohomology-nontrivial Kontsevich graph cocycle(s) built over five
or more internal vertices ? ∧
F.4. The tetrahedral flow Ṗ = Q1:6(P) preserves the space {P ∈ Γ( 2 TNn) | [[P ,P ]] =
0} of Poisson bi-vectors; this is guaranteed by Theorem 3 that asserts ∂P(Q
.
1:6) =
0 within the (graded-)commutative geometry of finite-dimensional affine real mani-
folds Nn.
Open problem 3. Does the proven property,
[[P ,Q1:6(P
.
)]] = 0 via [[P ,P ]] = 0, (2)
generalize to the formal noncommutative symplectic supergeometry [18], to the calcu-
lus of multivectors performed by using their necklace brackets (see [6] and references
therein), and to Poisson structures on the commutative non-associative unital algebras
of cyclic words (e. g., see [22]) ?
Chapter 16
Poisson brackets symmetry from the
pentagon-wheel cocycle in the graph
complex
This chapter is based on the peer-reviewed journal publication R. Buring, A. V. Kiselev,
and N. J. Rutten, Physics of Particles and Nuclei, 49(5): Supersymmetry and Quantum
Symmetries 2017, 924–928, 2018. (Preprint arXiv:1712.05259 [math-ph] – 4 p.)
Commentary. In reference to Part I of the dissertation, the material of this chapter
is used in §5.2, Chapter 6 (§6.2), and §7.4. In the Appendix within this chapter we
give the analytic formula of the pentagon-wheel flow on the spaces of Poisson structures.
Interestingly, an alternative solution ♢2 of the Poisson cocycle factorization problem
(via 8691 Leibniz graphs) was found long before the canonical Kontsevich solution ♢1
(consisting of only 3876 Leibniz graphs).
389
Poisson brackets symmetry from the
pentagon-wheel cocycle in the graph complex
R. Buring∗,,‡ A.V.Kiselev†,,§ N. J. Rutten†
E-mail: ‡ rburing@uni-mainz.de, § A.V.Kiselev@rug.nl
Abstract
Kontsevich designed a scheme to generate infinitesimal symmetries Ṗ = Q(P) of Pois-
son brackets P on all affine manifolds M r; every such deformation is encoded by
oriented graphs on n+2 vertices and 2n edges. In particular, these symmetries can be
obtained by orienting sums of non-oriented graphs γ on n vertices and 2n − 2 edges.
The bi-vector flow Ṗ = O⃗r(γ)(P) preserves the space of Poisson structures if γ is a
cocycle with respect to the vertex-expanding differential in the graph complex.
A class of such cocycles γ2ℓ+1 is known to exist: marked by ℓ ∈ N, each of them
contains a (2ℓ + 1)-gon wheel with a nonzero coefficient. At ℓ = 1 the tetrahedron
γ3 itself is a cocycle; at ℓ = 2 the Kontsevich–Willwacher pentagon-wheel cocycle γ5
consists of two graphs. We reconstruct the symmetry Q5(P) = O⃗r(γ5)(P) and verify
that Q5 is a Poisson cocycle indeed: [[P,Q5(P
.
)]] = 0 via [[P,P]] = 0.
Generic classical Poisson brackets P can be deformed along no less than countably many
directions (in the spaces of bi-vectors) such that they stay Poisson at least infinitesimally and
the change of brackets is not necessarily induced by a diffeomorphim along integral curves
of a vector field on the Poisson manifold at hand.1 The use of graphs converts this infinite
analytic problem into a set of finite combinatorial problems of finding cocycles γ ∈ ker d in
th∣∣e graph complex and orie∣nting th∣em: Q(P) = O⃗r(γ)(P), see the diagram.∣∣cocycles ∈ ∣ ∣ ∣ker d: sums of ∣ ∣sums of Kontsevich graphs Q on∣ put ∣bi-vector fields∣∣n-vertex∧(2n− 2)-edge non- ∣∣∣∣ ∣−−−O⃗−r→ ∣∣∣2 sinks, n internal vertices, and∣∣∣∣ ∣−−P−→ ∣∣∣
∣
Q(P) = O⃗r(γ)(P): ∣∣
oriented graphs with make 2n edges in n× (←− • −→) with into Poisson 2-cocycles ∣
E(γ) = ei and coeff ∈ R skew Left ≺ Right L R • ∈
∣
ker ∂P = [[P, ·]]
i
1. Graph complex theory. There are several ways to introduce a differential on the space
of non-oriented graphs (see [7, 8]). We consider the real vector space of finite non-oriented
graphs such t∧hat each of them is equipped with a wedge product of edges, i.e. we supposethat for every graph its edges ei are enumerated I, II, . . . and proclaimed parity-odd, so
that E(γ) := i ei and (γ, I ∧ II ∧ III ∧ . . .) = −(γ, II ∧ I ∧ III ∧ . . .), etc.
∗Mathematical Institute, Johannes Gutenberg University of Mainz, Staudingerweg 9, D-55128 Germany.
†Johann Bernoulli Institute for Mathematics & Computer Science, University of Groningen, P.O. Box 407,
9700 AK Groningen, The Netherlands. Partially supported by JBI RUG project 103511 (Groningen). A part
of this research was done while R.B. and A.V.K. were visiting at the IHÉS (Bures-sur-Yvette, France) and
A.V.K. was visiting at the University of Mainz.
1The dilation Ṗ = P is an example of symmetry for Jacobi identity; we study nonlinear flows Ṗ = Q(P)
which are universal w.r.t. all affine manifolds and should persist under the quantization ℏi {·, ·}P 7→ [·, ·].
390
Suppose also that all vertices are at least tri-valent (cf. [4, 9]). On this subspace (which
we study here), the differential amounts to a blow-up – via the Leibniz rule – of vertices
in a graph γ; every vertex v at hand is replaced by the new edge E such that every edge
which was incident to v in γ is now re-directed to one of the two ends of E. The choice
where to direct a given edge does not depend on a similar choice for other such edges, but
overall, the valency of either end of E must be at least two.2 By construction, the new edge
E is placed firstmost in the wedge product of edges in every graph g in d(γ): whenever
E(γ) = I ∧ II ∧ . . ., letq E(qg) = E ∧ I ∧ II ∧ . . .. Now one has d2 = 0.
Example(1. Letr w4 :=r @q r@)q , and let trhe edrge ordering in these graphs be lexicographic:r 1 r 4δ := d 3 r 6 r = 2 4 r
5 r r 3 5 r r r1 7 r r3 r2 r r12 36
6 r 2r + 4 3 r r r6 r + 4 4 r rr5 − 4 4 r r725 2 7 1 7 4 6 r r1 6 r r5
A flip over a diagonal in w4 swaps three pairs of edges; 3 is odd, so by this symmetry,
E(w4) = −E(w4), i.e. w4 is a zero graph.3 By this, d(wqq 4)q = 0. Because d
2 = 0, one has
d(δ6) = 0 for the coboundary δ6 ∈ im d. Put γ3 := @@q ; another example of nontrivial
cocycle, γ5 6∈ im d, also on n vertices and 2n− 2 edges, is given on p. 392.
The notion of oriented Kontsevich graphs from [7] was recalled in [1, 2, 5]. Every such
graph is built over m ordered sinks from n wedges ←−L • −→R : each top • of the wedge is the
source of exactly two arrows (which are ordered by Left ≺ Right). Let (M r, P) be a real affine
Poisson manifold of dimension r; let x1, . . ., xr be local coordinates. By decorating each edge
with its own summation index that runs from 1 to r, by identifying every such edge −→i with
∂/∂xi acting on the content of arrowhead vertex, and by placing a copy of the Poisson bi-
vector P = (P ij) at the top • of each wedge←−i • −→j , we associate a polydifferential operator
(e.g., an m-vector) with every such graph. The arguments of the operator are contained in
the m respective sinks. The resulting polydifferential operators are differential-polynomial
in the coefficients P ij of a given Poisson structure P . It is known that for P Poisson (hence
[[P ,P ]] = 0 under the Schouten bracket), its adjoint action ∂P := [[P , ·]] is a differential on
the space of multi-vectors. One can try finding Poisson cohomology cocycles Q ∈ ker ∂P by
assuming they are realized using the Kontsevich oriented graphs.
Now let us note that certain sums Q of oriented graphs built on two sinks from n wedges
can be obtained by taking all admissible ways to orient graphs γ on n vertices and 2n−2 edges
(clearly, two sinks and two edges into them are added). Moreover, suppose that γ ∈ ker d in
vertex-edge bi-grading (n, 2n − 2) is such that this sum of graphs can be oriented to yield
a sum of Kontsevich graphs on two sinks, n internal vertices and 2n edges. Then, in fact,
these oriented graphs, taken with suitable coefficients ∈ R, do assemble to a Poisson cocycle
Q(P) ∈ ker ∂P . Let this orientation mapping be denoted by O⃗r (cf. [7] and [1, 5]).4
2. The pentagon-wheel cocycle. The mechanism of factorization [[P ,Q P .( )]] = 0 via
[[P ,P ]] = 0 for the cocycle condition Q(P) ∈ ker ∂P is known from [2], where it is used in a
similar problem of the ⋆-product associativity (cf. [3]). In [1] this mechanism is applied to
the Kontsevich tetrahedral flow Q3(P) = O⃗r(γ3)(P). Would the mapping O⃗r be known, the
verification O⃗r(γ) ∈ ker ∂P is still compulsory (e.g., by using a fac(torization)via the Jacobi
identity for P). But for us now, the factorization [[P ,Q5(P)]] = ♢ P , [[P ,P ]] is the way to
2In earnest, graphs with valency 1 of an end of E cancel out in the action of this differential d, cf. [4, 8].
3One proves that d(zero graph) = sum of zero graphs and graphs with zero coefficients.
4The present paper is aimed to help us reveal the general formula of the morphism O⃗r which connects
the two graph complexes.
391
find the right formula of the flow Ṗ = Q5(P) that should correspond to the Kontsevich–
Willwacher pentagon-wheel cocycle γ5 under the orientation mapping, Q5 = O⃗r(γ5), giving
one solution Q5 yet not necessarily unique operator ♢.
Example 2. There are only two essentially different admissible ways to orient (and skew-
symmetrize with respect to sinks) the tetrahedron γ3 ∈ ker d. Each of the three oriented
graphs in the flow Q3 is encoded by the list of targets for the ordered pair of edges issued
from the ith vertex (m = 2 ⩽ i ⩽ 5 = m + n − 1), and a coefficient ∈ Z. Specifically,
we have that Q3 = 1 · (0, 1; 2, 4; 2, 5; 2, 3) − 3 · (0, 3; 1, 4; 2, 5; 2, 3 + 0, 3; 4, 5; 1, 2; 2, 4); the
analytic formula of the respective bi-differential operators acting on the sinks content f ,
g is Q (f, g) = ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq · ∂ f∂ g − 3∂ P ij∂ Pkℓ∂ Pmn∂ Ppq3 kmp q ℓ n i j mp jq ℓ n · ∂if∂kg −
3∂ ij kℓnpP ∂jP ∂ Pmn∂ Ppqkq ℓ · ∂if∂mg. A factorization of [[P ,Q3(P)]]
containing [[P ,P ]] is explained in [1], based on [2]. r r
via 8 tri-verctor graphsr
Now consider the pentagon-wheel cocycle γ r5 ∈ ker d, r 5 r r
see [4]. By orienting both graphs in γ5 (i.e. by shifting
γ5 = + 2 r r
the vertex labelling by +1 = m − 1, adding two edges r r  
to the sinks 0, 1, and keeping only those oriented graphs out of 1024 = 2#edges which are
built from ←− • −→) and skew-symmetrizing with respect to 0 ⇄ 1, we obtain 91 parameters
for Kontsevich graphs on 2 sinks, 6 internal vertices, and 12 (= 6 pairs) of edges. We take
the sum Q of these 91 bi-vector graphs (or skew differences of Kontsevich graphs) with
their undetermined coefficients, and for the set of tri-vector graphs occurring in [[P ,Q]],
we generate all the possibly needed tri-vector “Leibniz” graphs with [[P ,P ]] inside.5 This
yields 41031 such Leibniz graphs, which, with undetermined (coefficients), provide the ansatz
for the r.-h.s. of the factorization problem [[P ,Q(P)]] = ♢ P , [[P ,P ]] . This gives us an
inhomogeneous system of 463,344 linear algebraic equations for both the coefficients in Q
and ♢. In its l.-h.s., we fix the coefficient of one bi-vector graph6 by setting it to +2.
Claim. For γ5, the factorization problem [[P ,Q(P)]] = ♢(P , [[P ,P ]]) has a solution (Q5,♢5);
the sum Q5 of 167 Kontsevich graphs (on m = 2 sinks 0, 1 and n = 6 internal vertices 2, . . .,
7) with integer coefficients is given in the table below.7
0 1 2 4 2 5 3 6 4 7 2 4 10 0 3 4 5 1 2 2 6 2 7 2 4 −10 0 3 1 4 5 6 2 3 2 7 4 5 5 0 3 1 4 2 5 2 6 3 7 2 5 5
0 1 2 4 2 5 2 6 4 7 3 4 −10 0 3 1 4 5 6 2 3 3 7 2 3 −10 0 3 4 5 2 6 4 7 1 2 3 6 5 0 3 4 5 1 2 4 6 4 7 2 5 −5
0 3 1 4 2 5 6 7 2 4 3 4 10 0 3 4 5 2 6 2 7 1 2 2 6 10 0 3 1 4 5 6 2 3 5 7 2 4 −5 0 3 1 4 2 5 2 6 2 7 3 5 5
0 3 4 5 1 2 6 7 2 3 3 4 −10 0 1 2 4 2 5 2 6 2 7 2 3 2 0 3 4 5 1 2 6 7 2 4 4 6 −5 0 3 1 4 5 6 2 6 3 7 2 3 −5
0 3 1 4 2 5 2 6 4 7 3 4 10 0 1 2 4 2 5 2 6 3 7 3 4 −5 0 3 1 4 2 5 6 7 2 3 2 6 −5 0 3 4 5 2 6 4 7 2 7 1 2 −5
0 3 4 5 1 2 4 6 3 7 2 3 −10 0 1 2 4 2 5 3 6 3 7 2 4 5 0 3 1 4 5 6 2 3 5 7 2 3 −5 0 3 1 4 5 6 2 3 2 7 3 4 −5
0 3 1 4 2 5 3 6 4 7 2 4 −10 0 1 2 4 2 5 2 6 3 7 4 5 −5 0 3 4 5 2 6 4 7 1 2 2 6 5 0 3 4 5 2 6 6 7 1 2 2 3 −5
0 3 4 5 1 2 2 6 3 7 3 4 −10 0 1 2 4 2 5 2 6 4 7 3 5 −5 0 3 1 4 2 5 6 7 2 3 3 4 5 0 3 1 4 5 6 2 3 3 7 2 4 −5
0 3 1 4 5 6 2 3 5 7 2 5 −10 0 3 1 4 5 6 2 7 5 7 2 3 5 0 3 4 5 1 2 6 7 2 3 2 4 −5 0 3 4 5 2 6 2 7 1 2 3 6 5
0 3 4 5 2 6 4 7 1 2 4 6 10 0 3 4 5 5 6 6 7 2 7 1 2 5 0 3 1 4 2 5 3 6 4 7 2 3 −5 0 3 1 4 2 5 3 6 2 7 3 5 −5
0 3 4 5 1 6 2 4 5 7 2 5 10 0 3 1 4 2 5 6 7 2 4 3 6 5 0 3 4 5 1 2 2 6 3 7 2 4 −5 0 3 4 5 1 2 2 6 4 7 2 5 5
0 3 4 5 2 6 4 6 1 7 2 4 −10 0 3 4 5 1 2 6 7 2 7 3 4 −5 0 3 1 4 2 5 6 7 2 3 3 6 −5 0 3 1 4 2 5 3 6 3 7 2 5 −5
0 3 4 5 2 6 4 7 2 7 1 4 −10 0 3 1 4 2 5 2 6 3 7 4 5 5 0 3 4 5 1 2 6 7 2 4 2 6 −5 0 3 4 5 1 2 2 6 2 7 4 5 5
0 3 4 5 1 6 2 4 3 7 2 3 10 0 3 4 5 1 2 4 6 2 7 3 5 −5 0 3 4 5 1 2 6 7 2 4 3 4 −5 0 3 4 5 5 6 6 7 1 2 2 6 −5
0 3 4 5 2 6 6 7 1 3 2 3 −10 0 3 1 4 2 5 2 6 4 7 3 5 5 0 3 1 4 2 5 6 7 2 3 2 4 5 0 3 1 4 5 6 2 6 2 7 2 3 5
0 3 4 5 2 6 2 7 1 3 3 6 10 0 3 4 5 1 2 4 6 3 7 2 5 −5 0 3 4 5 1 2 4 6 3 7 2 4 −5 0 1 2 4 2 5 2 6 2 7 3 4 −5
0 3 4 5 1 6 4 7 2 3 2 3 −10 0 3 4 5 1 2 6 7 2 3 4 6 5 0 3 1 4 2 5 2 6 4 7 2 3 −5 0 1 2 4 2 5 2 6 3 7 2 5 −5
0 3 4 5 1 5 2 6 2 7 4 5 10 0 3 1 4 2 5 6 7 2 7 3 4 5 0 1 2 4 2 5 6 7 2 7 3 4 −5 0 1 2 4 2 5 2 6 2 7 3 5 −5
0 3 4 5 1 6 2 7 2 3 3 4 10 0 3 4 5 1 2 2 6 4 7 3 5 5 0 1 2 4 2 5 3 6 2 7 4 5 5 0 3 4 5 2 6 6 7 1 2 4 6 5
0 3 4 5 1 5 2 6 4 7 2 5 10 0 3 1 4 2 5 3 6 2 7 4 5 −5 0 1 2 4 2 5 3 6 4 7 2 5 5 0 3 1 4 5 6 2 3 2 7 2 5 −5
0 3 4 5 1 2 4 6 4 7 2 4 −10 0 3 4 5 1 2 2 6 3 7 4 5 5 0 1 2 4 2 5 3 6 2 7 3 5 5 0 3 4 5 1 2 4 6 4 7 2 3 −5
0 3 1 4 2 5 2 6 2 7 2 3 −10 0 3 1 4 2 5 3 6 4 7 2 5 −5 0 1 2 4 2 5 3 6 3 7 2 5 5 0 3 1 4 2 5 2 6 2 7 3 4 5
0 3 1 4 2 5 3 6 3 7 2 3 −10 0 3 4 5 2 6 6 7 1 2 3 4 5 0 3 4 5 1 2 4 6 2 7 4 5 −5 (see next page)
5The algorithm from [5, §1.2] produces 41031 Leibniz graphs in ν = 3 iterations and 56509 at ν ⩾ 7.
6This is done because it is anticipated that, counting the number of ways to obtain a given bi-vector
while orienting the nonzero cocycle γ5, none of the coefficients in a solution Q5 vanishes.
7The analytic formula of degree-six nonlinear differential polynomial Q5(P) is given in App. A. The
encoding of 8691 Leibniz tri-vector graphs containing the Jacobiator [[P,P]] for the Poisson structure P that
occur in the r.-h.s. ♢(P, [[P,P]]) is available at https://rburing.nl/Q5d5.txt. The machine format to
encode such graphs (with one tri-valent vertex for the Jacobiator) is explained in [5] (see also [1, 3]).
392
0 3 4 5 1 2 2 6 4 7 3 4 −5 0 3 4 5 1 5 6 7 2 4 2 6 5 0 3 4 5 1 6 2 7 2 3 4 6 −5 0 3 4 5 1 6 2 4 2 7 2 5 5
0 3 1 4 2 5 3 6 2 7 2 4 −5 0 3 4 5 2 6 2 7 1 5 3 6 5 0 3 4 5 1 5 2 6 4 7 2 3 5 0 3 4 5 1 6 4 6 2 7 2 4 5
0 3 1 4 5 6 2 3 3 7 2 5 −5 0 3 4 5 1 6 2 6 3 7 2 4 5 0 3 4 5 1 5 2 6 2 7 3 4 −5 0 3 4 5 1 6 2 4 2 7 2 3 5
0 3 4 5 2 6 2 7 1 2 4 6 5 0 3 4 5 2 6 2 6 1 7 3 4 −5 0 3 4 5 1 6 4 7 2 3 2 6 −5 0 3 4 5 2 6 4 7 5 7 1 2 5
0 3 1 4 5 6 2 7 3 7 2 3 −5 0 3 4 5 2 6 4 7 1 5 2 6 −5 0 3 4 5 1 6 2 4 2 7 4 5 −5 0 3 1 4 5 6 2 6 3 7 2 5 5
0 3 4 5 2 6 6 7 2 7 1 2 −5 0 3 4 5 1 6 2 7 2 5 3 4 5 0 3 4 5 1 6 2 7 2 7 2 4 −5 0 3 4 5 2 5 6 7 1 2 4 6 −5
0 3 1 4 2 5 3 6 3 7 2 4 −5 0 3 4 5 1 6 4 7 2 5 2 6 5 0 3 4 5 1 6 2 4 5 7 2 4 5 0 3 1 4 5 6 2 7 3 5 2 6 5
0 3 4 5 1 2 2 6 2 7 3 4 −5 0 3 4 5 1 6 4 7 2 7 2 3 −5 0 3 4 5 2 6 2 6 1 7 2 4 −5 0 3 4 5 2 5 6 7 1 2 3 6 −5
0 3 1 4 2 5 2 6 3 7 3 4 5 0 3 4 5 1 6 4 6 2 7 2 5 5 0 3 4 5 1 5 2 6 4 7 2 4 5 0 3 1 4 5 6 2 7 3 5 2 4 5
0 3 4 5 1 2 4 6 2 7 2 3 −5 0 3 4 5 1 6 2 7 3 5 2 4 −5 0 3 4 5 1 6 2 7 2 3 2 4 5 0 3 4 5 2 6 6 7 3 7 1 2 5
0 3 4 5 1 6 2 7 5 7 2 4 −5 0 3 4 5 2 5 6 7 1 4 2 6 −5 0 3 4 5 1 6 2 4 2 7 3 4 5 0 3 1 4 5 6 2 7 3 7 2 4 5
0 3 4 5 2 6 4 6 1 7 2 5 −5 0 3 4 5 2 6 4 7 2 7 1 3 −5 0 3 4 5 1 6 2 6 2 7 2 4 −5 0 3 4 5 5 6 6 7 1 2 2 3 5
0 3 4 5 1 6 2 7 2 5 4 6 5 0 3 4 5 2 5 6 7 1 3 2 6 −5 0 3 4 5 1 6 2 4 3 7 2 4 5 0 3 1 4 5 6 2 6 2 7 3 4 5
0 3 4 5 1 6 4 7 2 5 2 3 −5 0 3 4 5 2 6 6 7 1 7 2 4 5 0 3 4 5 2 6 2 7 1 5 2 6 5 0 3 4 5 1 2 2 6 4 7 2 4 −5
0 3 4 5 1 6 2 6 2 7 4 5 5 0 3 4 5 1 6 2 4 5 7 2 3 5 0 3 4 5 2 6 6 7 1 3 2 6 −5 0 3 1 4 2 5 3 6 2 7 2 3 −5
0 3 4 5 1 6 2 7 2 7 3 4 5 0 3 4 5 2 6 6 7 2 7 1 4 −5 0 3 4 5 2 6 2 7 1 3 2 6 5 0 3 4 5 2 6 6 7 1 2 2 6 5
0 3 4 5 2 6 6 7 1 7 2 3 −5 0 3 4 5 1 6 2 4 3 7 2 5 5 0 3 4 5 1 6 4 7 2 3 2 4 −5 0 3 1 4 5 6 2 3 2 7 2 3 −5
0 3 4 5 1 5 6 7 2 3 2 4 5 0 3 4 5 2 6 2 7 1 3 4 6 5 0 3 4 5 1 5 2 6 2 7 2 4 −5 0 3 4 5 1 2 4 6 2 7 2 4 −5
0 3 4 5 2 6 4 6 1 7 2 3 −5 0 3 4 5 2 6 6 7 1 3 2 4 −5 0 3 4 5 1 6 4 7 2 7 2 4 5 0 3 1 4 2 5 2 6 3 7 2 3 −5
Remark. To establish the formula for the morphism O⃗r that would be universal with respect
to all cocycles γ ∈ ker d, we are accumulating a sufficient number of pairs (d-cocycle γ,
∂P-cocycle Q), in which Q is built exactly from graphs that one obtains from orienting the
graphs in γ. Let us remember that not only nontrivial cocycles (e.g., γ3, γ5, or γ7 from [4],
cf. [6, 9]) but also d-trivial, like δ6 on p. 391, or even the ‘zero’ non-oriented graphs are
suited for this purpose: e.g., a unique O⃗r(w4)(P) ≡ 0 constrains O⃗r. In every such case,
the respective ∂P-cocycle is obtaineda by solving the factorization problem
.
[[P ,Q(P)]] = 0
via [[P ,P ]] = 0. The formula of the orientation morphism O⃗r will be the object of another
paper.
Acknowledgements. The authors thank M. Kontsevich and T. Willwacher for recalling the
existence of the orientation morphism O⃗r. A.V.K. thanks the organizers of international workshop
SQS’17 (July 31 – August 5, 2017 at JINR Dubna, Russia) for discussions.b
aThe actually found ∂P -cocycle Q might differ from the value O⃗r(γ) by ∂P -trivial or improper terms,
i.e. Q = O⃗r(γ) + ∂P(X) +∇(P, [[P,P]]) for some vector field X realized by Kontsevich graphs and for some
“Leibniz” bi-vector graphs ∇ vanishing identically at every Poisson structure P.
bAs soon as the expression of 167 Kontsevich graph coefficients in Q5 via the 91 integer parameters
was obtained, the linear system in factorization [[P,Q5(P)]] = ♢(P, [[P,P]]) for the pentagon-wheel flow
Ṗ = Q5(P) was solved independently by A. Steel (Sydney) using the Markowitz pivoting run in Magma.
The flow components Q5 of all the known solutions (Q5,♢5) match identically. (For the flow Ṗ = Q5(P) =
O⃗r(γ5)(P), uniqueness is not claimed for the operator ♢ in the r.-h.s. of the factorization.)
References ISQS’25. Prague, June 6–10, 2017. Preprint
arXiv:1710.02405
[1] Bouisaghouane A., Buring R., Kiselev A. The
Kontsevich tetrahedral flow revisited // J. Geom. [6] Dolgushev V. A., Rogers C. L., Willwacher T. H.
Phys. 2017. V. 119.P. 272–285. Kontsevich’s graph complex, GRT, and the defor-mation complex of the sheaf of polyvector fields
[2] Buring R., Kiselev A.V. On the Kontsevich ⋆- // Annals of Math. 2015. V. 182(3). P. 855–943;
product associativity mechanism // PEPAN Let- Willwacher T., Živković M. Multiple edges in
ters. 2017. V. 14(2). P. 403–407. M. Kontsevich’s graph complexes and computa-
[3] Buring R., Kiselev A.V. Software modules tions of the dimensions and Euler characteristics
and computer-assisted proof schemes in the // Advances of Math. 2015. V. 272. P. 553–578.
Kontsevich deformation quantization. Preprint [7] Kontsevich M. Formality conjecture // Proc.
IHÉS/M/17/05. 2017. of Conf. “Deformation theory and symplectic
[4] Buring R., Kiselev A.V., Rutten N. J. The hep- geometry”. Ascona, June 17–21, 1996. Dordrecht:
tagon-wheel cocycle in the Kontsevich graph com- Kluwer Acad. Publ., 1997. P. 139–156;
plex // J. Nonlin. Math. Phys. 2017. V. 24, Kontsevich M. Derived Grothendieck–
Suppl. 1. P. 157–173. Teichmüller group and graph complexes [afterT. Willwacher] // Séminaire Bourbaki (69ème
[5] Buring R., Kiselev A.V., Rutten N. J. Infinites- année, Janvier 2017). 2017. No. 1126. P. 1–26.
imal deformations of Poisson bi-vectors using [8] Khoroshkin A., Willwacher T., Živković M. Dif-
the Kontsevich graph calculus // Proc. of Conf. ferentials on graph complexes // Advances of
Math. 2017. V. 307. P. 1184–1214.
393
[9] Willwacher T. M.Kontsevich’s graph complex
and the Grothendieck–Teichmüller Lie algebra //
Invent. Math. 2015. V. 200(3). P. 671–760.
394
A The pentagon-wheel flow: analytic formula
Here is the value Q5(P)(f, g) of bi-vector Q5 at two functions f, g:
10∂t∂
ij
m∂kP ∂pPkℓ∂ ∂ ∂ Pmn∂ Ppqv r ℓ n ∂ Prsq ∂ P tvs ∂if∂jg
−10∂ ∂ ∂ P ij∂ kℓ mn pq rs tvp m k tP ∂v∂r∂ℓP ∂nP ∂qP ∂sP ∂if∂jg
+10∂ ij kℓ mn pq rs tvr∂mP ∂t∂jP ∂v∂s∂ℓP ∂nP ∂pP ∂qP ∂if∂kg
−10∂ ∂ P ij∂ ∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppqr n t s j v k ℓ ∂pPrs∂qP tv∂if∂mg
+10∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppq rsp m t j v r ℓ n ∂qP ∂ P tvs ∂if∂kg
−10∂ ∂ P ij∂ kℓ mnt n v∂r∂jP ∂p∂kP ∂ℓPpq∂ Prsq ∂sP tv∂if∂mg
−10∂t∂mP ij∂ ∂ Pkℓp j ∂v∂ mn pqr∂ℓP ∂nP ∂qPrs∂sP tv∂if∂kg
−10∂p∂ P ij∂ ∂ kℓ mn pq rs tvn t r∂jP ∂v∂kP ∂ℓP ∂qP ∂sP ∂if∂mg
−10∂t∂ P ij∂ kℓp q∂jP ∂ Pmn∂ ∂ ∂ Ppqℓ v r m ∂nPrs∂ tvsP ∂if∂kg
+10∂ ∂ P ij∂ kℓ mn pq rs tvs m jP ∂t∂p∂kP ∂ℓP ∂v∂nP ∂qP ∂if∂rg
+10∂ ∂ P ij∂ Pkℓ∂ ∂ Pmnt p j q k ∂ pq rs tvv∂r∂ℓP ∂nP ∂sP ∂if∂mg
−10∂t∂ ijmP ∂ Pkℓj ∂v∂p∂ Pmn∂ Ppqk ℓ ∂q∂ rs tvnP ∂sP ∂if∂rg
−10∂ ij kℓ mn pq rs tvr∂mP ∂jP ∂v∂p∂kP ∂ℓP ∂nP ∂s∂qP ∂if∂tg
+10∂ ∂ P ijt p ∂v∂r∂jPkℓ∂ mn pqq∂kP ∂ℓP ∂nPrs∂ P tvs ∂if∂mg
−10∂ ijt∂mP ∂v∂ ∂ Pkℓs j ∂kPmn∂ pq rsℓP ∂p∂nP ∂qP tv∂if∂rg
+10∂ ∂ P ij∂ ∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ Prs∂ P tvp m t s j k ℓ v n q ∂if∂rg
−10∂ ∂ P ijt r ∂v∂s∂jPkℓ∂ ∂ Pmn∂ Ppq∂ Prs∂ P tvp k ℓ n q ∂if∂mg
+10∂ ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ ∂ ∂ Ppq∂ rs tvr p j t k v n ℓ qP ∂sP ∂if∂mg
+10∂ ∂ P ijr p ∂t∂s∂ Pkℓj ∂v∂ Pmn∂ Ppq∂ Prs∂ P tvk ℓ n q ∂if∂mg
+10∂ ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ ∂ ∂ Ppq∂ Prst p j r k v n ℓ q ∂ P tvs ∂if∂mg
−10∂ ∂ ij kℓ mn pq rs tvt nP ∂jP ∂v∂r∂p∂kP ∂ℓP ∂qP ∂sP ∂if∂mg
−10∂ ∂ ∂ ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ Prs∂ P tvt r p m v j ℓ n q s ∂if∂kg
−10∂t∂ P ijm ∂v∂r∂p∂jPkℓ∂ℓPmn∂ pq rs tvnP ∂qP ∂sP ∂if∂kg
−10∂t∂r∂p∂ P ijn ∂jPkℓ∂v∂kPmn∂ pqℓP ∂ Prsq ∂sP tv∂if∂mg
−10∂t∂ P ij∂ ∂ ∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ Prs∂ P tvp v r q j ℓ m n s ∂if∂kg
+10∂t∂s∂ ∂ P ij∂ Pkℓ∂ Pmn∂ Ppqp m j k ℓ ∂ ∂ rs tvv nP ∂qP ∂if∂rg
+2∂ ∂ ∂ ∂ ∂ P ij∂ Pkℓ∂ Pmn∂ Ppq∂ Prs tvt r p m k v ℓ n q ∂sP ∂if∂jg
−5∂ ∂ ∂ P ij∂ ∂ Pkℓp m k t r ∂v∂ Pmnℓ ∂nPpq∂ rsqP ∂ P tvs ∂if∂jg
+5∂t∂m∂kP ij∂r∂pPkℓ∂v∂ℓPmn∂ Ppq∂ Prs∂ P tvn q s ∂if∂jg
−5∂p∂m∂ P ij∂ Pkℓk r ∂t∂ mn pqℓP ∂v∂nP ∂qPrs∂sP tv∂if∂jg
−5∂p∂ ∂ P ij∂ kℓ mn pq rs tvm k tP ∂r∂ℓP ∂v∂nP ∂qP ∂sP ∂if∂jg
+5∂ ∂ ij kℓt pP ∂v∂jP ∂ℓPmn∂ ∂ Ppq∂ Prsr m n ∂ ∂ tvs qP ∂if∂kg
+5∂ ∂ P ij∂ Pkℓv r j ∂ mnkP ∂ ∂ pq rs tvm ℓP ∂p∂nP ∂s∂qP ∂if∂tg
i
+5∂ ijr∂mP ∂t∂ kℓ mn pq rs tvjP ∂s∂ℓP ∂nP ∂v∂pP ∂qP ∂if∂kg
−5∂ ij kℓ mn pq rs tvr∂nP ∂t∂jP ∂v∂kP ∂ℓP ∂pP ∂s∂qP ∂if∂mg
+5∂ ∂ P ijp m ∂r∂ Pkℓ∂ ∂ Pmn∂ ∂ pq rs tvj t ℓ v nP ∂qP ∂sP ∂if∂kg
−5∂ ∂ P ij∂ kℓ mn pq rs tvr n t∂jP ∂p∂kP ∂v∂ℓP ∂qP ∂sP ∂if∂mg
+5∂p∂
ij
mP ∂ ∂ kℓ mnt jP ∂r∂ℓP ∂ ∂ Ppq∂ Prs∂ P tvv n q s ∂if∂kg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq rs tvt n r j p k v ℓ ∂qP ∂sP ∂if∂mg
+5∂ ∂ ij kℓ mn pq rsr nP ∂s∂jP ∂t∂kP ∂ℓP ∂v∂pP ∂ P tvq ∂if∂mg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppqr m t j v ℓ n ∂pPrs∂ tvs∂qP ∂if∂kg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmnp n t j r k ∂v∂ Ppqℓ ∂ rs tvqP ∂sP ∂if∂mg
−5∂ ∂ P ijr m ∂p∂ Pkℓj ∂t∂ℓPmn∂ pq rs tvv∂nP ∂qP ∂sP ∂if∂kg
+5∂p∂nP ij∂ ∂ Pkℓr j ∂t∂ mn pqkP ∂v∂ℓP ∂qPrs∂ P tvs ∂if∂mg
−5∂ ∂ P ij∂ ∂ Pkℓt m p j ∂r∂ Pmnℓ ∂v∂ Ppqn ∂qPrs∂ tvsP ∂if∂kg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmns m t j v k ∂ℓPpq∂p∂ rs tvnP ∂qP ∂if∂rg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmnr p q j t ℓ ∂ pq rs tvv∂mP ∂nP ∂sP ∂if∂kg
+5∂ ∂ P ijs m ∂ kℓt∂jP ∂p∂ Pmn∂ pqk ℓP ∂v∂ Prs∂ tvn qP ∂if∂rg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ P tvt p q j v ℓ r m n s ∂if∂kg
−5∂ ∂ P ij∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppq∂ ∂ Prs∂ P tvr n j t s k ℓ v p q ∂if∂mg
−5∂t∂ ∂ ijr mP ∂s∂jPkℓ∂ℓPmn∂nPpq∂v∂ rs tvpP ∂qP ∂if∂kg
−5∂ ∂ P ij∂ ∂ ∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ Prst p v q j ℓ r m n ∂ P tvs ∂if∂kg
+5∂ ∂ ∂ ij kℓ mn pq rs tvt s mP ∂jP ∂p∂kP ∂ℓP ∂v∂nP ∂qP ∂if∂rg
+5∂ ∂ ijr mP ∂t∂s∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prs tvj v ℓ n p ∂qP ∂if∂kg
−5∂ ∂ ∂ P ij∂ kℓ mn pq rs tvt r n s∂jP ∂v∂kP ∂ℓP ∂pP ∂qP ∂if∂mg
−5∂ ij kℓ mnt∂mP ∂v∂p∂jP ∂r∂ℓP ∂ Ppqn ∂ Prsq ∂ P tvs ∂if∂kg
−5∂ ∂ ∂ ij kℓ mn pq rs tvt p nP ∂r∂jP ∂v∂kP ∂ℓP ∂qP ∂sP ∂if∂mg
−5∂ ∂ P ijr m ∂t∂s∂ Pkℓ∂ mnj ℓP ∂nPpq∂ rsv∂pP ∂qP tv∂if∂kg
−5∂t∂r∂nP ij∂ Pkℓ∂ mn pq rs tvj s∂kP ∂ℓP ∂v∂pP ∂qP ∂if∂mg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ ∂ Pmnr n t j v s k ∂ Ppqℓ ∂pPrs∂ tvqP ∂if∂mg
+5∂ ∂ ∂ P ijt r m ∂s∂jPkℓ∂ mn pq rs tvv∂ℓP ∂nP ∂pP ∂qP ∂if∂kg
−5∂ ∂ P ijt n ∂r∂jPkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prsv p k ℓ q ∂ P tvs ∂if∂mg
−5∂ ∂ ∂ P ij∂ ∂ kℓ mn pq rs tvt p m v jP ∂r∂ℓP ∂nP ∂qP ∂sP ∂if∂kg
−5∂ ∂ ∂ P ij∂ Pkℓr m k t ∂v∂ Pmnℓ ∂nPpq∂ Prsp ∂s∂qP tv∂if∂jg
+5∂ ∂ ∂ P ij∂ Pkℓr m k p ∂t∂ Pmnℓ ∂v∂ Ppqn ∂ Prsq ∂ P tvs ∂if∂jg
+5∂ ∂ ∂ ij kℓ mn pq rs tvt m kP ∂pP ∂r∂ℓP ∂v∂nP ∂qP ∂sP ∂if∂jg
+5∂r∂
ij
m∂kP ∂t∂ Pkℓ∂ mnp ℓP ∂v∂ Ppq∂ rsn qP ∂sP tv∂if∂jg
+5∂ ∂ ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ ∂ Ppqt m k r p ℓ v n ∂qPrs∂ P tvs ∂if∂jg
−5∂r∂nP ij∂ kℓ mnjP ∂t∂p∂kP ∂ ∂ Ppq∂ rs tvv ℓ qP ∂sP ∂if∂mg
ii
+5∂ ∂ ∂ P ij∂ ∂ Pkℓ∂ Pmnt p m r j ℓ ∂ pq rs tvv∂nP ∂qP ∂sP ∂if∂kg
−5∂ ij kℓ mn pq rs tvt∂nP ∂jP ∂r∂p∂kP ∂v∂ℓP ∂qP ∂sP ∂if∂mg
+5∂r∂p∂
ij
mP ∂ ∂ Pkℓt j ∂ mnℓP ∂ ∂ Ppqv n ∂ rsqP ∂ P tvs ∂if∂kg
−5∂ ∂ P ij∂ ∂ ∂ Pkℓ∂ Pmn∂ Ppqt p v r j ℓ m ∂ ∂ rs tvq nP ∂sP ∂if∂kg
−5∂ ∂ ∂ P ij∂ Pkℓ mnv r m j ∂p∂kP ∂ Ppqℓ ∂nPrs∂ tvs∂qP ∂if∂tg
−5∂ ∂ P ij∂ ∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq rs tvr p t q j v ℓ m ∂nP ∂sP ∂if∂kg
−5∂ ∂ ij kℓ mn pq rst s∂mP ∂v∂jP ∂kP ∂ℓP ∂p∂nP ∂ P tvq ∂if∂rg
−5∂t∂pP ij∂r∂q∂jPkℓ∂ ∂ Pmn∂ Ppq∂ Prsv ℓ m n ∂ P tvs ∂if∂kg
+5∂ ∂ ∂ P ij∂ ∂ Pkℓ∂ Pmns p m t j k ∂ℓPpq∂v∂ rs tvnP ∂qP ∂if∂rg
−5∂ ∂ P ij∂ ∂ ∂ Pkℓ∂ Pmn pqr m t p j ℓ ∂v∂nP ∂qPrs∂ tvsP ∂if∂kg
+5∂ ∂ ij kℓ mn pq rs tvt p∂nP ∂jP ∂r∂kP ∂v∂ℓP ∂qP ∂sP ∂if∂mg
−5∂t∂ ijmP ∂ ∂ ∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ Prs tvr p j ℓ v n q ∂sP ∂if∂kg
+5∂ ∂ ∂ P ij∂ Pkℓr p n j ∂t∂ mn pq rskP ∂v∂ℓP ∂qP ∂sP tv∂if∂mg
−5∂ ∂ P ij∂ Pkℓ∂ Pmn∂ ∂ Ppqt s j k m ℓ ∂ ∂ rs tvv p∂nP ∂qP ∂if∂rg
+5∂ ∂ ∂ P ij∂ ∂ Pkℓ∂ Pmnt r p v j ℓ ∂ pq rs tvmP ∂q∂nP ∂sP ∂if∂kg
−5∂ ∂ ∂ ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prs∂ P tvr p m k t v ℓ n q s ∂if∂jg
−5∂t∂p∂m∂ P ij∂ Pkℓk r ∂ mnℓP ∂ ∂ Ppq∂ rs tvv n qP ∂sP ∂if∂jg
−5∂r∂p∂m∂kP ij∂ Pkℓt ∂ℓPmn∂ ∂ Ppq∂ Prs∂ P tvv n q s ∂if∂jg
+5∂ ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ ∂ Prss m j t k ℓ v p n ∂ P tvq ∂if∂rg
−5∂ ∂ ∂ ij kℓ mn pq rs tvt r pP ∂q∂jP ∂ℓP ∂v∂mP ∂nP ∂sP ∂if∂kg
−5∂ ∂ ijt nP ∂v∂jPkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prs tvr p k ℓ q ∂sP ∂if∂mg
+5∂ ∂ ∂ P ij∂ kℓ mn pq rs tvr p m t∂jP ∂v∂ℓP ∂nP ∂qP ∂sP ∂if∂kg
−5∂ ij kℓ mnp∂nP ∂t∂jP ∂v∂r∂kP ∂ Ppqℓ ∂ Prsq ∂ P tvs ∂if∂mg
−5∂ ∂ ∂ ij kℓ mn pq rs tvt r mP ∂p∂jP ∂v∂ℓP ∂nP ∂qP ∂sP ∂if∂kg
−5∂ ∂ P ijt p ∂r∂q∂ Pkℓ∂ mnj ℓP ∂v∂ Ppq∂ rsm nP ∂sP tv∂if∂kg
+5∂s∂p∂mP ij∂ Pkℓ∂ mn pq rs tvj t∂kP ∂ℓP ∂v∂nP ∂qP ∂if∂rg
−5∂t∂pP ij∂ kℓ mnv∂r∂jP ∂ℓP ∂ Ppq∂ Prsm n ∂ tvs∂qP ∂if∂kg
−5∂ ∂ ∂ P ij∂ Pkℓ∂ mn pq rs tvv r m j kP ∂ℓP ∂p∂nP ∂s∂qP ∂if∂tg
−5∂ ∂ P ijt m ∂r∂p∂ kℓ mn pqjP ∂v∂ℓP ∂nP ∂qPrs∂ P tvs ∂if∂kg
−5∂ ∂ ∂ P ij∂ ∂ Pkℓr p n t j ∂v∂kPmn∂ pq rs tvℓP ∂qP ∂sP ∂if∂mg
+5∂ ∂ P ijp m ∂t∂ ∂ kℓr jP ∂v∂ℓPmn∂ pqnP ∂ Prs∂ tvq sP ∂if∂kg
−5∂t∂r∂nP ij∂v∂jPkℓ∂p∂kPmn∂ pq rs tvℓP ∂qP ∂sP ∂if∂mg
−5∂ ∂ P ij∂ Pkℓt p j ∂v∂ Pmnk ∂r∂ Ppqℓ ∂nPrs∂s∂qP tv∂if∂mg
−5∂t∂ P ij∂ kℓm jP ∂p∂ Pmnk ∂v∂ℓPpq∂ rs tvq∂nP ∂sP ∂if∂rg
+5∂ ∂ P ij∂ Pkℓ∂ ∂ Pmnr p j t k ∂s∂ Ppqℓ ∂v∂ Prs∂ P tvn q ∂if∂mg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmnt r v j p k ∂ pq rs tvs∂ℓP ∂nP ∂qP ∂if∂mg
iii
+5∂ ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ rs tvr p j t k v ℓ q∂nP ∂sP ∂if∂mg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmnr p t j v k ∂ℓPpq∂nPrs∂s∂ P tvq ∂if∂mg
−5∂ ij kℓ mn pq rs tvt∂mP ∂v∂jP ∂kP ∂ℓP ∂p∂nP ∂s∂qP ∂if∂rg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs tvt r s j v k n ℓ p ∂qP ∂if∂mg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ Prs tvt m v j p k ℓ q n ∂sP ∂if∂rg
+5∂ ∂ P ij∂ Pkℓt r j ∂s∂kPmn∂ pq rsn∂ℓP ∂v∂pP ∂ P tvq ∂if∂mg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ ∂ Prs∂ P tvp m t j k s ℓ v n q ∂if∂rg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmnt p r j v k ∂ Ppqℓ ∂q∂ Prs∂ P tvn s ∂if∂mg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ Prs∂ P tvp m t j v k ℓ q n s ∂if∂rg
−5∂ ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ ∂ Prs tvt m j p k s ℓ v n ∂qP ∂if∂rg
+5∂ ∂ P ij∂ kℓ mn pq rs tvr p t∂jP ∂v∂kP ∂s∂ℓP ∂nP ∂qP ∂if∂mg
+5∂ ∂ P ij∂ Pkℓ∂ ∂ Pmnt r j p k ∂s∂ℓPpq∂v∂ rs tvnP ∂qP ∂if∂mg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ rs tvt r v j p k ℓ nP ∂s∂qP ∂if∂mg
+5∂t∂rP ij∂jPkℓ∂p∂ Pmnk ∂ pq rsv∂ℓP ∂q∂nP ∂sP tv∂if∂mg
−5∂ ∂ P ij∂ ∂ kℓ mn pq rs tvt p r jP ∂v∂kP ∂s∂ℓP ∂nP ∂qP ∂if∂mg
−5∂ ∂ P ij∂ Pkℓt m j ∂s∂kPmn∂ ∂ Ppq∂ ∂ Prs∂ P tvn ℓ v p q ∂if∂rg
−5∂r∂ ijmP ∂v∂jPkℓ∂p∂kPmn∂ pq rs tvℓP ∂nP ∂s∂qP ∂if∂tg
−5∂ ∂ P ijt m ∂ kℓs∂jP ∂kPmn∂ ∂ pq rs tvn ℓP ∂v∂pP ∂qP ∂if∂rg
+5∂ ijt∂mP ∂ Pkℓ∂ mnj v∂kP ∂ Ppqℓ ∂ rs tvp∂nP ∂s∂qP ∂if∂rg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ P tvt p v j q k r ℓ n s ∂if∂mg
−5∂r∂ ijmP ∂ kℓ mn pq rsjP ∂v∂kP ∂ℓP ∂p∂nP ∂s∂qP tv∂if∂tg
+5∂t∂
ij
pP ∂ ∂ Pkℓr j ∂q∂ mn pq rs tvkP ∂v∂ℓP ∂nP ∂sP ∂if∂mg
+5∂p∂
ij
mP ∂ ∂ Pkℓs j ∂t∂ Pmn∂ Ppqk ℓ ∂v∂ rs tvnP ∂qP ∂if∂rg
−5∂ ij kℓ mn pq rs tvt∂mP ∂s∂jP ∂v∂kP ∂ℓP ∂p∂nP ∂qP ∂if∂rg
−5∂ ∂ P ij∂ ∂ Pkℓr p s j ∂ mn pqt∂kP ∂ℓP ∂v∂nPrs∂ P tvq ∂if∂mg
+5∂ ijt∂pP ∂v∂ Pkℓj ∂r∂kPmn∂ pq rsn∂ℓP ∂qP ∂sP tv∂if∂mg
−5∂r∂ P ij∂ kℓp t∂jP ∂v∂kPmn∂n∂ℓPpq∂ Prsq ∂sP tv∂if∂mg
−5∂t∂rP ij∂ kℓ mns∂jP ∂p∂kP ∂ Ppqℓ ∂v∂ Prsn ∂qP tv∂if∂mg
−5∂r∂ P ijp ∂ Pkℓj ∂ mn pq rs tvt∂q∂kP ∂v∂ℓP ∂nP ∂sP ∂if∂mg
−5∂ ∂ ∂ P ij∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prst r p j v k ℓ n ∂s∂qP tv∂if∂mg
+5∂t∂
ij kℓ mn pq
pP ∂jP ∂v∂q∂kP ∂r∂ℓP ∂nPrs∂sP tv∂if∂mg
−5∂t∂ ∂ P ij∂ kℓ mn pq rs tvp m jP ∂v∂kP ∂ℓP ∂q∂nP ∂sP ∂if∂rg
+5∂t∂ P ij∂ kℓp jP ∂v∂ ∂ Pmn∂ ∂ Ppqr k n ℓ ∂qPrs∂ tvsP ∂if∂mg
+5∂ ijt∂r∂pP ∂ ∂ Pkℓ∂ mn pq rs tvs j v∂kP ∂ℓP ∂nP ∂qP ∂if∂mg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ ∂ Pmnr p t j v q k ∂ Ppqℓ ∂nPrs∂ tvsP ∂if∂mg
iv
−5∂ ∂ ∂ P ij∂ Pkℓt r p j ∂v∂kPmn∂ pqℓP ∂ ∂ rs tvq nP ∂sP ∂if∂mg
+5∂t∂pP ij∂r∂ Pkℓj ∂v∂q∂kPmn∂ℓPpq∂nPrs∂sP tv∂if∂mg
+5∂ ∂ ∂ P ij∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ ∂ Prs tvt p m j k s ℓ v n ∂qP ∂if∂rg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ ∂ Prs∂ P tvt m s j k ℓ v p n q ∂if∂rg
+5∂t∂ ∂ P ijp m ∂s∂jPkℓ∂kPmn∂ℓPpq∂v∂nPrs∂ tvqP ∂if∂rg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppqt r s j v p k ℓ ∂ Prs∂ tvn qP ∂if∂mg
−5∂t∂r∂pP ij∂ Pkℓj ∂v∂ Pmnk ∂n∂ pq rs tvℓP ∂qP ∂sP ∂if∂mg
+5∂ ∂ P ij∂ Pkℓt r j ∂v∂p∂ Pmn∂ pqk ℓP ∂nPrs∂ tvs∂qP ∂if∂mg
+5∂t∂r∂pP ij∂ Pkℓj ∂q∂kPmn∂ pqv∂ℓP ∂nPrs∂ tvsP ∂if∂mg
+5∂ ij kℓ mn pq rs tvt∂rP ∂jP ∂v∂p∂kP ∂ℓP ∂q∂nP ∂sP ∂if∂mg
+5∂ ijt∂r∂pP ∂ ∂ Pkℓ∂ ∂ Pmnv j q k ∂ℓPpq∂ rs tvnP ∂sP ∂if∂mg
+5∂v∂ P ijm ∂ Pkℓ∂ ∂ Pmn∂ pq rs tvj p k r∂ℓP ∂nP ∂s∂qP ∂if∂tg
+5∂t∂ P ij∂ ∂ Pkℓ∂ Pmnp r j ℓ ∂v∂mPpq∂q∂nPrs∂ tvsP ∂if∂kg
−5∂s∂mP ij∂jPkℓ∂t∂ Pmnk ∂ ∂ Ppqn ℓ ∂v∂pPrs∂ P tvq ∂if∂rg
+5∂ ∂ P ijt p ∂ kℓr∂jP ∂ℓPmn∂ ∂ Ppq∂ ∂ Prs∂ P tvs m v n q ∂if∂kg
−5∂ ∂ P ij∂ ∂ Pkℓ∂ Pmns m t j k ∂n∂ℓPpq∂v∂ Prsp ∂ P tvq ∂if∂rg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ mnt p r j v∂ℓP ∂ ∂ Ppqs m ∂ rsnP ∂ P tvq ∂if∂kg
+5∂ ∂ P ijv m ∂r∂jPkℓ∂ mnkP ∂ pq rs tvℓP ∂p∂nP ∂s∂qP ∂if∂tg
+5∂t∂pP ij∂ ∂ kℓ mn pq rs tvr jP ∂v∂ℓP ∂mP ∂nP ∂s∂qP ∂if∂kg
+5∂ ∂ P ij∂ ∂ Pkℓ∂ Pmn∂ ∂ Ppqt s v j k m ℓ ∂p∂ Prsn ∂ P tvq ∂if∂rg
+5∂ ∂ P ij∂ ∂ kℓ mn pq rs tvr p t jP ∂v∂ℓP ∂mP ∂q∂nP ∂sP ∂if∂kg
−5∂t∂ ∂ P ijp n ∂jPkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prsv r k ℓ q ∂ P tvs ∂if∂mg
−5∂ ∂ ∂ P ij∂ ∂ kℓ mn pq rs tvt r m v p∂jP ∂ℓP ∂nP ∂qP ∂sP ∂if∂kg
+5∂ ij kℓ mn pq rs tvt∂s∂mP ∂jP ∂kP ∂ℓP ∂v∂p∂nP ∂qP ∂if∂rg
−5∂t∂r∂ ijpP ∂v∂q∂jPkℓ∂ℓPmn∂mPpq∂nPrs∂ P tvs ∂if∂kg
−5∂ ∂ ∂ P ij∂ Pkℓt r n j ∂ mn pqv∂p∂kP ∂ℓP ∂ Prs∂ tvq sP ∂if∂mg
−5∂t∂p∂ P ij∂ ∂ ∂ Pkℓ∂ Pmn∂ Ppqm v r j ℓ n ∂ Prs∂ tvq sP ∂if∂kg.
In every term, the Einstein summation convention works for each repeated index (i.e. once
upper and another time lower), the indices running from 1 to the dimension r < ∞ of the
affine Poisson manifold M r at hand.
v

Chapter 17
The orientation morphism: from
graph cocycles to deformations of
Poisson structures
This chapter is based on the peer-reviewed conference proceedings R. Buring and A. V.
Kiselev, J. Phys.: Conf. Ser. 1194, Paper 012017, 2019. (Preprint arXiv:1811.07878
[math.CO] – 10 p.) That paper follows the talk given by the dissertant at the 32nd
International colloquium on Group-theoretical methods in Physics: Group32 (9–13 July
2018, CVUT Prague, Czech Republic).
Commentary. In reference to Part I of the dissertation, the material of this chapter is
used in Chapter 4, Chapter 5, and Chapter 6 (the Nijenhuis–Richardson bracket shows
up in §6.3). The explanations in this chapter build on a paper by Jost (2013), which
itself follows an outline in a paper by Willwacher (2010-15), which in turn comments on
the seminal paper by Kontsevich (1996).
401
THE ORIENTATION MORPHISM: FROM GRAPH COCYCLES TO
DEFORMATIONS OF POISSON STRUCTURES
R. BURING‡ AND A.V.KISELEV§
Abstract. We recall the construction of the Kontsevich graph orientation morphism
γ 7→ O⃗r(γ) which maps cocycles γ in the non-oriented graph complex to infinitesi-
mal symmetries Ṗ = O⃗r(γ)(P) of Poisson bi-vectors on affine manifolds. We reveal
in particular why there always exists a factorization of the Poisson cocycle condi-
tion [[P ., O⃗r(γ)(P)]] = 0 through the differential consequences of the Jacobi identity
[[P,P]] = 0 for Poisson bi-vectors P. To illustrate the reasoning, we use the Kontsevich
tetrahedral flow Ṗ = O⃗r(γ3)(P), as well as the flow produced from the Kontsevich–
Willwacher pentagon-wheel cocycle γ5 and the new flow obtained from the heptagon-
wheel cocycle γ7 in the unoriented graph complex.
Introduction. On an affine manifold M r, the Poisson bi-vector fields are those satis-
fying the Jacobi identity [[P ,P ]] = 0, where [[·, ·]] is the Schouten bracket ([12], see also
Example 1 below). A deformation P 7→ P+εQ+ ō(ε) of a Poisson bi-vector P preserves
the Jacobi identity infinitesimally if [[P ,Q]] = 0. If, by assumption, the deformation
term Q (itself not necessarily Poisson) depends on the bi-vector P , then the equation
.
[[P ,Q(P)]] = 0 must be satisfied by force of [[P ,P ]] = 0. In [10] Kontsevich designed a
way to produce infinitesimal deformations Ṗ = Q(P) which are universal with respect
to all Poisson structures on all affine manifolds: for a given bi-vector P , the coefficients
of bi-vector Q(P) are differential polynomial in the coefficients of P .
The original co(nstru∧ction fro)m [10] goes in three steps, as follows. First, recall that
the vector space i edgeGra i#Vert=:n⩾1 of unoriented finite graphs with unlabelled verticesSn
and wedge ordering on the set of edges carries the structure of a complex with respect
to th∑e vertex-expanding differential d. In fact, this space is a differential graded Liealgebra such that the differential d is the Lie bracket with a single edge, d = [•−•, ·]. Let
γ = ii c γi be a sum of graphs with n vertices and 2n − 2 edges, satisfying d(γ) = 0.
Then let us sum –with signs, which will be discussed in §1.2 below – over all possible
ways to orient the graphs γi in the cocycle γ such that each vertex is the arrowtail
for two outgoing edges; create two extra edges going to two new vertices, the sinks.
Secondly, skew-symmetrize (w.r.t. the sinks) the resulting sum of Kontsevich oriented
graphs. Finally, insert a Poisson bi-vector P into each vertex of every γi in the sum of
Kontsevich graphs at hand. Now, every oriented graph built of the decorated wedges
←−i−•−−j−→ determines a differential-polynomial expression in the coefficients P ij(x1,
Left Right
Date: 2 December 2018.
2010 Mathematics Subject Classification. 05C22, 68R10, 16E45, 53D17, 81R60.
‡Address: Institut für Mathematik, Johannes Gutenberg–Universität, Staudingerweg 9, D-55128
Mainz, Germany. E-mail: rburing@uni-mainz.de.
§Address: Bernoulli Institute for Mathematics, Computer Science & Artificial Intelligence, Univer-
sity of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands. E-mail: A.V.Kiselev@rug.nl.
402
THE ORIENTATION MORPHISM 403
. . ., xr) of a bivector P whenever the arrows −→a denote derivatives ∂/∂xa in a local
coordinate chart, each vertex • at the top of a wedge contains a copy of P , and one
takes the product of vertex contents and sums up over all the indexes. The right-hand
side of the symmetry flow Ṗ = Q(P) is obtained!
We give an explicit, relatively elementary proof that this recipe does the job, i.e. why
the Poisson cocycle condition .[[P ,Q(P)]] = 0 is satisfied for every Poisson structure
P , and for every Q = O⃗r(γ) obtained from a graph cocycle γ ∈ ker d in this way.
The reasoning is based on that given by Jost [9], which in turn follows an outline by
Willwacher [15], itself referring to the seminal paper [10] by Kontsevich.
At the same time, the present text conclud(es a series)of papers [1, 2, 5] with an empiric
search for the factorizations [[P ,Q(P)]] = ♢ P , [[P ,P ]] using the Jacobiator [[P ,P ]], as
well as containing an independent verification of the numerous rules of signs for many
graded objects under study — the ultimate aim being to understand the morphism O⃗r.
Section 1.2 establishes the formula1 of Poisson cocycle factorization through the Ja-
cobiator [[P ,P ]]:
2 · [[P , O⃗r(γ)(P , . . . ,P)]] = O⃗r(γ)([[P ,P ]],P , . . . ,P) + . . .+
+ O⃗r(γ)(P , . . . ,P , [[P ,P ]],P , . . . ,P) + . . .+ O⃗r(γ)(P , . . . ,P , [[P ,P ]]), (1)
where the r.-h.s. consists of oriented graphs with one copy of the tri-vector [[P ,P ]]
inserted consecutively into a vertex of the graph(s) γ.
We illustrate the work of orientation morphism O⃗r which maps ker d 3 γ 7→ Q(P) ∈
ker[[P , ·]] by using four examples, which include in particular the first elements γ3, γ5,
γ7 ∈ ker d of nontrivial graph cocycles found by Willwacher in [15]: the Kontsevich
tetrahedral flow Ṗ = O⃗r(γ3)(P) (see [10] and [1, 2]), the Kontsevich–Willwacher pen-
tagon wheel cocycle γ5 and the respective flow Ṗ = O⃗r(γ5)(P) (here, see [6] and [15]),
and similarly, the heptagon-wheel cocycle γ7 and its flow. In each case, the reason-
ing reveals a factorization [[P , O⃗r(γ)(P)]] = ♢(P , [[P ,P ]]) through the Jacobi identity
[[P ,P ]] = 0. For the tetrahedral flow Ṗ = O⃗r(γ3)(P) we thus recover the factoriza-
tion of [[P , Ṗ ]] – in terms of the “Leibniz” graphs with the tri-vector [[P ,P ]] inside –
which had been obtained in [2] by a brute force calculation. Let it be noted that such
factorizations, [[P , Ṗ ]] = ♢(P , [[P ,P ]]), are known to be non-unique for a given flow Ṗ ;
the scheme which we presently consider provides one such operator ♢ (out of many,
possibly).
Trivial graph cocycles, i.e. d-coboundaries γ = d(β) also serve as an illustration.
Under the orientation mapping O⃗r their “potentials” β (sums of graphs with n − 1
vertices and 2n − 3 edges) are transformed into the vector fields X, also codified by
the Kontsevich oriented graphs, which trivialize the respective flows Ṗ = O⃗r(γ)(P) in
the space of bi-vectors: namely, O⃗r(d(β))(P) = [[P , O⃗r(β)(P)]] so that the resulting
1The existence of this formula with some vanishing right-hand side is implied in [10, 15, 8] where
it is stated that there is an action of the graph complex on Poisson structures (or Maurer–Cartan
elements of Tpoly(M)). The precise right-hand side is all but written in [9]; still to the best of our
knowledge, the exact formula is presented here and on p. 408 below for the first time. — The same
applies to Jacobi identity (2) for the Lie bracket of graphs (cf. [14]).
404 R. BURING AND A. V. KISELEV
flow Ṗ = Q(P) = [[P ,X(P)]] is trivial in Poisson cohomology. We offer an example on
p. 409: here, X(P) = 2O⃗r(β6)(∑P).This paper continues in §1.3 with some statistics about the number of graphs (i)
in the “known” cocycles γ = ciγi ∈ ker d, (ii) in the respective flows Q = O⃗r(γ)
which consist of the oriented Kontsevich graphs, (iii) in the factorizing operators ♢
(provided by the proof) which are encoded by the Leibniz graphs (see [3, 2]), and (iv)
in the cocycle equations .[[P , O⃗r(γ)(P)]] = 0. We see that for thousands and millions of
oriented graphs in the left- and right-hand sides of (1) the coefficients match perfectly.
1. The parallel worlds of graphs and endomorphisms
The universal deformations Ṗ = Q(P) which we consider will be given by certain
endomorphisms evaluated at copies of a given Poisson structure P . In particular, the
resulting expressions will be differential polynomials in the coefficients of P . Moreover,
such expressions will be built using graphs, so that properties of objects in the graph
complex are translated into properties of the objects realized by the graphs in the
Poisson complex. To this end, let us recall and compare the notions of operads of non-
oriented graphs and of endomorphisms of multi-vector fields on affine manifolds. This
material is standard; we follow [10, 9, 15, 13].
1.1. Endomorphisms End(Tpoly(M)[1]) (e.g., the Schouten bracket [[·, ·]]). Denote
the shifted-graded vector space o⊕f all multi-vector fields on the manifold M r by2
Tpoly(M)[1] = T
ℓ
poly(M) where ℓ = ℓ̄+ 1.
ℓ̄⩾−1
The grading in ↓[1]Tpoly(M)[1] = Tpoly(M) is shifted down so that, by definition, a bi-
vector P has degree |P| = 2 but P̄ = 1, etc. We let the multi-vectors be encoded
in a standard way using a local coordinate chart x1, . . ., xr on M r and the respective
parity-odd variables ξ1, . . ., ξ along the reverse-parity fibre∑s of ΠT ∗M rr over that chart.
For example, a bi-vector is written in coordinates as P = ij1⩽i<j⩽r P (x)ξ ξ .3i j
An endomorphism of Tpoly(M)[1] of arity k and degree d̄ is a k-linear (over the field R)
map θ : Tpoly(M)[1]⊗ . . .⊗ Tpoly(M)[1]→ Tpoly(M)[1], not necessarily (graded-)skew in
its k arguments, and such that for grading-homogeneous arguments we have that
θ : T d̄1poly(M)⊗ . . .⊗ T
d̄k (M)→ T d̄1+...+d̄k+d̄poly poly (M),
i.e. θ restricts to a map of degree d̄.
Example 1. The Schouten bracket [[·, ·]] :∣∣T d̄1poly∣∣(M)⊗ T d̄2 (M)→ T d̄1+d̄2poly poly (M) has arity 2and shifted degree deg ([[·, ·]]) = 0 (note [[·, ·]] = −1). It is expressed in coordinates by
the formula ∑r ⃗
P Q ∂ ∂⃗ ∂
⃗ ∂⃗
[[ , ]] = (P) · (Q)− (P) · (Q).
∂ξ ℓℓ ∂x ∂xℓ ∂ξℓ
ℓ=1
2This notation for the space of multi-vectors should not be confused with a similar notation for the
space of vector fields with polynomial coefficients on an affine manifold Mr. Nor should it be read as
the space of multi-vectors on a super-manifold.
3Our notation is such that the wedge product of multi-vectors does not include any constant factor.
THE ORIENTATION MORPHISM 405
Notation. The bi-graded( vector spac)e of⊕endomorp(hisms under)study is denoted by
End Tpoly(M)[1] = End
k,d̄ Tpoly(M)[1] .
d̄∈Z
k⩾1
This space has the structure of an operad (with an action by the permutation group Sk
on the part of arity k): indeed, endomorphisms can be inserted one into another.
Let θa and θb be two endomorphisms of respective arities ka and kb. The inser-
tion of θa into the ith argument of θb is denoted by θa ◦⃗i θb. For instance, (θa ◦⃗1
θb)(p1, . . . , pka+k −1) = θb(θa(p1, . . ., pka), pka+1, . . ., pka+k −1). Likewise, the notationb b
θa ◦i⃗θb means the insertion of the succeeding object θb into the preceding θa, whence
(θa ◦1⃗θb)(p) = θa(θb(p1, . . ., pk ), pk +1, . . ., pka+k −1). Without an arrow pointing left,b b b
this notation ◦i⃗is used in other papers; it is also natural because the graded objects θa
and θb are not swapped.
Definition 1. The insertion ◦⃗ of an endomo∑rphism θa into an endomorphism θb of
arity kb is the sum of insertions: θ kba ◦⃗ θb = i=1 θa ◦⃗i θb. The graded commutator
of endomorphisms of degrees da and db is [θa, θb] = θa ◦⃗ θb − (−)|θa|·|θb|θb ◦⃗ θa. An
endomorphism θ of arity k is skew with respect to permutations of its graded arguments
if it acquires the Koszul sign, θ(p1, . . . , pk) = ϵp(σ)θ(pσ(1), . . . , pσ(k)) under σ ∈ Sk. Here
ϵp((1 2)) = (−)(1 2)(−)p̄1·p̄2 and similarly for all other transpositions which generate the
permutation group Sk. Suppose that both of the endomorphisms θa and θb from the
above are graded skew-symmetric. The Nijenhuis–Richardson bracket [θa, θb]NR of those
skew endomorphisms (of degrees da and db respectively) is the skew-symmetrization of
[θa, θb] with respect to the permutations, graded by the Koszul signs.
Example 2. The shifted-graded skew-symmetric Schouten bracket
π (p |p1|−1S 1, p2) := (−) [[p1, p2]] ∈ End2(Tpoly(M)[1])
of multivectors a, b, c of respective homogeneities satisfies the shifted-graded Jacobi
identity
[[a, [[b, c]]]]− (−)āb̄[[b, [[a, c]]]] = [[[[a, b]], c]] = 0,
or equivalently,
[[a, [[b, c]]]] + (−)āb̄+āc̄[[b, [[c, a]]]] + (−)c̄ā+c̄b̄[[c, [[a, b]]]] = 0.
Taken four times, [πS, πS]NR evaluated (with Koszul signs shifted by deg[[·, ·]] = −1) at
a, b, c yields the l.-h.s. of the Jacobi identity for [[·, ·]]. This shows that [πS, πS]NR = 0.
Proposition 1. The Nijenhuis–Richardson bracket (of homogeneous arguments of re-
spective degrees) itself satisfies the graded Jacobi identity
[a, [b, c]NR]NR − (−)|a|·|b|[b, [a, c]NR]NR = [[a, b]NR, c]NR, (2)
or equivalently,
[a, [b, c]NR]NR + (−)|a|·|b|+|a|·|c|[b, [c, a] ] + (−)|c|·|a|+|c|·|b|NR NR [c, [a, b]NR]NR = 0.
Corollary 1. The map ∂ := [πS, ·]NR is a differential on the space of skew endomorph-
isms.
406 R. BURING AND A. V. KISELEV
1.2. Graphs vs endomorphisms. Having studied the natural differential graded
Lie algebra (dgLa) structure o⊕n th(e space of graded skew-symmetric endomorphismsEnd∗,∗skew(Tpoly(M)[1]), we observe that i∧ts constredge )uction goes in parallel with the dgLastructure on the vector space Gra i ik #Vert=:k⩾1 of finite non-oriented graphs withSk
wedge ordering of edges (and without leaves). Referring to [8, 9, 10, 15] (and references
therein), as well as to [6, 7, 14] with explicit examples of calculations in the graph
complex, we summarize the set of analogous objects and structures in Table 1 below.
Table 1. From graphs to endomorphisms: the respective objects or structures.
World of graphs World of endomorphisms
Graphs (γ,E(γ)) Endomorphisms
Insertion ◦⃗i of graph into ith vertex Insertion of endomorphism into ith argu-
ment
Insertion ◦⃗ of graph into graph Insertion ◦⃗
Bracket [a, b] = a ◦⃗ b− (−)|E(a)|·|E(b)|b ◦⃗ a Bracket [a, b] = a ◦⃗ b− (−)|a|·|b|b ◦⃗ a
Lie bracket ([a, b], E([a, b]) := E(a)∧E(b)) Nijenhuis-Richardson bracket [a, b]NR on
the space of skew endomorphisms
The stick •−• The Schouten bracket πS = ± [[·, ·]]
Master equation [•−•, •−•] = 0 Master equation [πS, πS]NR = 0
Graded Jacobi identity for [·, ·] Graded Jacobi identity for [·, ·]NR
Differential d = [•−•, ·] Differential ∂ = [πS, ·]NR
The orientation morphism O⃗r, which we presently discuss, provides a transition “=⇒”
f{ro⊕m g(raph∧s to endo)morphisms. Our}goal is to{have a Lie algebra morphism }
i edgeGra i
O⃗r
#Vert=:k⩾1 , d = [•−•, ·] −−−−→ End
∗,∗
skew(Tpoly(M)[1]), ∂ = [πS, ·]S NR ,k
k
hence a dgLa morphism because the differentials d = [•−•, ·] and ∂ = [πS, ·]NR are the
adjoint actions of the Maurer–Cartan elements.
In the meantime, we claim without proof that the edge •−• is taken to the Schouten
bracket πS = ± [[·, ·]] by O⃗r: namely, •−• 7→ πS = ± [[·, ·]] (see (3) below). So, having a
Lie algebra morphism implies that O⃗r([•−•, γ]) = [πS, O⃗r(γ)]NR for a graph γ with edge
ordering E(γ), i.e. the following diagram is commutative:
O⃗r
(γ,E(γ)) - O⃗r(γ)
d ∂
? ?
[•−• O⃗r, γ] - [πS, O⃗r(γ)]NR.
When this diagram is reached, it will be seen – by evaluating the endomorphisms at
copies of P – why the mapping of d-cocycles in the graph complex to Poisson cocycles ∈
ker [[P , ·]] is well defined. This will solve the problem of producing universal infinitesimal
symmetries Ṗ = O⃗r(γ)(P) of Poisson brackets P from d-cocycles γ ∈ ker d.
Let γ be an unoriented graph on k vertices and let p1, . . ., pk ∈ Tpoly(M) be a k-tuple
of multivectors. Not yet at the level of Lie algebras but at the level of two operads
THE ORIENTATION MORPHISM 407
with the respective graph- and endomorphism insertions ◦⃗, let the linear mapping o⃗r
be given by the formula [10] ( ∏ )
o⃗r(γ)(p1, . . . , pk)(x, ξ) := multk ∆⃗ij(p1 ⊗ . . .⊗ pk) (x, ξ),
(i,j)∈E(γ) ( ) ( )
where for each edge (i, j) = eij in∑the(graph γ, the operator∆)ij : eij 7→ −→
ℓ
i j + i←ℓ− j ,
r
∂⃗ ∂⃗ ∂⃗ ∂⃗
∆⃗ij = + ,
∂xℓ (i) (j) ℓ
ℓ=1 (j) ∂ξ ∂ξ ∂xℓ ℓ (i)
acts on the ith and jth factors in the ordered tensor product of arguments p1, . . . , pk. By
construction, the right-to-left ordering of the operators ∆⃗ij is inherited from the wedge
ordering of edges E(γ) in the graph γ: the operator corresponding to the firstmost edge
acts first.4 The operatorm∏ultk, acting at the end of the day, is the ordered multiplication
of the resulting terms in (i,j) ∆⃗ij(p1 ⊗ . . .⊗ pk).
It can be seen ([9, 15]) that the graph insertions ◦⃗i are mapped by o⃗r to the insertions
◦⃗i of endomorphisms: o⃗r(γ1 ◦⃗i γ2) = o⃗r(γ1) ◦⃗i o⃗r(γ2). Consequently, the sum of insertions
◦⃗ goes – under o⃗r – to the sum of insertions ◦⃗. The mapping o⃗r induces the linear map-
ping O⃗r taking graphs to the space of graded-skew endomorphisms Endskew(Tpoly(M)[1]).
We reach the important equality:
O⃗r(•−•) = πS, i.e. O⃗r(•−•)(p1, p2) = (−)p̄1 [[p1, p2]] for p1, p2 ∈ Tpoly(M)[1]. (3)
Recall also that both the domain and image of O⃗r, i.e. graphs with wedge ordering of
edges and their skew-symmetrized images in the space Endskew(Tpoly(M)[1]) carry the
respective Lie algebra structures. Th⊕e co(nclu∧sion is th)is:
Proposition 2. The mapping O⃗r : Gra i edgei ∗,∗k #Vert=:k⩾1 → EndS skew(Tpoly(M)[1]) is ak
Lie algebra morphism: O⃗r([γ, β]) = [O⃗r(γ), O⃗r(β)]NR.
Corollary 2. O⃗r(d(γ)) = O⃗r([•−•, γ]) = [O⃗r(•−•), O⃗r(γ)]NR = [πS, O⃗r(γ)]NR.
Let there be k vertices and 2k−2 edges in γ, whence k+1 vertices in d(γ). Evaluating
both sides of the endomorphism equality O⃗r(d(γ)) = [πS, O⃗r(γ)]NR at a tuple of Poisson
bi-vectors P , we have that O⃗r([•−•, γ])(P ⊗ . . .⊗ P) =
= (π ◦⃗ O⃗r(γ))(P ⊗ . . .⊗ P)− (−)(|πS |=−1)·(|O⃗r(γ)|=−|E(γ)|)S (O⃗r(γ) ◦⃗ πS)(P ⊗ . . .⊗ P)
= O⃗r(γ)(πS(P ,P),P , . . . ,P − ) + . . .+ O⃗r(γ)(P , . . . ,P , πk 1 k−1 S(P ,P))−
− πS(O⃗r(γ)(P , . . . ,P ),P)− πS(P , O⃗r(γ)(P , . . . ,P )). (4)k k
Theorem 1. Whenever P is a Poisson bi-vector so that πS(P ,P) = 0 = [[P ,P ]], and
whenever γ ∈ ker d is a cocycle on k vertices and 2k−2 edges (so that [•−•, γ] = 0), then
O⃗r(γ)(P ⊗ . . .⊗ P k) is a Poisson cocycle (so that [[P
.
, O⃗r(γ)(P , . . . ,P)]] = 0 modulo
the Jacobi identity [[P ,P ]] = 0 for the Poisson structure).
4By construction, own grading of the endomorphism o⃗r(γ) equals minus the number of edges in γ
(because each edge differentiates one ξℓ): |o⃗r(γ)| = −|E(γ)|, cf. Table 1.
408 R. BURING AND A. V. KISELEV
Proof. This is immediate from (4): its l.-h.s. vanishes by γ ∈ ker d; in its right-hand
side, the Jacobiator πS(P ,P) is an argument of endomorphisms which are linear, hence
all the k terms in the minuend vanish. The subtrahend remains, it yields 2 times the
cocycle condition O⃗r(γ)(P , . . . ,P) ∈ ker [[P , ·]]. □
Corollary 3 (A realization of ♢ by Leibniz graphs). The operator ♢ in the factorization
problem
∂P(O⃗r(γ)(P , . . . ,P)) = ♢(P , [[P ,P ]]), γ ∈ ker d,
is the sum of Leibniz graphs obtained from γ by inserting the Jacobiator [[P ,P ]] into
one of its vertices (by the Leibniz rule) and skew-symmetrizing w.r.t. the sinks.
Constructive proof. Indeed{, as (4) yields (with πS(P ,P) = (−)2−1[[P ,P ]]) equality (1}),
[[P , O⃗r(γ)(P , . . . ,P)]] = 1 O⃗r(γ)([[P ,P ]],P , . . . ,P) + . . .+ O⃗r(γ)(P , . . . ,P , [[P ,P ]]) ,
2
the left-hand side of the cocycle condition factors, in particular, through the explicitly
given set of Leibniz graphs with [[P ,P ]] in one vertex in the right-hand side. □
Corollary 4. Suppose that δ = d(γ) is a trivial d-cocycle in the graph complex: let
there be k vertices and 2k − 1 edges in γ. Then, reading (4) again, we have that for P
Poisson,
O⃗r(δ)(P , . . . ,P) = 0+ . . .+0− πS(︸O⃗r(γ)(P︷︷, . . . ,P︸),P)− πS(P , O⃗︸ r(γ)(P︷︷, . . . ,P︸)) =
1-vector 1-vector
−(−)1−1[[O⃗r(γ)(P , . . . ,P),P ]]−(−)2−1[[P , O⃗r(γ)(P , . . . ,P)]] = 2 [[P , O⃗r(γ)(P , . . . ,P)]].
(5)
Equality (5) provides the composition of the 1-vector field X(P) := 2 O⃗r(γ)(P , . . . ,P)
trivializing O⃗r(δ)(P , . . . ,P) = [[P ,X(P)]] in the Poisson cohomology.
Remark 1. From the above proof we also recognize the composition of Leibniz graphs
(i.e. improper terms which vanish by the Jacobi identity [[P ,P ]] = 0) in the factorization
problem
O⃗r(δ)(P , . . . ,P)− [[P ,X(P .)]] = ∇(P , [[P ,P ]]).
Namely, it is the terms O⃗r(γ)(πS(P ,P),P , . . . ,P) + . . . + O⃗r(γ)(P , . . . ,P , πS(P ,P))
from (4).
1.3. The morphism O⃗r at work: examples. The following collection of examples
illustrates (i) the construction of infinitesimal symmetries Ṗ = Q(P) for Poisson struc-
tures P by orienting cocycles γ ∈ ker d, so that Q = O⃗r(γ), and (ii) the construction
of trivializing vector fields X = 2O⃗r(γ) in Q = O⃗r(d(γ)). At the same time, we detect
(iii) the non-uniqueness of factorizations [[P ,Q(P)]] = ♢(P , [[P ,P ]]) for such cocycles
and flows.
We remember that the (iterated commutators of the) infinite sequence of d-cocycles
γ2ℓ+1, marked by (2ℓ + 1)-gon wheel graphs (see [15]), is a regular source of universal
symmetries for Poisson structures. Moreover, no flows Ṗ = Q(P) other than these ones,
Q(P) = O⃗r(γ)(P), are currently known (under the assumption that the cocycles γ be
sums of connected graphs).
THE ORIENTATION MORPHISM 409
Let us remark finally that it is also an open problem whether these flows, Q(P) =
O⃗r(γ)(P), can be Poisson cohomology nontrivial, that is Q 6= [[P ,X]] for some Poisson
structure P and a globally defined vector field X on an affine manifold M .
Eqq xaqqmple 3 (The tetrahedron γ3). For the tetrahedron γ3 ∈ ker d, i.e. the full graph@@ on 4 vertices and 6 edges (see [10]), both the Kontsevich flow Ṗ = Q1:6/2(P) and
the factorizing operator ♢ in the problem [[P ,Q1:6/2(P)]] = ♢(P , [[P ,P ]]) are presented
in [2] (cf. [11]). The operator ♢ is of the form given by Corollary 3.
Example 4 (The pentagon-wheel cocycle γ5 ∈ ker d).
r r  r
For the pentagon-wheel cocycle, the set of ori- r r r 5γ = + r r
ented Kontsevich graphs that encode the flow Ṗ 5= 2 r r
O⃗r(γ5)(P) is listed in [5]. The resulting differential
r r  
polynomial expression of this infinitesimal symmetry
is available in Appendix A below. But the factorizing operator for O⃗r(γ5)(P) reported
in [5], i.e. expressing [[P , O⃗r(γ5)(P)]] as a sum of Leibniz graphs, is different from the
operator ♢ which Corollary 3 provides for the cocycle γ5. This demonstrates that such
operators can be non-unique (as one obtains it in this particular example).5 r r
Example 5 (Coboundary δ6 = d(β6)). Take the only nonzero (with β6 = r r
respect to the wedge ordering of edges) connected graph β r r6 on six
vertices and 11 edges, and put δ6 = d(β6) ∈ ker d (indeed, d2 = 0). In view of
Corollary 4 and Remark 1, we verify the decomposition,
O⃗r(d(β6))(P) = [[P ,X(P)]] +∇(P , [[P ,P ]]),
into the Poisson cohomology trivial and improper terms. Indeed, the vector field X
stems from O⃗r(β6)(P , . . . ,P) and the improper part comes from the terms like O⃗r(β6)(
[[P ,P ]], . . . ,P). Interestingly, all the graphs from the ∂P-exact term [[P ,X]] also appear
in the improper terms, and in fact they cancel. (There are 598 graphs in the former
and 2098 in the latter; 2098− 598 = 1500, cf. Table 2 below.)
Example 6 (The heptagon-wheel cocycle γ7 ∈ ker d). The d-cocycle starting with the
heptagon-wheel graph is presented in [6]. The flow Ṗ = O⃗r(γ7)(P) is realized by
37,185 Kontsevich graphs on 2 sinks; they are listed in a standard format (see [2,
Implementation 1]) at http://rburing.nl/gamma7.zip. The factorizing operator ♢
is provided by Corollary 3 so that the validity of cocycle equation [[P , O⃗r(γ7)(P)]] =
♢(P , [[P ,P ]]) is verified experimentally. (It would be unfeasible to solve this equation
w.r.t. the unknown coefficients of the Leibniz graphs in the right-hand side, pretending
that a solution is not known from §1.2. Note however that no uniqueness is claimed for
this ♢.)
5We say that two Leibniz graphs (i.e. graphs with a tri-vector [[P,P]] in a vertex) are adjacent
vertices in the Leibniz meta-graph if the(expansions of these Leibniz graphs have at least one Kontsevichoriented graph in co(mmon. (I)n the meta-graphs), multip(le edges ar)e allowed.) The known existenceof several factorizations, [[P,Q]] = ♢1 P, [[P,P]] = ♢2 P, [[P,P]] , into Leibniz graphs reveals the
identities (♢1 − ♢2) P, [[P,P]] ≡ 0 for ♢1 6= ♢2, that is, a nontrivial topology of the meta-graph. Its
study is an open problem.
410 R. BURING AND A. V. KISELEV
Table 2. The number of graphs in the problem [[P , O⃗r(γ)(P)]] = ♢(P , [[P ,P ]]).
Cocycle: γ3 γ5 δ6 = d(β6) γ7
#vertices: 4 6 7 8
#edges: 6 10 12 14
#graphs: 1 2 4 46
#or.graphs in Q(P) = O⃗r(γ)(P , . . . ,P): 3 167 1,500 37,185
#or.graphs in [[P ,Q(P)]]: 39 3,495 35,949 1,003,611
#skew Leibniz graphs in ♢(P , [[P ,P ]]): 8 843 9,556 293,654
Implementation. All calculations above were performed by using the software pack-
ages graph_complex-cpp and kontsevich_graph_series-cpp, which are released un-
der the MIT free software license and available from https://github.com/rburing.
Specifically, the programs expanding_differential and kernel have been used to find
non-oriented graph cocycles γ, orient yields the sums of Kontsevich oriented graphs
O⃗r(γ)(P , . . . ,P) and sums of Leibniz graphs O⃗r([[P ,P ]], . . . ,P), and schouten_bracket
implements the Schouten bracket. The program leibniz_expand expands sums of Leib-
niz graphs into Kontsevich graphs, and reduce_mod_skew reduces sums of Kontsevich
oriented graphs modulo skew-symmetry, L ≺ R = −R ≺ L, of the Left ≺ Right
mark-up of outgoing edges.
Acknowledgements. The research of RB was supported in part by IM JGU Mainz; AVK was
partially supported by JBI RUG project 106552. The authors are grateful to the organizers
of the 32nd International Colloquium on Group Theoretical Methods in Physics (Group32,
9 – 13 July 2018, CVUT Prague, Czech Republic) for partial financial support (for RB) and
warm hospitality. The authors thank M. Kontsevich, A. Sharapov, and T. Willwacher for
helpful discussions. A part of this research was done while RB was visiting at RUG and AVK
was visiting at JGU Mainz (supported by IM via project 5020).
THE ORIENTATION MORPHISM 411
Appendix A. The differential polynomial flow Ṗ = O⃗r(γ5)(P)
Here is the value Q5(P)(f, g) of the bi-vector Q5(P) = O⃗r(γ5)(P) at two functions f, g:6
10∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prs∂ Ptv∂ f∂ g − 10∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prs∂ Ptvt m k p v r ℓ n q s i j p m k t v r ℓ n q s ∂if∂jg
+10∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prs tv ijr m t j v s ℓ n p ∂qP ∂if∂kg − 10∂r∂nP ∂t∂s∂jPkℓ∂ ∂ Pmn∂ Ppq∂ Prsv k ℓ p ∂ Ptvq ∂if∂mg
+10∂p∂ Pijm ∂t∂jPkℓ∂v∂r∂ℓPmn∂ Ppq∂ rs tvn qP ∂sP ∂if∂kg − 10∂ ∂ Pijt n ∂v∂ ∂ Pkℓr j ∂p∂ mn pq rs tvkP ∂ℓP ∂qP ∂sP ∂if∂mg
−10∂ ∂ Pij∂ kℓ mn pq rs tv ij kℓ mn pq rs tvt m p∂jP ∂v∂r∂ℓP ∂nP ∂qP ∂sP ∂if∂kg − 10∂p∂nP ∂t∂r∂jP ∂v∂kP ∂ℓP ∂qP ∂sP ∂if∂mg
−10∂ ∂ Pij∂ ∂ Pkℓ∂ Pmn∂ ∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 10∂ ∂ Pijt p q j ℓ v r m n s i k s m ∂ Pkℓj ∂t∂p∂kPmn∂ Ppq∂ ∂ Prs∂ Ptvℓ v n q ∂if∂rg
+10∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ ∂ Ppq∂ Prs∂ Ptvt p j q k v r ℓ n s ∂if∂mg − 10∂ ∂ Pij∂ Pkℓ∂ mn pq rs tvt m j v∂p∂kP ∂ℓP ∂q∂nP ∂sP ∂if∂rg
−10∂ ∂ Pijr m ∂jPkℓ∂ mn pqv∂p∂kP ∂ℓP ∂ Prs∂ ∂ Ptv∂ f∂ g + 10∂ ∂ Pij∂ ∂ ∂ Pkℓ mn pqn s q i t t p v r j ∂q∂kP ∂ℓP ∂nPrs∂sPtv∂if∂mg
−10∂ ∂ Pijt m ∂ kℓv∂s∂jP ∂ Pmn∂ Ppq∂ ∂ Prs∂ tv ij kℓ mn pq rs tvk ℓ p n qP ∂if∂rg + 10∂p∂mP ∂t∂s∂jP ∂kP ∂ℓP ∂v∂nP ∂qP ∂if∂rg
−10∂ ∂ Pijt r ∂v∂s∂ Pkℓj ∂p∂ Pmn∂ Ppq∂ Prs∂ Ptvk ℓ n q ∂if∂mg + 10∂ ∂ Pijr p ∂jPkℓ∂t∂ Pmnk ∂v∂ pq rs tvn∂ℓP ∂qP ∂sP ∂if∂mg
+10∂ ∂ Pij∂ ∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 10∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ ∂ Ppq∂ Prs tvr p t s j v k ℓ n q i m t p j r k v n ℓ q ∂sP ∂if∂mg
−10∂ ∂ Pij∂ Pkℓt n j ∂v∂r∂p∂ Pmn∂ Ppq∂ Prs∂ Ptvk ℓ q s ∂if∂mg − 10∂t∂r∂p∂mPij∂ kℓv∂jP ∂ℓPmn∂ Ppqn ∂qPrs∂ Ptvs ∂if∂kg
−10∂ ∂ Pij∂ ∂ ∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ Prs∂ Ptvt m v r p j ℓ n q s ∂if∂kg − 10∂t∂r∂ ∂ Pij∂ Pkℓ∂ ∂ mn pq rsp n j v kP ∂ℓP ∂qP ∂ Ptvs ∂if∂mg
−10∂ ∂ Pij∂ ∂ ∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 10∂ ∂ ∂ ∂ Pij∂ Pkℓ mn pq rs tvt p v r q j ℓ m n s i k t s p m j ∂kP ∂ℓP ∂v∂nP ∂qP ∂if∂rg
+2∂ ∂ ∂ ∂ ∂ Pij∂ Pkℓ∂ Pmn∂ Ppq∂ Prs∂ Ptv∂ f∂ g − 5∂ ∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn pq rs tvt r p m k v ℓ n q s i j p m k t r v ℓ ∂nP ∂qP ∂sP ∂if∂jg
+5∂ ∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ pq rs tv ij kℓ mn pq rs tvt m k r p v ℓ nP ∂qP ∂sP ∂if∂jg − 5∂p∂m∂kP ∂rP ∂t∂ℓP ∂v∂nP ∂qP ∂sP ∂if∂jg
−5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs tvp m k t r ℓ v n q ∂sP ∂if∂jg + 5∂ ∂ Pij∂ kℓ mn pq rs tvt p v∂jP ∂ℓP ∂r∂mP ∂nP ∂s∂qP ∂if∂kg
+5∂ ∂ Pij∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ ∂ Prs∂ ∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ Prs tvv r j k m ℓ p n s q i t r m t j s ℓ n v p ∂qP ∂if∂kg
−5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppqr n t j v k ℓ ∂ Prs∂ tvp s∂qP ∂if∂mg + 5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptvp m r j t ℓ v n q s ∂if∂kg
−5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prsr n t j p k v ℓ q ∂ tv ij kℓ mnsP ∂if∂mg + 5∂p∂mP ∂t∂jP ∂r∂ℓP ∂v∂ Ppq∂ Prsn q ∂ tvsP ∂if∂kg
−5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ rs tv ij kℓ mn pq rs tvt n r j p k v ℓ qP ∂sP ∂if∂mg + 5∂r∂nP ∂s∂jP ∂t∂kP ∂ℓP ∂v∂pP ∂qP ∂if∂mg
+5∂ ∂ Pijr m ∂ kℓt∂jP ∂v∂ Pmn∂ Ppqℓ n ∂pPrs∂s∂ Ptvq ∂if∂kg + 5∂p∂nPij∂ ∂ kℓ mn pq rs tvt jP ∂r∂kP ∂v∂ℓP ∂qP ∂sP ∂if∂mg
−5∂ ∂ Pijr m ∂p∂ kℓ mnjP ∂t∂ℓP ∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prsv n q s i k p n r j t k v ℓ q ∂sPtv∂if∂mg
−5∂t∂ ijmP ∂ ∂ Pkℓp j ∂r∂ Pmnℓ ∂v∂nPpq∂ Prsq ∂ tvsP ∂if∂kg + 5∂ ijs∂mP ∂t∂ Pkℓ∂ mnj v∂kP ∂ Ppq∂ rsℓ p∂nP ∂ Ptvq ∂if∂rg
+5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn pq rs tvr p q j t ℓ v m n s i k s m t j p k ∂ℓP ∂v∂nP ∂qP ∂if∂rg
−5∂ ∂ Pij∂ ∂ Pkℓ mnt p q j ∂v∂ℓP ∂ ∂ Ppq∂ Prs∂ Ptvr m n s ∂if∂kg − 5∂ ∂ Pij∂ kℓ mn pq rs tvr n jP ∂t∂s∂kP ∂ℓP ∂v∂pP ∂qP ∂if∂mg
−5∂t∂r∂mPij∂s∂ Pkℓj ∂ Pmnℓ ∂ Ppqn ∂ rs tv ij kℓv∂pP ∂qP ∂if∂kg − 5∂t∂pP ∂v∂q∂jP ∂ Pmnℓ ∂r∂mPpq∂ rsnP ∂ Ptvs ∂if∂kg
+5∂t∂s∂ Pij∂ Pkℓ∂ ∂ mn pq rs tv ij kℓ mn pq rs tvm j p kP ∂ℓP ∂v∂nP ∂qP ∂if∂rg + 5∂r∂mP ∂t∂s∂jP ∂v∂ℓP ∂nP ∂pP ∂qP ∂if∂kg
−5∂t∂r∂ Pij∂ kℓn s∂jP ∂ ∂ mn pq rs tv ij kℓ mn pq rs tvv kP ∂ℓP ∂pP ∂qP ∂if∂mg − 5∂t∂mP ∂v∂p∂jP ∂r∂ℓP ∂nP ∂qP ∂sP ∂if∂kg
−5∂ ∂ ∂ Pijt p n ∂r∂ kℓ mnjP ∂v∂kP ∂ℓPpq∂ Prsq ∂sPtv∂if∂mg − 5∂r∂mPij∂ kℓt∂s∂jP ∂ Pmn∂ pqℓ nP ∂v∂ Prs∂ tvp qP ∂if∂kg
−5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ Prs∂ Ptv∂ f∂ g − 5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prs∂ Ptvt r n j s k ℓ v p q i m r n t j v s k ℓ p q ∂if∂mg
+5∂ ∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prs∂ Ptv∂ f∂ g − 5∂ ∂ Pij∂ ∂ Pkℓ mnt r m s j v ℓ n p q i k t n r j ∂v∂p∂kP ∂ Ppq∂ Prs∂ Ptvℓ q s ∂if∂mg
−5∂t∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prs∂ Ptv∂ f∂ g − 5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn pq rs tvp m v j r ℓ n q s i k r m k t v ℓ ∂nP ∂pP ∂s∂qP ∂if∂jg
+5∂ ∂ ∂ Pij∂ Pkℓ∂ mn pq rs tv ij kℓ mn pq rs tvr m k p t∂ℓP ∂v∂nP ∂qP ∂sP ∂if∂jg + 5∂t∂m∂kP ∂pP ∂r∂ℓP ∂v∂nP ∂qP ∂sP ∂if∂jg
+5∂ ij kℓ mn pq rs tv ij kℓ mn pq rs tvr∂m∂kP ∂t∂pP ∂ℓP ∂v∂nP ∂qP ∂sP ∂if∂jg + 5∂t∂m∂kP ∂r∂pP ∂ℓP ∂v∂nP ∂qP ∂sP ∂if∂jg
−5∂r∂ Pijn ∂jPkℓ∂ mn pq rs tv ijt∂p∂kP ∂v∂ℓP ∂qP ∂sP ∂if∂mg + 5∂t∂p∂mP ∂r∂jPkℓ∂ Pmn∂ ∂ Ppq∂ Prsℓ v n q ∂ Ptvs ∂if∂kg
6In every term, the Einstein summation convention works for each repeated index (i.e. once upper
and another time lower), the indices running from 1 to the dimension dimM <∞ of the affine Poisson
manifold M at hand.
412 R. BURING AND A. V. KISELEV
−5∂ ∂ Pij∂ Pkℓt n j ∂r∂p∂kPmn∂v∂ Ppqℓ ∂ Prs∂ tv ij kℓ mn pq rs tvq sP ∂if∂mg + 5∂r∂p∂mP ∂t∂jP ∂ℓP ∂v∂nP ∂qP ∂sP ∂if∂kg
−5∂ ∂ Pij∂ ∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ Prs∂ Ptv∂ f∂ g − 5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ Pmnt p v r j ℓ m q n s i k v r m j p k ∂ℓPpq∂nPrs∂s∂ Ptvq ∂if∂tg
−5∂ ∂ Pij∂ ∂ ∂ Pkℓ∂ mn pq rs tv ij kℓ mn pq rs tvr p t q j v∂ℓP ∂mP ∂nP ∂sP ∂if∂kg − 5∂t∂s∂mP ∂v∂jP ∂kP ∂ℓP ∂p∂nP ∂qP ∂if∂rg
−5∂ ∂ ij kℓ mn pq rs tv ij kℓ mn pq rs tvt pP ∂r∂q∂jP ∂v∂ℓP ∂mP ∂nP ∂sP ∂if∂kg + 5∂s∂p∂mP ∂t∂jP ∂kP ∂ℓP ∂v∂nP ∂qP ∂if∂rg
−5∂ ∂ Pij∂ ∂ ∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptvr m t p j ℓ v n q s i k t p n j r k v ℓ q s ∂if∂mg
−5∂ ∂ Pij∂ ∂ ∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ ∂ Pij∂ Pkℓ mnt m r p j ℓ v n q s i k r p n j ∂t∂kP ∂v∂ Ppq∂ Prs∂ Ptvℓ q s ∂if∂mg
−5∂t∂ Pij∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ ∂ ∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ ∂ Pij∂ ∂ Pkℓ∂ Pmn pq rs tvs j k m ℓ v p n q i r t r p v j ℓ ∂mP ∂q∂nP ∂sP ∂if∂kg
−5∂r∂p∂m∂ Pijk ∂ Pkℓt ∂ ∂ Pmn∂ Ppqv ℓ n ∂ rsqP ∂sPtv∂if∂jg − 5∂ ∂ ij kℓt p∂m∂kP ∂rP ∂ Pmnℓ ∂v∂ Ppqn ∂qPrs∂ Ptvs ∂if∂jg
−5∂ ∂ ∂ ∂ Pij∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ Ppq rs tvr p m k t ℓ v n q s i j s m j t k ℓ ∂v∂p∂nP ∂qP ∂if∂rg
−5∂t∂r∂pPij∂ ∂ kℓ mn pqq jP ∂ℓP ∂v∂mP ∂nPrs∂sPtv∂if∂kg − 5∂t∂ Pijn ∂v∂jPkℓ∂r∂p∂kPmn∂ Ppq∂ rs tvℓ qP ∂sP ∂if∂mg
+5∂ ∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prsr p m t j v ℓ n q ∂ Ptvs ∂if∂kg − 5∂ ∂ Pijp n ∂t∂ Pkℓj ∂ ∂ ∂ Pmn∂ pq rs tvv r k ℓP ∂qP ∂sP ∂if∂mg
−5∂t∂ ijr∂mP ∂p∂ Pkℓj ∂v∂ Pmnℓ ∂nPpq∂ Prsq ∂sPtv∂if∂kg − 5∂t∂pPij∂ ∂ ∂ Pkℓr q j ∂ Pmnℓ ∂ ∂ Ppq∂ Prs∂ Ptvv m n s ∂if∂kg
+5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ Prs∂ Ptvs p m j t k ℓ v n q ∂if∂ g − 5∂ ij kℓ mnr t∂pP ∂v∂r∂jP ∂ℓP ∂ Ppq∂ Prsm n ∂ tvs∂qP ∂if∂kg
−5∂ ∂ ∂ Pij∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ Prs∂ ∂ Ptv∂ f∂ g − 5∂ ∂ Pij∂ ∂ ∂ Pkℓ mn pq rs tvv r m j k ℓ p n s q i t t m r p j ∂v∂ℓP ∂nP ∂qP ∂sP ∂if∂kg
−5∂r∂p∂ Pij∂ kℓn t∂jP ∂ ∂ Pmnv k ∂ pqℓP ∂ Prs∂ Ptv∂ ij kℓ mn pq rs tvq s if∂mg + 5∂p∂mP ∂t∂r∂jP ∂v∂ℓP ∂nP ∂qP ∂sP ∂if∂kg
−5∂ ∂ ∂ Pij∂ ∂ Pkℓt r n v j ∂p∂ mn pq rskP ∂ℓP ∂qP ∂ Ptv∂ f∂ g − 5∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prss i m t p j v k r ℓ n ∂s∂ Ptvq ∂if∂mg
−5∂t∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ ∂ Prs tvm j p k v ℓ q n ∂sP ∂if∂rg + 5∂r∂ Pijp ∂ Pkℓj ∂t∂kPmn∂ ∂ Ppqs ℓ ∂v∂ rs tvnP ∂qP ∂if∂mg
−5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ Pkℓ mnt r v j p k s ℓ n q i m r p j ∂t∂kP ∂v∂ Ppq∂ rsℓ q∂nP ∂ Ptvs ∂if∂mg
+5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prs tvr p t j v k ℓ n ∂s∂qP ∂if∂ g − 5∂ ∂ Pijm t m ∂ kℓv∂jP ∂ Pmn∂ Ppq∂ ∂ Prsk ℓ p n ∂s∂qPtv∂if∂rg
+5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmnt r s j v k ∂n∂ℓPpq∂ Prs∂ Ptv∂ f∂ g − 5∂ ∂ Pij∂ ∂ Pkℓp q i m t m v j ∂ ∂ Pmnp k ∂ℓPpq∂q∂ Prsn ∂ Ptvs ∂if∂rg
+5∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ ∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ ∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ ∂ Prs tvt r j s k n ℓ v p q i m p m t j k s ℓ v n ∂qP ∂if∂rg
+5∂t∂pPij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ rs tv ij kℓ mn pq rs tvr j v k ℓ q∂nP ∂sP ∂if∂mg − 5∂p∂mP ∂t∂jP ∂v∂kP ∂ℓP ∂q∂nP ∂sP ∂if∂rg
−5∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppqt m j p k s ℓ ∂v∂ rs tv ij kℓnP ∂qP ∂if∂rg + 5∂r∂pP ∂t∂jP ∂ ∂ Pmnv k ∂ pqs∂ℓP ∂nPrs∂ tvqP ∂if∂mg
+5∂t∂rPij∂jPkℓ∂p∂ mn pqkP ∂s∂ℓP ∂v∂nPrs∂ Ptv∂ f∂ g − 5∂ ∂ Pij∂ ∂ Pkℓq i m t r v j ∂p∂ mn pq rs tvkP ∂ℓP ∂nP ∂s∂qP ∂if∂mg
+5∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ ∂ Prs∂ Ptvt r j p k v ℓ q n s ∂if∂mg − 5∂ ∂ ij kℓ mn pq rs tvt pP ∂r∂jP ∂v∂kP ∂s∂ℓP ∂nP ∂qP ∂if∂mg
−5∂t∂ Pijm ∂jPkℓ∂ ∂ Pmn∂ ∂ Ppqs k n ℓ ∂ rs tvv∂pP ∂qP ∂if∂rg − 5∂r∂mPij∂v∂jPkℓ∂p∂kPmn∂ Ppq∂ Prs∂ ∂ Ptvℓ n s q ∂if∂tg
−5∂t∂mPij∂ ∂ Pkℓs j ∂ mnkP ∂n∂ℓPpq∂v∂pPrs∂ Ptvq ∂if∂rg + 5∂t∂ Pijm ∂jPkℓ∂v∂kPmn∂ pqℓP ∂p∂nPrs∂s∂qPtv∂if∂rg
+5∂t∂
ij
pP ∂v∂jPkℓ∂q∂kPmn∂r∂ℓPpq∂ Prs∂ Ptv∂ f∂ g − 5∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ Ppqn s i m r m j v k ℓ ∂p∂ Prsn ∂s∂ Ptvq ∂if∂tg
+5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppqt p r j q k v ℓ n s i m p m s j t k ℓ ∂ ∂ Prs∂ Ptvv n q ∂if∂rg
−5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppqt m s j v k ℓ ∂ ∂ Prsp n ∂ tvqP ∂ f∂ g − 5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ Prs tvi r r p s j t k ℓ v n ∂qP ∂if∂mg
+5∂ ij kℓ mn pq rs tv ij kℓ mn pq rs tvt∂pP ∂v∂jP ∂r∂kP ∂n∂ℓP ∂qP ∂sP ∂if∂mg − 5∂r∂pP ∂t∂jP ∂v∂kP ∂n∂ℓP ∂qP ∂sP ∂if∂mg
−5∂t∂ Pij∂ kℓr s∂jP ∂ ∂ mn pq rs tv ij kℓ mn pq rs tvp kP ∂ℓP ∂v∂nP ∂qP ∂if∂mg − 5∂r∂pP ∂jP ∂t∂q∂kP ∂v∂ℓP ∂nP ∂sP ∂if∂mg
−5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prs∂ ∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ Pkℓ∂ ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptvt r p j v k ℓ n s q i m t p j v q k r ℓ n s ∂if∂mg
−5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ Prs tv ij kℓt p m j v k ℓ q n ∂sP ∂if∂rg + 5∂t∂pP ∂jP ∂v∂r∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptvk n ℓ q s ∂if∂mg
+5∂ ijt∂r∂pP ∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prss j v k ℓ n ∂qPtv∂if∂mg + 5∂ ∂ Pijr p ∂t∂ Pkℓj ∂ ∂ ∂ Pmn∂ pq rs tvv q k ℓP ∂nP ∂sP ∂if∂mg
−5∂t∂ ijr∂pP ∂jPkℓ∂ ∂ Pmn∂ Ppqv k ℓ ∂q∂nPrs∂ tvsP ∂ f∂ g + 5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prsi m t p r j v q k ℓ n ∂sPtv∂if∂mg
+5∂ ∂ ∂ Pij∂ Pkℓ∂ Pmn∂ ∂ Ppq∂ ∂ Prs∂ Ptvt p m j k s ℓ v n q ∂if∂ g − 5∂ ij kℓ mnr t∂mP ∂s∂jP ∂kP ∂ℓPpq∂v∂ rs tvp∂nP ∂qP ∂if∂rg
+5∂ ∂ ∂ Pij∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ ∂ Prs∂ Ptvt p m s j k ℓ v n q ∂if∂rg − 5∂t∂ Pij∂ ∂ Pkℓ∂ mn pq rs tvr s j v∂p∂kP ∂ℓP ∂nP ∂qP ∂if∂mg
−5∂t∂r∂ Pij∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ Pkℓ∂ ∂ ∂ Pmn pq rs tvp j v k n ℓ q s i m t r j v p k ∂ℓP ∂nP ∂s∂qP ∂if∂mg
+5∂t∂ ∂
ij kℓ
r pP ∂jP ∂q∂kPmn∂ ∂ pq rsv ℓP ∂nP ∂ Ptvs ∂if∂mg + 5∂t∂ ij kℓrP ∂jP ∂v∂p∂kPmn∂ pqℓP ∂q∂nPrs∂ Ptvs ∂if∂mg
+5∂t∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prs tvr p v j q k ℓ n ∂sP ∂if∂mg + 5∂v∂ Pijm ∂ Pkℓj ∂p∂kPmn∂ ∂ Ppqr ℓ ∂ rsnP ∂s∂qPtv∂if∂tg
+5∂t∂pPij∂ kℓ mn pqr∂jP ∂ℓP ∂v∂mP ∂q∂nPrs∂ tvsP ∂if∂kg − 5∂s∂ ijmP ∂ Pkℓj ∂t∂ Pmn∂ ∂ Ppqk n ℓ ∂ rs tvv∂pP ∂qP ∂if∂rg
THE ORIENTATION MORPHISM 413
+5∂t∂pPij∂r∂ Pkℓj ∂ Pmn∂ ∂ Ppqℓ s m ∂ rsv∂nP ∂qPtv∂if∂kg − 5∂s∂mPij∂t∂ Pkℓ∂ mnj kP ∂n∂ Ppqℓ ∂ rs tvv∂pP ∂qP ∂if∂rg
+5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ ∂ Ppq∂ Prs∂ Ptv∂ f∂ g + 5∂ ∂ Pij∂ ∂ Pkℓ∂ Pmn pqt p r j v ℓ s m n q i k v m r j k ∂ℓP ∂p∂nPrs∂s∂qPtv∂if∂tg
+5∂ ijt∂pP ∂r∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ Prs∂ ∂ Ptvj v ℓ m n s q ∂if∂kg + 5∂ ij kℓ mn pq rs tvt∂sP ∂v∂jP ∂kP ∂m∂ℓP ∂p∂nP ∂qP ∂if∂rg
+5∂ ∂ Pij∂ ∂ Pkℓ∂ ∂ Pmn∂ Ppq∂ ∂ Prs∂ Ptv∂ f∂ g − 5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prs∂ Ptvr p t j v ℓ m q n s i k t p n j v r k ℓ q s ∂if∂mg
−5∂t∂r∂ Pij∂ ∂ ∂ Pkℓ∂ mn pq rs tv ij kℓ mn pq rs tvm v p j ℓP ∂nP ∂qP ∂sP ∂if∂kg + 5∂t∂s∂mP ∂jP ∂kP ∂ℓP ∂v∂p∂nP ∂qP ∂if∂rg
−5∂ ∂ ∂ Pij∂ ∂ ∂ Pkℓ∂ Pmn∂ Ppq∂ Prs∂ Ptv∂ f∂ g − 5∂ ∂ ∂ Pij∂ Pkℓ∂ ∂ ∂ Pmn∂ Ppq∂ Prs∂ Ptvt r p v q j ℓ m n s i k t r n j v p k ℓ q s ∂if∂mg
−5∂t∂p∂ Pijm ∂v∂r∂jPkℓ∂ Pmnℓ ∂nPpq∂ Prsq ∂sPtv∂if∂kg.
References
[1] Bouisaghouane A., Kiselev A.V. (2017) Do the Kontsevich tetrahedral flows preserve or destroy
the space of Poisson bi-vectors ? J. Phys.: Conf. Ser. 804 Proc. XXIV Int. conf. ‘Integrable
Systems and Quantum Symmetries’ (14–18 June 2016, ČVUT Prague, Czech Republic), Pa-
per 012008, 10 p. (Preprint arXiv:1609.06677 [q-alg])
[2] Bouisaghouane A., Buring R., Kiselev A. (2017) The Kontsevich tetrahedral flow revisited,
J. Geom. Phys. 119, 272–285. (Preprint arXiv:1608.01710 [q-alg])
[3] Buring R., Kiselev A.V. (2017) On the Kontsevich ⋆-product associativity mechanism, PEPAN
Letters 14:2, 403–407. (Preprint arXiv:1602.09036 [q-alg])
[4] Buring R., Kiselev A.V. (2017) The expansion ⋆ mod ō(ℏ4) and computer-assisted proof schemes
in the Kontsevich deformation quantization, Preprint arXiv:1702.00681 [math.CO]
[5] Buring R., Kiselev A.V., Rutten N. J. (2018) Poisson brackets symmetry from the pentagon-
wheel cocycle in the graph complex, PEPAN Letters 49:5, 924–928. (Preprint arXiv:1712.05259
[math-ph])
[6] Buring R., Kiselev A.V., Rutten N. J. (2017) The heptagon-wheel cocycle in the Kontsevich graph
complex, J. Nonlin. Math. Phys. 24, Suppl. 1, 157–173. (Preprint arXiv:1710.00658 [math.CO])
[7] Buring R., Kiselev A.V., Rutten N. J. (2017) Infinitesimal deformations of Poisson bi-vectors
using the Kontsevich graph calculus, J. Phys.: Conf. Ser. 965, Paper 012010, 12 pp. (Preprint
arXiv:1710.02405 [math.CO])
[8] Dolgushev V. A., Rogers C. L., Willwacher T. H. (2015) Kontsevich’s graph complex, GRT, and
the deformation complex of the sheaf of polyvector fields, Ann. Math. 182:3, 855–943. (Preprint
arXiv:1211.4230 [math.KT])
[9] Jost C. (2013) Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fields,
Differ. Geom. Appl. 31:2, 239–247. (Preprint arXiv:1201.1392 [math.QA])
[10] Kontsevich M. (1997) Formality conjecture. Deformation theory and symplectic geometry (Ascona
1996, D. Sternheimer, J. Rawnsley and S.Gutt, eds), Math. Phys. Stud. 20, Kluwer Acad. Publ.,
Dordrecht, 139–156.
[11] Kontsevich M. (2017) Derived Grothendieck–Teichmüller group and graph complexes [after
T. Willwacher], Séminaire Bourbaki (69ème année, Janvier 2017), no. 1126, 26 p.
[12] Laurent-Gengoux C., Pichereau M., Vanhaecke P. (2013) Poisson Structures. Grundlehren der
mathematischen Wissenschaften 347, Springer-Verlag Berlin Heidelberg.
[13] Merkulov S., Willwacher T. (2014) Grothendieck--Teichmüller and Batalin–Vilkovisky, Lett. Math.
Phys. 104:5, 625–634. (Preprint arXiv:1012:2467 [math.QA])
[14] Rutten N. J., Kiselev A.V. (2018) The defining properties of the Kontsevich unoriented graph
complex, Preprint arXiv:1811.10638 [math.CO], 10 p.
[15] Willwacher T. (2015) M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie al-
gebra, Invent. Math. 200:3, 671–760. (Preprint arXiv:1009.1654 [q-alg])

Chapter 18
The Kontsevich graph orientation
morphism revisited
This chapter is based on the peer-reviewed journal publication A.V. Kiselev and R. Bur-
ing, Banach Center Publ. 123 Homotopy algebras, deformation theory & quantization,
123–139, 2021. (Preprint arXiv:1904.13293 [math.CO] – 18 p.)
Commentary. In reference to Part I of the dissertation, the material of this chapter is
used in Chapter 5 (particularly, the attach_to_ground method). All the examples of
universal flows encoded by graphs are borrowed in this chapter from Chapters 15 and 16.
415
416 A. V. KISELEV AND R. BURING
THE KONTSEVICH GRAPH ORIENTATION MORPHISM
REVISITED
ARTHEMY V. KISELEV
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence,
University of Groningen, P.O.Box 407, 9700AK Groningen, The Netherlands.
E-mail: A.V.Kiselev@rug.nl (corresponding author)
RICARDO BURING
Institut für Mathematik, Johannes Gutenberg–Universität,
Staudingerweg 9, D-55128 Mainz, Germany.
E-mail: rburing@uni-mainz.de
Abstract. The orientation morphism O⃗r(·)(P) : γ →7 Ṗ associates differential-polynomial flows
Ṗ = Q(P) on spaces of bi-vectors P on finite-dimensional affine manifolds Nd with (sums of)
finite unoriented graphs γ with ordered sets of edges and without multiple edges and one-cycles.
It is known that d-cocycles γ ∈ ker d with respect to the vertex-expanding differential d = [•−−•, ·]
are mapped by O⃗r to Poisson cocycles Q(P) ∈ ker [[P, ·]], that is, to infinitesimal symmetries of
Poisson bi-vectors P. The formula of orientation morphism O⃗r was expressed in terms of the edge
orderings as well as parity-odd and parity-even derivations on the odd cotangent bundle ΠT ∗Nd
over any d-dimensional affine real Poisson manifold Nd. We express this formula in terms of
(un)oriented graphs themselves, i.e. without explicit reference to supermathematics on ΠT ∗Nd.
Introduction. A differential graded Lie algebra structure on the vector space of un-
oriented graphs with ordered sets of parity-odd edges was introduced by Kontsevich
in [14, 15]. The vertex-expanding differential d = [•−•, ·] is the adjoint action by the
edge •−•, which itself satisfies the master equation [•−•, •−•] = (const 6= 0) · 0 ∈ Graph
complex. Examples of d-cocycles are discussed in [16, 22] or [9]. Properties of the un-
oriented graph complex and its relation to the Grothendieck–Teichmüller group were
explored by Willwacher (see [21]).
2010 Mathematics Subject Classification: Primary 05C22, 16E45, 53D17; Secondary 68R10,
81R60.
Key words and phrases: Graph complex, vertex-expanding differential, orientation morphism,
Poisson bracket, deformation
The paper is in final form and no version of it will be published elsewhere.
THE GRAPH ORIENTATION MORPHISM REVISITED 417
The la(nguage of )graphs allows encoding (poly)differential operators on affine manifolds(let us consider Rd for simplicity), as well as encoding graded-symmetric endomorphisms
π ∈ End T (Rdpoly ) on the spaces of totally skew-symmetric multivecto∧r fields on Rd.
For example, a Poisson bivector P on Rd is represented by the wedge with Left <
Right edge ordering, whereas the Sc(houten bracket is an endomorphism:(
•1−−•2
) ∂⃗
πS(F,G) = O⃗r (F⊗G) = 12! ∣∣∣ ⊗ ∂⃗ ∣∣∣ ∂⃗ ∣∣∣ ∂⃗ ∣∣∣ )( ∣∣ ∣∣+ ⊗ F ·G +F ∣∣ ∣·G∣ ){ =∂ξ 1 ∂x 2 ∂ξ 2 ∂x 1 1 2 2 1∂⃗ ∂⃗ ∂⃗ ∂⃗ ∂ ⃗ ∂⃗ ∂ ⃗ ∂⃗ }
= (F )· (G)+(−)|F | (F )· (G) = (−)|F |−1 (F{) · (G)−(F ) · (G) =∂ξ ∂x ∂x ∂ξ ∂ξ ∂x ∂x ∂ξ }
= (−)|F |−1[[F,G]] = (−)|F |−1 F −−−−→ G − F ←−−−− G ,
for any homogeneous multivectors F and G. It is readily seen from the definition that
the endomorphism πS is graded-symmetric,
πS(F,G) = (−)|F |·|G|πS(G,F ),
and the usual variant of the Schouten bracket is shifted-graded skew-symmetric,
[[F,G]] = −(−)(|F |−1)·(|G|−1)[[G,F ]].
This language of oriented graphs is used in Kontsevich’s solution of the problem of defor-
mation quantisation on finite-dimensional Poisson manifolds (see [16, 18] and [4, 6, 7]). It
provides also the construction of universal –with respect to all affine Poisson manifolds –
infinitesimal symmetries Ṗ = Q(P) of classical Poisson structures (see [16, 19] and [2]).
The orientation morphism O⃗r associates Poisson cocycles Q(P) ∈ ker[[P, ·]] with
[•−•, ·]-cocycles in the unoriented graph complex. The algebraic formula of O⃗r was given
in [16]; its constructions was discussed in more detail by Jost in [12], by Willwacher
in [21], and by the authors in a recent paper [5] (cf. [8]). This morphism is a tool which
produces symmetries of Poisson brackets. If such a symmetry is not Poisson-exact in the
second Poisson cohomology, then it yields deformations of classical Liouville-integrable
systems, preserving the property that the dynamics is Poisson. If the symmetry of a given
bracket is Poisson-exact, then its flow produces a family of diffeomorphisms – typically,
nonlinear – of the affine Poisson manifold under study. A c∑onstruction of universal Poisson
1-cocycles n−1X⃗ = O⃗r(γ)(V⃗ ,P⊗ ) from graph cocycles γ = a ca·γa with n vertices in each
term γa is introduced in [8] for homogeneous Poisson bi-vectors P = LV⃗ (P) = [[V⃗ ,P]].
(The domain of definition of O⃗r is of course larger than the homogeneous component
of unoriented graphs on n vertices and 2n − 2 edges, n ∈ N.) So far, one example of
its work, namely O⃗r : γ3 ∈ ker d 7→ Q1: 6 (P) ∈ ker ∂P , was known from the seminal2
paper [16] in whi(ch the tetr)ahedral flow was introduced (cf. [19] and [2]). The pentagon-wheel flow O⃗r(γ5)(P) has been obtained in [10] by solving the factorisation problem
[[P,Q5(P)]] = ♢ P, [[P,P]] with respect to the Leibniz graphs in ♢ and the Kontse-
vich bivector graphs in Q5 = O⃗r(γ5) on 6 internal vertices. The heptagon-wheel flow
Q7 = O⃗r(γ7) on 8 internal vertices is described in [5].
The algebraic formula of the orientation morphism amounts to a simple but extensive
calculation using Z/2Z-gradings, i.e. supermathematics. The morphism O⃗r determines
both the orgraph multiplicities and the signs of all the Kontsevich graphs Γ (which, we
418 A. V. KISELEV AND R. BURING
recall, (are equ∣∣ip)ped wit(h an or∣∣de)ring Left < Right at all their internal vertices v, suchthat Γ L < R = −Γ R < L ). Nevertheless, the algebraic formula is external with
v v
respect to the (un)oriented graph complexes. We pose the question whether doing this
Z2-graded computation – involving thousands and millions of graphs – is the only way to
build each element of a known infinite sequence Ṗ = Q2ℓ+1(P) of flows.1
The aim of this note is to express the orientation morphism in geometric terms,
that is, by using combinatorial data which are intrinsic with respect to the (un)oriented
graphs γa equipped with edge orderings I ≺ II ≺ . . . in γa∑and wedge orderings Li < Ri
at vertices vi in O⃗r(γa), respectively. We let a cocycle γ = a ca ·γa ∈ ker d on n vertices
and with a given ordering I ≺ II ≺ III ≺ . . . of 2n − 2 edges in each graph γa be
the initial datum. The problem is to reveal the few rules of matching for the Kontsevich
oriented graphs: whenever a sign is fixed in front of just one Kontsevich graph which is
obtained by orienting a given graph γa in a cocycle γ = c1γ1 + . . . + cmγm ∈ ker d, the
signs in front of all admissible orientations of all the graphs in their linear combination γ
are then determined without calculation of O⃗r for each of them.
This paper contains three sections. For consistency, in section 1 we recall the algebraic
formula of the orientation morphism O⃗r for the Kontsevich graph complex; the approach is
operadic (cf. [17]). In section 2, we establish the main rule for the count of multiplicities
and signs of orgraphs. In the final section, we expand on the rule of signs purely in
terms of oriented Kontsevich graphs; to this end, we analyse their admissible types and
combinatorial properties of transitions between them.
1. The alg∑ebraic formula of graph orientation morphism O⃗r. Denote by Grathe real vector space of (formal sums2 of) unoriented, – by default, connected – finite
graphs γ = a ca · γa equipped with wedge ordering of edges E(γa) = I ∧ II ∧ . . . in each
graph γa; by definition, put (γ, I ∧ II ∧ . . .) = −(γ, II ∧ I ∧ . . .), etc. The vertices of a
graph γa are not ordered; one is free to choose any labelling of these #Vert(γa) vertices
by using 1, . . ., n = #Vert(γa) because the assignment ℘i 7→ vσ(i) of an n-tuple ℘1, . . .,
℘n of multivectors, which form the ordered set of graded arguments for an endomorphism
defined in formula (1) below, will be provided by the entire permutation group Sn 3 σ:
the ith multivector is placed into the vertex vσ(i).
A zero unoriented graph 0 is a graph which is equal to minus itself under an auto-
morphism (that induces a permutation of edges whose parity we thus inspect).
Example 1. The two-edge connected graph •−•−• is a zero graph because the flip, a
symmetry of this graph, swaps the edges which anticommute, I ∧ II = −II ∧ I, whence
the graph at hand equals minus its image under its own automorphism, i.e. minus itself.
1Willwacher showed in [21] that at every ℓ ∈ N, the (2ℓ+ 1)-wheel graph marks a nontrivial
d-cocycle in the unoriented graph complex, and those are expected to yield – for generic Pois-
son structures P – nontrivial terms in the respective Poisson cohomology under the orientation
morphism. It remains an open problem to find a graph cocycle γ and Poisson bracket P on
an affine manifold Nd such that the respective Poisson cocycle O⃗r(γ)(P) is nontrivial. Iterated
commutators of the cocycles γ2ℓ+1 yield Poisson cocycles as well.
2For the sake of definition, we assume by default that all terms in each sum are homogeneous
w.r.t. the various gradings.
THE GRAPH ORIENTATION MORPHISM REVISITED 419
The operadic insertion ◦i : (γ1, E(γ1))⊗ (γ2, E(γ2)) 7→ (γ1 ◦i γ2, E(γ1) ∧ E(γ2)) is the
Leibniz-rule sum of insertions of γ1 into vertices of γ2 such that every edge which was
incident to a vertex v in γ2 is redirected to vertices in γ1 in all possible ways according to
another Leibniz rule. The super Lie bracket [γ , γ #E(γ1)·#E(γ2)1 2] = γ1 ◦i γ2 − (−) γ2 ◦i γ1
on Gra yields the ve(rtex-expan)ding differential d = [•−•, ·].
Denote by End T dpoly(R ) the vector space of graded-symmetric endomorphisms
on the space of multivector fields on the space Rd (which we view here as an affine
manifold). In a full analogy with graphs (see [5]), one inserts a given endomorphism
into an argument slot of another endomorphism, proceeding over the slots consecu-
tively and ta(king the sum. The Richardson–Nijenhuis bracket [·, ·]RN is the graded sym-metrisation (with respect to its multivector arguments) of the difference of insertions:
[π, ρ] = Sym π ◦ ρ− (−)|π|·|ρ|
)
ρ ◦ π . The Schouten bracket π (R,S) = (−)|R|−1S [[R,S]] of
multivectors R, S is a graded symmetric endomorphism which satisfies the (shifted-)
graded Jacobi identity [πS , πS ]RN = 0; hence by the graded Jacobi identity for the
Richardson–Nijenhuis bracket (see [20]), the operation [πS , ·]RN is a differential.
For a given graph γ on n vertices a(nd for an)ordered n-tuple ℘ of multivectors ℘1,
. . ., ℘n, the morphism O⃗r : Gra→ End T d∑poly∏(R ) is e(nco∣ded by the fo∣rmula
O⃗r(γ)(℘ ⊗ · · · ⊗ ℘ ) = 1 ∆⃗ ℘ ∣ · . . . · ℘ ∣ )1 n n! ij 1 n , (1)vσ(1) vσ(n)
σ∈Sn eij∈E(γ)
where, in the ordered product of edges, the first edge I ∈ E(γ) acts first and the last edge
acts last. For every edge eij connecting the vertices vi and vj in the graph γ, the edge
operator,
di∑mRd{ } di∑mRd( )
∆⃗ij = i −→
α ←−α ∂⃗ ⊗ ∂⃗ ∂⃗j + i j = + ⊗ ∂⃗
(i) ∂xα ∂xα (j)
α=1 α=1 ∂ξα (j) (i) ∂ξα
acts from left to right by the (graded-) derivations along the ordered sequence of multi-
vectors ℘1 ⊗ . . . ⊗ ℘n, now placed by a permutation σ ∈ Sn into the respective vertices
vσ(1), . . ., vσ(n). In this way, the derivations in each edge operator ∆⃗ij reach the content
℘σ−1(i) and ℘σ−1(j) of the vertices vi and vj . Summarizing, the graph with an edge order-
ing (γ,E(γ)) is the set of topological data which determine all the portraits of couplings
within the n-tuple of multivectors: the above formula is referred to a d-tuple {xα} of
affine coordinates on Rd and the canonical conjugate d-tuple {ξα} of∑fibre coordinatesd
on the parity-odd cotangent bundle ΠT ∗Rd, so that the summation dimRα =1 gives theij
coupling along the edge eij . (Clearly, one proceeds by linearity over sums of graphs γ
and over sums of multivectors ℘i at every i running from 1 to n.)
Lemma 1. Suppose that at most one – in the rest of this paper, none usually, still exactly
one only in Corollary 3 below and likewise in [8] – multivector ℘1, . . ., ℘#Vert(Z∈Gra) is
odd-graded, while all the rest are even-grading (so that all are permutable without signs
appearing). Then, for the sets of endomorphisms’ arguments restricted in this way, the
420 A. V. KISELEV AND R. BURING
morphism O⃗r in (1) is well defined on the space Gra, that is, its action gives
O⃗r (zero graph Z) =
= (0 ∈ R) · (nonzero oriented graphs) + (coeffs ∈ R) · (zero Kontsevich orgraphs).
Proof. By definition, a zero unoriented graph Z has a symmetry σ ∈ Aut(Z) which acts
by a parity-odd permutation σE on the wedge-ordered set E(Z) of its edges and by a
permu∏tation σV on the set Vert(Z) of its vertices. The permutation of edges yields thesame parity-odd permutation σE of the edge operators ∆ij in their ordered product
∆⃗ = e ∈E(Z) ∆⃗ij . The permutation σV of the content of vertices does not yield anyij
signs because by our assumption, all the multivectors are pairwise permutable. Put n =
#Vert(Z). Acting by the symmetry σ of unoriented graph Z on the anticommuting edge
operators and on the multivectors ℘1, . . ., ℘n, which are assigned consecutively by all
the n! permutations τ ∈ Sn to the vertices of Z (and the ordered product of which is the
argu∑ment∏of ∆⃗), we obt(ain∏– for t(he isom)o)rphic unoriented graph σ(Z) – the expression
σE(∆⃗ij) τ σV (℘) =
τ∈Sn eij∈E(Z) ∑ ∏Vert(Z) ( ∏ ( )) ∑ ( ( ∏ )
= ∆⃗σE(eij) τ σV (℘) = −∆⃗) + τ(℘) .
τ∈Sn eij∈E(Z) Vert(Z) τ∈Sn Vert(Z)
This confirms that O⃗r(Z)(℘1, . . ., ℘n) is a differential polynomial (w.r.t. the components
of multivectors ℘i) which equals minus itself, hence it is identically zero. But this defining
property of the differential-polynomial expression means that the Kontsevich orgraph that
encodes the endomorphism O⃗r(Z) is a zero orgraph.
Proposition 2. Under the assumption – similar to the above in Lemma 1 – that all the
multivector arguments ℘i of the endomorphisms are in fact copies of a given Poisson
bi-vector P, the morphism O⃗r “respects the operadic insertions” by sending the super Lie
bracket of graphs to the Richa(rdson–N)ijen[huis bracket of]endomorphisms:
O⃗r [γ1, γ2] = O⃗r(γ1), O⃗r(γ2) RN,
where the right-hand side is graded symmetrised by construction.
In particular, O⃗r(•−•) = πS, i.e. the orientation morphism sends the edge to the
Schouten bracket, whence ( ) [ ]
O⃗r d(γ) = πS , O⃗r(γ) RN. (2)
This is standard (see Appendix A below and also [12, 16, 17] or [5]).
Corollary 3. Evaluating the endomorphisms in both sides of Lie superalgebra mor-
phis(m (2) a)t n+ 1 copies of a given Poisson bivect∑or P sati(sfying [[P,P]] = 0 on Rd, )
O⃗r [•−•, γ] (P, . . . ,P) = 2[[P, O⃗r(γ)(P, . . . ,P)]]− O⃗r(γ) P, . . . ,P, ︸[[P︷,︷P]]︸,P, . . . ,P ,
i
ith slot
one obtains – for a cocycle γ on n vertices and 2n− 2 edges – an explicit solution of the
factorisation problem, ( )
[[P, O⃗r(γ)(P)]] = ♢ P, [[P,P]] , (3)
THE GRAPH ORIENTATION MORPHISM REVISITED 421
for the Poisson cocycles Q(P) = O⃗r(γ)(P) corresponding to [•−•, ·]-cocycles in the unori-
ented graph complex.
Otherwise speaking, the orientation morphism O⃗r sends d-cocycle graphs to ∂P -cocy-
cles (i.e. Poisson cocycles). Moreover, d-exact unoriented graphs γ = d(ζ) yield ∂P -co-
boundaries Q(P) = [[P,X(P)]] with one-vectors X(P) = 2 ·O⃗r(ζ)(P) (modulo the improper
terms which, by definition, vanish on the entire Rd whenever a bivector P is Poisson).
Example 2. The factorisation operators ♢ in (3) for the tetrahedral flow Ṗ = Q1: 6 (P) =2
O⃗r(γ3)(P), pentagon-wheel flow O⃗r(γ5)(P), and for the heptagon-wheel flow O⃗r(γ7)(P)
have been presented in [2], [10], and [5], respectively.3
Proposition 4 (proved in App. A). For a given Poisson bi-vector P, the graph orien-
tation mapping, ∣
O⃗r(·)(P) : ker d∣ − 3 γ →7 Q(P) ∈ ker ∂P ,(n,2n 2)
is a Lie superalgebra morphism that takes the bracket of two cocycles in bi-grading (n, 2n−
2) to the commutator [ d , ddε dε ](P) of two symmetries
d 4
1 2 dε
(P) = Qi(P).
i
2. Kontsevich orgraphs: The count of multiplicicities and signs. In this section
we derive the rule for calculation of orgraph multiplicities and matching the signs of
oriented Kontsevich graphs. This defining property completely determines the evalua-
tion O⃗r(γ)(P) of the orientation morphism for cocycles γ ∈ ker d at (tuples of) Poisson
bivectors P. ∑
Convention. In the sequel, for a given graph γa in a cocycle a ca · γa = γ ∈ ker d,
sums are taken over the big set of 2#E(γa) of ways to orient edges of γa by using the
operator ∆⃗.
Not every such way to orient edges would yield a Kontsevich orgraph built of n wedges
(hence, having n internal vertices, 2n arrow edges formed by 2n−2 old edges of γa and 2
new edges S0, S1 to the new sink vertices 0, 1 with their content f , g). But the Kontsevich
orgraphs are selected automatically whenever a copy of bi-vector P = 12P ij(x) ξiξj is
placed in each internal vertex of every orgraph, so that every vertex of γa is the arrowtail
for exactly two oriented edges.
Convention. Every Kontsevich orgraph on n internal vertices and 2n edges is encoded
by the ordered (using an arbitrary fixed labelling of internal vertices) list of ordered
(w.r.t. Left < Right at a vertex) pairs of target vertices for the respective outgoing edges.
Definition 1. AKontsevich orgraph (built of wedges) is a Λ-shaped orgraph if one vertex
is the arrowtail of both edges directed to the sinks. Otherwise, a Kontsevich orgraph – in
3Let us emphasize that such operators are in general not unique. For instance, there are
known solutions ♢ besides the subtrahend in the right-hand side of (2). E.g., the operator ♢
in (3) for the orgraph sum O⃗r(γ5)(P) is different from the factorisation found in [10]. Likewise,
two linearly independent solutions of (3) for the tetrahedral flow O⃗r(γ3)(P) are built in [2].
4By Brown [3], the commutator does in general – for a generic Poisson structure P – not
vanish for Willwacher’s odd-sided wheel cocycles.
422 A. V. KISELEV AND R. BURING
which the edges to sinks are issued from different vertices – is called a Π-shaped orgraph
(see Fig. 1).
5 5 5
(a) So (b) So (c) 7S
 S  S  S
 S  S  S
V III SVI V III SVI V III SVI
 SIV 
S  S
3 / -S 4 −3· 3 / IV -SQ  Qk  4 −3· 3
 IV -Sw
Qk  4.Q   
I Q ? 
Q Q
S
Qs + II 1 I Q ?
 II I Q  II S1
?1 Q+ Q?+ ?
2S 2 2
1
S0 SS1 S0 S0
0/ Sw 1 0? 0?
Fig. 1. The right-hand side Q1: 6 of the tetrahedral flow is encoded by the Λ-shaped and two
2
Π-shaped orgraphs; the edge ordering in the wedges is (4), cf. Example 4 below.
Example 3. In the tetrahedral flow
Q1: 6 = 1 · (0, 1; 2, 4; 2, 5; 2, 3)− 3 · (0, 3; 1, 4; 2, 5; 2, 3+ 0, 3; 4, 5; 1, 2; 2, 4), (4)2
the Λ-shaped orgraph and the Π-shaped orgraph, skew-symmetrised over its sinks, occur
in proportion 8 : 24 = 1 : 3 (see [2] and [19]).5
Remark 1. Each encoding of a way to orient a graph γa to a Kontsevich orgraph on 2n
edges yields a permutation of the ordered set E(γa)∧S0 ∧S1 = +S0 ∧S1 ∧E(γa) of these
edges. Namely, the permutation is encoded by the ordered (using any given labelling of
internal vertices) list of ordered (w.r.t. Left < Right at a vertex) pairs of outgoing edges.
Convention. We let the default, occurring with ‘+’ sign, ordering of oriented edges in
a Kontsevich orgraph under study be S0 ≺ S1 ≺ E(γa) = S0 ≺ S1 ≺ I ≺ II ≺ · · · .
Example 4. Th(e tetrahedral graph flow )is
Q = (+1) · 0 1 2 4 [(2 5 2 31: 62 S0 S1 I IV II V I III V ) ( )]
− 3 · 0 3 1 4 2 5 2 3 0 3 4 5 1 2 2 4S0 I S1 IV II V I III V + S0 I IV V S1 II III V I .
The parity sign of edge permutation in the Λ-shaped graph is (−)4 = (+), here we count
the transpositions w.r.t. S0 ≺ S1 ≺ I ≺ . . . ≺ V I; the permutations of edges in the
second and third, Π-shaped graphs are parity-odd: (−)5 = (−) = (−)7, respectively.
Using the set of all the 2#E(γa) orientations of edges in a single graph (γa, E(γa)) from
a cocycle γ, select all topologically isomorphic Kontsevich orgraphs.
Theorem 5. Every orgraph Γ, isomorphic to a Kontsevich orgraph Γ0, acquires under (1)
the sign
sign(Γ) = (−)σE · sign(Γ0), (5)
5We verify in App. B that the third graph is equal to minus the second term with edges
S0 ⇄ S1 interchanged.
THE GRAPH ORIENTATION MORPHISM REVISITED 423
where the permutation σE : Edge(Γ) ' Edge(Γ0) of oriented edges is induced by the
orgraph isomorphism σ : Γ ' Γ0.
This is the rule for the count of orgraph multiplicities and signs. For instance, the
multiplicity +8 of the first Kontsevich orgraph in the tetrahedral flow (cf. [16, 19] and [2])
is obtained in Example 5 at the end of this section. Examples illustrating how rule (5)
works in the various counts of signs will be given in section 3 (see p. 426 below, as well
as Example 4 earlier in this section).6
Proof. Represent the αth copy of Poisson bi-vector P(α) using · 1 p(α) q(α)ξp(α)ξq(α) 2P(α) (x),
here 1 ⩽ α ⩽ n, and collect the set of parity-even pairs (ξξ)α to the left of the product
of bivector coefficients. The edge orienting operator ∆⃗ acts from the left on the string
(ξξ)1 · . . . · (ξξ)n; the topology of unoriented graph γa and the choice of (j)∂⃗/∂xν(i)⊗ ∂⃗/∂ξν
versus (i)∂⃗/∂xν(j)⊗ ∂⃗/∂ξν at each pair (ij) specify the nth degree differential monomial in
coefficients p(α) q(α)P(α) . (Note that the index ν for every edge eij is a summation index.)
Under the orgraph isomorphism σ, which unshuffles the vertex pairs (ξξ)α by an
even-parity permutation of the letters ξ, the parity-odd edge operators ∆ij and the two
new edges S0, S1 are permuted by σE . This permutation of edges corresponds to a
permutation of the parity-odd letters ξ that indicate the tails of those edges. Therefore,
the two Kontsevich orgraph encodings differ by the sign factor (+) · (−)σE = (−)σE .
Corollary 6. A skew-symmetry w.r.t. the sinks is guaranteed for bi-vector orgraphs.
Indeed, for orgraphs with swapped new edges S0 and S1 issued to the sinks f , g from some
vertex (or two distinct vertices) of γa, the sign of transposition S0 ⇄ S1 balances the two
types of orgraphs: with S0 to f and S1 to g against orgraphs with the edge S1 now issued
to g from the old source of S0, and the new edge S0 issued to f from the old source of the
old edge S1.
Lemma 7. The sign of edge permutation in (5) trivializes the contribution from any zero
unoriented graph Z to the sum over all possible ways to orient its edges.
Proof. Run over the entire sum of 2#E(Z) ways to orient edges in a zero graph Z, and select
all admissible not identically zero differential monomials similar to a given one, i.e. pick
all Kontsevich orgraphs Γ isomorphic to a given one (denote it by Γ0). Each isomorphism
Γ ' Γ0 represents an element of the automorphism group Aut(Z). By the definition of
zero unoriented graph, the subgroup H ⊴S2n−2 of unoriented edge permutations – under
the action of the group Aut(Z) – contains at least one parity-odd element σE . Therefore,
the numbers of parity-even and parity-odd elements inH coincide. This implies that these
#Aut(Z) elements cancel in disjoint pairs, so that the contribution of each topological
profile in O⃗r(Z)(P, . . ., P) comes with zero coefficient.
Remark 2. The wedge orderings Left < Right of outgoing edges at internal vertices of
the Kontsevich orgraphs under study play no role in the count of multiplicities and signs !
6The secret confined in Theorem 5 is that the calculation of edge permutation parities is in
fact redundant for finding the signs in front of the Kontsevich orgraphs; this will be seen in the
next section.
424 A. V. KISELEV AND R. BURING
Example 5. Indeed, were the Left < Right oriented edge orderings contributing with
a ‘−’ sign factor per vertex whenever Right ≺ Left in the edge ordering S0 ≺ S1 ≺
I ≺ II ≺ . . ., the Λ-shaped graph of differential orders (3, 1, 1, 1) in the tetrahedral flow
O⃗r(γ3)(P) would accumulate the wrong coefficient 4 − 4 = 0 instead of the true value
4 + 4 = 8 in the balance 8 : 24 for the linear combination Q1: 6 (P) of the Kontsevich2
graphs (see Fig. 2 and Table 1). For instance, Ia ' V h and IV a ' Ih so that in the
4
{a} So {b} 7S {c} 6S
 S  S  S
 S  S  S
V III SVI V III SVI V III SVI
 S 
2 / IV
S  S
-S 3  IV Sw / IV -wSQ  Q  Q
Q  Q  Q 3 CI QQs ?
 Q  Q
+ II I Qs ?+ II I Qs  II
 C
  CW
1S S f gS0 SS1 S0 SS1
f / Sw g f / wS g
{d} 7S {e} So {f} 6S S  S  S
 S  S  S
V III SVI V III SVI V III SVI
 S
 IV
S   S
-wS / IV S / IV wS
kQ 3
Q  C C
Qk 3 CQk Q  Q I Q  II  C  C I Q  II  C I Q  II
Q?  WC  CW Q?  CW Q +
f g f g f g
f OC g f CO g
C  C 
{ C Cg} 7S6o
{h} 76So S  S
 S  S
V III SVI V III SVI
 S  S
 IV -S  IV S
Qk  Q Q 3QI Q  II I Q

Q + Qs
 II

Fig. 2. The eight Λ-shaped ways to orient the tetrahedron: {a} ÷ {h} yield +8.
orgraph {a}, Ia < IV a, but under the orgraph isomorphism {a} ' {h}, one obtains the
inversion V h < Ih at a vertex.7
7The full list of inversions is this: {c} V > V I, I > IV ; {d} I > II, III > V I; {e} III > V ;
{f} II > IV ; {g} IV > V , II > V I, I > III; {h} I > V , II > III, IV > V I.
THE GRAPH ORIENTATION MORPHISM REVISITED 425
Table 1. Permutations of six edges I, . . ., V I in the Λ-shaped orientations of the tetrahedron.
(−)σE #(L > R) (−)#(L>R)
{a} I II III IV V VI + 0 +
{b} I III II V IV VI + 0 +
{c} II VI IV III I V + 2 +
{d} II IV VI I III V + 2 +
{e} IV I V II VI III + 1 −
{f} I V IV III II VI + 1 −
{g} V VI III IV I II + 3 −
{h} V III VI I IV II + 3 −
3. The rule of signs in terms of Kontsevich orgraphs. In this section we derive two
rules which allow the matching of signs in front of Kontsevich’s orgraphs by simply looking
at these graphs (or their encodings), so that no calculation of parities for permutations
of all the edges is needed. The first rule is specific to Π-shaped orgraphs. The second rule
describes the sign factors which are gained in the course of transitions Π ⇄ Π, Λ ⇄ Π,
and Λ ⇄ Λ between the orgraphs of respective shapes, as long as they ∑are taken from
the set of all admissible ways to orient a given graph γa in a cocycle γ = a ca · γa. The
work of both rules is illustrated using the tetrahedral and pentagon-wheel cocycle flows
O⃗r(γ3)(P) and O⃗r(γ5)(P) from [2] and [10], respectively.
Rule 1. A Π-shaped orgraph with ordered edge pairs (S0, A)(S1, B) · · · issued from two
distinct vertices acquires under (1) the extra sign factor (−), compared with a graph with
the ordered edge pairs (S0, B)(S1, A) · · · , if A ≺ B in the edge ordering E(γa) of a graph
to orient.
Proof. Indeed, S0∧S1∧A∧B = −S ∧A∧S ∧B = (−)20 1 S0∧B∧S1∧A = +S1∧A∧S0∧B.
Example 6. Both Π-shaped graphs in the tetrahedral flow (see its encoding in Example 4
in section 2) do acquire a sign factor by Rule 1.
Definition 2. The body of a Kontsevich orgraph which is obtained by orienting γa in
a cocycle γ is the set of oriented edges inherited from γa, i.e. excluding the new edges Si
to the sinks.
Rule 2. Let Γ1 and Γ2 be two topologically nonisomorphic orgraphs which are obtained
by orienting the same graph γa in a cocycle γ.8
Π ⇄ Π If both the orgraphs are Π-shaped, then the sign in front of (the multiplicity of)
the orgraph Γ2 is determined from such sign given by (1) for Γ1 by now using the
formula
sign(Γ ) = (−)#{ reverses of arrows in the body as Γ1 → Γ2}2 · sign(Γ1). (6)
8For instance, such obviously are all the terms in the Kontsevich flow O⃗r(γ3)(P) where the
tetrahedron γ3 ∈ ker d is oriented, or the orgraphs which one obtains by orienting the pentagon
wheel and the prism graph in the Kontsevich–Willwacher cocycle γ5, cf. [9, 10].
426 A. V. KISELEV AND R. BURING
Λ ⇄ Π Transitions Λ ⇄ Π yield the product of sign factors (−) × formula (6), i.e. the
extra (−) is universal, distinguishing between the shapes.
Λ ⇄ Λ Same-shape transitions Λ ⇄ Λ acquire only the sign factor (6).
In other words, the transition Λ ⇄ Π signals the sign factor (−), and the number of body
arrow reversals contributes in all cases.
Proof. Case Π ⇄ Π. For the sake of clarity, assume at once that the edge operators ∆⃗ij
corresponding to the edges whose orientation is not reversed have already acted on the
argument of two operators ∆⃗ corresponding to the two graphs, Γ1 and Γ2. There remain
κ edge operators acting on the product of κ + 2 comultiples ξ · · · ξ times an even factor
formed by the coefficients P pq(α)(x) of bi-vector copies.
Consider the righmost operator ∆⃗ij from what remains; it is the sum ∂⃗/∂ξoldtail ⊗
∂⃗/∂xoldhead and ∂⃗/∂ξoldhead ⊗ ∂⃗/∂xold new newtail = ∂⃗/∂ξtail ⊗ ∂⃗/∂xhead. Because the derivatives ∂⃗/∂x
have even parity, we focus on the choice of superderivation to orient the edge (resp., fix
and then reverse its orientation). In the ordered string ξ · · · ξ, let us bring next to each
other the symbols ξi and ξj from the copies P(i) and P(j) contained in the ith and jth
vertices. It is obvious that the action by ∂⃗/∂ξ on one such comultiple instead of the other
creates the sign factor (−). Doing this κ times counts the number of arrow reversions in
the body of Π-shaped orgraph, whence (−)κ.
Case Λ ⇄ Π. To avoid an agglomeration of symbols, we omit the letters ξ and display
their subscripts, thus indicating either which body edge it is (say A or B, A ≺ B) or where
it goes to (S0 := F to the argument f in the sink 0 and S1 := G to the argument g in the
sink 1). Remember that the edge letters A, B, F , and G are parity-odd by construction.
Without loss of generality, let us assume that in the string of 2n comultiples the four
rightmost are,
for the Λ-shaped orgraph: AB F G,
for the Π-shaped orgraph: A F B G.
We see that (AB) (F G) = −(AF ) (BG), whence we obtain the sought-for universal sign
factor (−) for any transitions between the different shapes Λ ⇄ Π (see Examples 7 and 8
in what follows). Now, the count of body edge reversals goes exactly as before.
Case Λ ⇄ Λ. There remains almost nothing to prove: in the above notation, we have
that AB F G = F G AB, hence no extra sign factor is produced when the wedge of two
edges directed to sinks is transported from one internal vertex to another.9
9This will presently be illustrated in Example 9 by using topologically nonisomorphic Λ-
shaped orgraphs in the set of admissible orientations of the pentagon wheel in the Kontsevich–
Willwacher cocycle γ5.
THE GRAPH ORIENTATION MORPHISM REVISITED 427
Example 7. Consider the r.-h.s. Q1: 6 = O⃗r(γ3)(P) of the Kontsevich tetrahedral flow,( 2 )
Q = (+1) · ︸ 0 1 2 4 ︷︷2 5 2 31: 62 S0 S1 I IV II V I III V ︸
Λ-sh[a(ped ) ( )]
− 3 · ︸ 0 3 1 4 ︷︷2 5 2 3S + 0 3 4 5 1 2 2 4 .0 I S1 IV II V I III V ︸ ︸ S0 I IV V ︷S︷1 II III V I ︸
minuend subtrahend
Using Rules 1 and 2, let us show why the sign which relates the Λ-shaped orgraph to
the skew-symmetrisation of Π-shaped orgraph is equal to (−); the count of multiplicities,
8 : 24 = 1 : 3, is standard.10
• In the minuend, which is a Π-shaped orgraph, Rule 1 contributes – for the edge
pairs (S0 I) (S1 IV ) · · · – with the first factor (−).
• In the course of transition Λ ⇄ Π to the minuend, one arrow in the body of orgraph
is reversed (namely, it is the edge I bridging the edges S0 and S1 issued from the
vertices 2 and 3), whence another minus sign, (−) = (−)1.
• The transition Λ ⇄ Π itself contributes with a universal sign (−), see Rule 2 again.
In total, we accumulate the sign factor (−) · (−) · (−) = (−), which indeed is the sign
that relates the skew-symmetric orgraphs in the flow Ṗ = O⃗r(γ3)(P).
Example 8. Consider two Λ-shaped terms and a Π-shaped term – in Fig. 3 – from the
{a} r4 {b} r6 {c} r5
r II }I II }I II 6 I= VII = VII = VII ~5 0 ?r 1 r 3 7 r r 5 6 r r 7VqIII BM  ) VqIII ?r 1 VqIIIB r4 )
VI  VI VI2 4
III r IX ] r V IIIN X N r IX X] r V III IX ] V
 N  X 
- 2 6 7 3 2 r - r 3
IV 0
BBN1 IV ?0 IV 1?
Fig. 3. Several Λ-shaped and Π-shaped terms from the result O⃗r(γ5) of orienting to Kontsevich
orgraphs the pentagon wheel graph in the cocycle γ5.
right-hand side Q5 = O⃗r(γ5)(P) of the flow determined by the Kontsevich–Willwacher
10The admissible Λ-shaped orientations of the tetrahedron are obtained by attaching the
wedge S0S1 to one of the four vertices and orienting the opposite face using one of two admissible
ways, so that 4 · 2 = 8. The Π-shaped Kontsevich graphs are obtained by selecting an edge from
six of them, directing it in one of the two ways, and orienting the opposite edge also in one of
two ways, whence 6 · 2 · 2 = 24.
428 A. V. KISELEV AND R. BURING
pentagon-whee(l cocycle γ5 ∈ ker d (see [5, 10] and [9]): )
Q5 = (+2) · 0 1 2S0 S1 V I (4 2 5 2 6 2 7 2 3I V II II V III III IX IV X V +( )+ 10 · 0 1 2 4 2 5 3 6 4 7 2 4S0 S1 IV X IX V I V I V II II III V III + )
+ 10 · 0 3 1 4 2 5 6 7 2 4 3 4S0 IV S1 X IX V II II I III V III V V I + · · · .
The first and second graphs, which we denote by {a} and {b}, are Λ-shaped whereas the
third graph {c} is Π-shaped; there are 167 terms in Q5, of which some are grouped in
pairs so that there are 91 bi-vector terms in total: of them, 15 orgraphs are Λ-shaped and
the rest, Π-shaped, undergo the skew-symmetrisation.
The transition {a} −7 → {c} employs the following sign matching factors:11
• Rule 1 for {c} having (S0 IV ) (S1 X) · · · contributes with (−).
• The number of arrow reversals in the body of orgraph in the course of transi-
tion {a} 7−→ {c} equals 4 (specifically, these are edges I, V , V II, and IX),
whence (−)4 = (+) by Rule 2.
• The transition Λ ⇄ Π between different shapes yields the universal sign factor (−).
In total, we have that for {a} 7−→ {c}, the overall sign is (−) · (+) · (−) = (+).
Counting the parity of three permutations of the edges S0 ≺ S1 ≺ I ≺ . . . ≺ X in the
graphs {a}, {b}, {c} is left as an exercise,12 cf. (5).
Example 9. The same-shape Λ ⇄ Λ-transition {a} ⇄ {b} in the pentagon-wheel
flow Q5 = O⃗r(γ5)(P), see previous example, amounts to the reversal of four arrows
in the body of orgraph (specifically, the edges IV {b} = 2-3, V {b} = 3-5, V I{b} = 4-5,
and IX{b} = 2-4). Rule 2 tells us at once that the orgraph multiplicities, 2 for {a} and
10 for {b}, are taken with equal signs (here, +2 : +10).
Rules 1 a∑nd 2 completely determine the signs of all Kontsevich orgraphs (countedwith their multiplicities) as long as they are obtained by orienting a given graph γa in a
cocycle γ = a ca · γa.
Finally, let γa and γb be topologically nonisomorphic unoriented graphs in a cocy-
cle γ ∈ ker d such that the differentials d(γa) and d(γb) have a least one nonzero unori-
ented graph in common.
Rule 3. The matc∑hing of signs for – clearly, topologically nonisomorphic – Kontsevichorgraphs which appear under (1) in the course of orienting different terms, γa and γb,
in a cocycle γ = s cs · γs ∈ ker d is provided by the cocycle itself, that is, by the
coefficients cs and respective edge orderings E(γa) and E(γb).
The signs in front of encodings of all the Kontsevich orgraphs are thus determined
for the linear combination O⃗r(γ)(P). Now, in each orgraph (and – independently from
other orgraphs), one can swap, at a price of the minus sign factor, the Left and Right
11This example of transition between orgraphs, {a} ⇄ {c} as well as {b} ⇄ {c}, is instructive
also in that the number of inversions, i.e. outgoing edge pairs Left < Right such that Left 
Right, does change parity in the course of {a}, {b} ⇄ {c} (specifically, from 5 and 3 to 2) but
does not contribute to the signs in front of the orgraph multiplicities.
12The respective numbers of elementary transpositions are 10, 24, and 26.
THE GRAPH ORIENTATION MORPHISM REVISITED 429
outgoing edges issued from any vertex. For example, this is done during the normalisation
of encodings, when an orgraph, given in terms of n pairs of n+2 target vertices, is realised
by using a minimal base-(n+ 2) positive number.13
Definition 3. An inversion is a situation where, at a vertex of a Kontsevich orgraph,
Left  Right in the overall edge ordering S0 ≺ S1 ≺ I ≺ II ≺ · · · .
Rule 4. For any Kontsevich orgraph Γ obtained from Γ0 by relabelling vertices and
possibly, for some of the internal vertices, swapping the consecutive order of two edges
issued from any such vertex, we have
(−)#inversions (Γ)
sign(Γ) =
(−)#inversions
· sign(Γ0).(Γ0)
Indeed, permutations of vertices induce parity-even permutations in the ordered string
of edges, whereas each elementary transposition –within a pair of edges referred to a
specific vertex – is parity-odd.
Remark 3. Apart from the ∂P -nontrivial linear scaling Ṗ = P, the only ∂P -(non)trivi-
al, nonlinear and proper ( 6≡ 0) flows Ṗ = Q(P) on spaces of Poisson structures which
are known so far (cf. [13]) are only those Q = O⃗r(γ) which are obtained by orienting
d-cocycles, that is, graphs γ ∈ ker d without multiple edges. In consequence, none of the
known orgraphs Q(P) contains any two-cycles • ⇄ •. All the more surprising it is that
orgraphs which do contain such two-cycles are dominant at the order ℏ4 in the expansion
of Kontsevich ⋆-product (presumably, so they are at higher orders of the parameter ℏ),
see [6] and [11, 18].
It would also be interesting to apply the technique of infinitesimal deformations,
Ṗ = O⃗r(γ)(P), of Poisson structures P by using graph complex cocycles γ and the
orientation morphism O⃗r, and the technique of formal deformations P 7−→ P[ℏ] of Poisson
structures by using the noncommutative ⋆-product (see [16, 18] and [11]) to deformations
and deformation quantisation of minimal surfaces which are specified by the Schild action
functional [1].
( )
A. T(he proo)f of Proposition 4. The Lie bracket of unoriented graphs γ1, E(γ1)
and γ2, E(γ2) on ni verti(ces and 2ni − 2 edges in each term) is, effectively,
[ γ1 ◦⃗]γ2 − γ1 ◦⃗γ2, E(γ1) ∧ E(γ2) . ( )
The commutator nid⃗/dε1, d⃗/dε2 (P) of the flows ddε (P) = Qi(P) = O⃗r(γi) P⊗ amo-i
unts to the consecutive insertions of the bi-vector Qi(P) instead of a copy of the bi-
vector P in[ one ve(rtex o)f the org(raph Q2−i. The claim is thatn n )] ( )( n +n −1)
O⃗r(γ1) P⊗
1 2
, O⃗r(γ2) P⊗ flows = O⃗r [γ1, γ2]graphs P
⊗ 1 2 . (7)
13The normalisation of orgraph encodings, which can be performed independently for different
graphs, can actually make it harder to count the number of arrow reverses in the course of
transitions which are controlled by Rule 2.
430 A. V. KISELEV AND R. BURING
Let us show that the minuend in the left-hand side is equal to the minuend in the
right-hand side, and the same for the subtrahends.
The left-hand side. The two orgraphs γi are oriented independently from each
other. The resulting orgraphs are built of wedges (such that the body edges get ori-
ented in all admissible ways). Every such Kontsevich orgraph either is automatically
skew-symmetric w.r.t. the content f , g of sinks (i.e. with respect to the ordered pair of
arguments of this bi-vector) or it is skew-symmetrised by the mechanism which already
worked in Corollary 6. Namely, the difference of orgraphs with the identical labelling of
ordered edges, S0 ≺ S1 ≺E(γ) or E(γ) ≺S0 ≺ S1, but with thecontentof the two sinkr r 
s
γ − r r = r r + r rγ  γ  γ@  
@
S0 ? ?S1 S0 ? ?S1 S0 ? ?S1 S0 	 @@RS1
f g g f f g f g
swapped is equal to the sum of orgraphs with the identical labelling of the body edges
but with the tails of the arrows S0 (heading to f) and S1 (heading to g) swapped.
The right-hand side: minuend. Summing over vertices and attachments, we re-
place a vertex v0 in the “victim” graph γ2 by the graph γ1 on n1 vertices and 2n1 − 2
edges. In the graph γ2, the edges which were incident to the blown-up vertex v0 are now
attached – in all possible ways – to some vertices of the inserted graph γ1. Note that if
two such edges, vu and v′u now connect two distinct vertices, v and v′, of the victim
graph γ2 with the same vertex u of the graph γ1, then one of the two edges precedes the
other with respect to the old edge ordering in the victim graph. Likewise, if two such
edges, vv ′0 and v v0 (for which the ordering was defined), now connect by vu and v′u′
two distinct vertices, v and v′, in the victim graph γ2 with two distinct vertices, u and u′
in the graph γ1, then the insertion γ1 ◦⃗ γ2 contains another graph in which the only
difference from the above is that the two edges vv0 and v′v0 become vu′ and v′u (but all
the other edges v0w in the body of the victim graph γ2 are attached to vertices of the
graph γ1 in the same way as they are in the former case).
In every term of the graph γ1 ◦⃗ γ2, consider the subgraph γ1; it remains intact in
the course of insertion ◦⃗. When the big graph γ1 ◦⃗ γ2 is oriented by O⃗r(·)(P), so is the
subgraph γ1. There were 2n1−2 edges in the (body of the) graph γ1; none of these edges,
still between two vertices of the (sub)graph γ1, can be oriented using any wedge issued
from a vertex of the outer graph γ2. This implies that exactly 2n1 − 2 arrows belonging
to the n1 bi-vector wedges are spent on orienting the body of the subgraph γ1 in the
big graph γ1 ◦⃗ γ2. Only two arrows leave the subgraph γ1: they head either to one or
two sinks of the orgraph O⃗r(γ1 ◦⃗ γ2) or to a vertex14 or two vertices in the rest of the
victim graph γ2, i.e. excluding the blown-up vertex v0. All the other edges which were of
14 −→It cannot be that two arrows, −u→v and u′v, from the subgraph γ1 head towards the same
vertex v in the victim graph γ2 because, with regards to the old topology of γ2 in which a
vertex v0 will be replaced by the graph γ1, this would mean a double edge v0v, hence γ2 was a
zero graph.
THE GRAPH ORIENTATION MORPHISM REVISITED 431
the form v0v in the graph γ2 now become arrows −v→u heading towards vertices u of the
subgraph γ1 in the big graph γ1 ◦⃗ γ2.
To establish the equality of the minuend in the left-hand side of (7) to the minuend
in the right-hand side of that formula, it remains to recall that by construction, all body
edges of the graph γ2 antecede those of γ1 (and vice versa: body edges of the graph γ1
precede those of the graph γ2), so that now, the ordering (1) (1)S0 ≺ S1 of the arrows which
are issued to the arguments of the bi-vector Q1(P) = O⃗r(γ1)(P) is always dictated by
the ordering E(γ2) ∧ (2) (2)S0 ∧ S1 of two edges from the (or)graph O⃗r(γ2).
The subtrahends in which the graph γ2 is inserted into some vertex of the graph γ1
are processed in an analogous way. The proof is complete.
B. The tetrah(edron: its Π-shape)d orientation (skew-symmetrized). The edge or-
derings (c)E(c) = S0∧S1∧ I ∧ . . .∧V I and E(b) = S0∧S1∧ I ∧ . . .∧
(b)
V I are related
5 120◦ 5
(c) 7S (c) So (b) So
 S  S  S
 S  S  S
V III SVI III VI SV VI III SV
 S  S  S
3  IV -Sw 4 −7 → 2 / I -S / IV -SQk  kQ  3 '  Qk QI Q Q

Q ?  II S1 S0 II Q ?

 IV S1 I Q
 II
Q+ 1? ? Q+ ? Q?+2 0 4 1
S0 S1 S0
0? 1? 0?
by the equalities (c) (b), (c) (b)S = S S = S , I(c) = IV (b), II(c) = I(b)0 1 1 0 , III(c) = V (b),
IV (c) = II(b), V (c) = V I(b), and V I(c) = III(b), whence one easily verifies that
E(c) = −(−)6 E(b),
the leading minus coming from the relabelling S0 ⇄ S1 and the rest from the permutation
of body edges. We conclude that the arithmetic sum of two Kontsevich orgraphs (b–
c) in Fig. 1 on p. 422 is the skew-symmetrisation of the Π-shaped orientation of the
tetrahedron γ3 by using arrow wedges.
Acknowledgements. The authors thank the Organisers of international workshop ‘Ho-
motopy algebras, deformation theory and quantization’ (16–22 September 2018 in Bedle-
wo, Poland) for helpful discussions and warm atmosphere during the meeting. The authors
are grateful to the anonymous referee for remarks and suggestions, and to G. Felder,
S. Gutt, and M. Kontsevich for helpful discussion. A part of this research was done
while RB was visiting at RUG and AVK was visiting at the IHÉS in Bures-sur-Yvette,
France and at the JGU Mainz (supported by IM JGU via project 5020 and JBI RUG
project 106552). The research of AVK was supported by the IHÉS (partially, by the
Nokia Fund).
432 A. V. KISELEV AND R. BURING
References
[1] J. Arnlind, J. Hoppe, M. Kontsevich, Quantum minimal surfaces, arXiv:1903.10792 [math-
ph]
[2] A. Bouisaghouane, R. Buring, A. Kiselev, The Kontsevich tetrahedral flow revisited,
J. Geom. Phys. 119 (2017), 272–285. (Preprint arXiv:1608.01710 [q-alg])
[3] F. Brown, Mixed Tate motives over Z, Annals of Math. 175 (2012), 949–976.
[4] R. Buring, A. V. Kiselev, On the Kontsevich ⋆-product associativity mechanism, PEPAN
Letters 14:2 Supersymmetry and Quantum Symmetries’2015 (2017), 403–407. (Preprint
arXiv:1602.09036 [q-alg])
[5] R. Buring, A. V. Kiselev, The orientation morphism: from graph cocycles to deformations
of Poisson structures, in: J. Phys.: Conf. Ser. 1194 (2019), Proc. 32nd Int. colloquium on
Group-theoretical methods in Physics: Group32 (9–13 July 2018, CVUT Prague, Czech
Republic), Paper 012017, 1–10. (Preprint arXiv:1811.07878 [math.CO])
[6] R. Buring, A. V. Kiselev, The expansion ⋆ mod ō(ℏ4) and computer-assisted
proof schemes in the Kontsevich deformation quantization, Experimental Math.
(2019), 54 p., in press, doi:10.1080/10586458.2019.1680463. (Preprint IHÉS/M/17/05,
arXiv:1702.00681 [math.CO])
[7] R. Buring, A. V. Kiselev, Formality morphism as the mechanism of ⋆-product associativity:
how it works, Collection of works Inst. Math., Kyiv 16:1 (2019), Symmetry & Integrability
of Equations of Mathematical Physics, 22–43. (Preprint arXiv:1907.00639 [q-alg])
[8] R. Buring, A. V. Kiselev, Universal cocycles and the graph complex action on homogeneous
Poisson brackets by diffeomorphisms, PEPAN Letters (in press) Supersymmetry and Quan-
tum Symmetries’2019 (2020), 8 pages. (Preprint IHÉS/M/19/20 (2019), arXiv:1912.12664
[math.SG])
[9] R. Buring, A. V. Kiselev, N. J. Rutten, The heptagon-wheel cocycle in the Kontsevich
graph complex, J. Nonlin. Math. Phys. 24 (2017), Suppl. 1 ‘Local & Nonlocal Symmetries
in Mathematical Physics’, 157–173. (Preprint arXiv:1710.00658 [math.CO])
[10] R. Buring, A. V. Kiselev, N. J. Rutten, Poisson brackets symmetry from the pentagon-
wheel cocycle in the graph complex, Physics of Particles and Nuclei 49:5 Supersymmetry
and Quantum Symmetries’2017 (2018), 924–928. (Preprint arXiv:1712.05259 [math-ph])
[11] A. S. Cattaneo, G. Felder, A path integral approach to the Kontsevich quantization formula,
Comm. Math. Phys. 212:3 (2000), 591–611. (Preprint arXiv:q-alg/9902090)
[12] C. Jost, Globalizing L∞-automorphisms of the Schouten algebra of polyvector fields, Dif-
ferential Geom. Appl. 31:2 (2013), 239–247. (Preprint arXiv:1201.1392 [q-alg])
[13] A. V. Kiselev, Open problems in the Kontsevich graph construction of Poisson bracket
symmetries, in: J. Phys.: Conf. Ser. 1416 (2019), Proc. XXVI Int. conf. ‘Integrable Systems
& Quantum Symmetries’ (8–12 July 2019, CVUT Prague, Czech Republic), Paper 012018,
1–8. (Preprint arXiv:1910.05844 [math-ph])
[14] M. Kontsevich, Feynman diagrams and low-dimensional topology, in: First Europ. Congr.
of Math. 2 (Paris, 1992), Progr. Math. 120, Birkhäuser, Basel, 1994, 97–121.
[15] M. Kontsevich, Homological algebra of mirror symmetry, in: Proc. Intern. Congr. Math. 1
(Zürich, 1994), Birkhäuser, Basel, 1995, 120–139.
[16] M. Kontsevich, Formality conjecture, in: Deformation theory and symplectic geometry
(Ascona 1996), D. Sternheimer, J.Rawnsley and S.Gutt (eds.), Math. Phys. Stud. 20,
Kluwer Acad. Publ., Dordrecht, 1997, 139–156.
[17] M. Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48:1
THE GRAPH ORIENTATION MORPHISM REVISITED 433
(1999), Moshé Flato (1937–1998), 35–72. (Preprint arXiv:math.QA/9904055)
[18] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66:3
(2003), 157–216. (Preprint arXiv:q-alg/9709040)
[19] M. Kontsevich, Derived Grothendieck–Teichmüller group and graph complexes [after
T. Willwacher], in: Séminaire Bourbaki (69ème année, 2016–2017), Janvier 2017, No. 1126
(2017), 183–212.
[20] N. J. Rutten, A. V. Kiselev, The defining properties of the Kontsevich unoriented graph
complex, in: J. Phys.: Conf. Ser. 1194 (2019), Proc. 32nd Int. colloquium on Group-
theoretical methods in Physics: Group32 (9–13 July 2018, CVUT Prague, Czech Repub-
lic), Paper 012095, 1–10. (Preprint arXiv:1811.10638 [math.CO])
[21] T. Willwacher, M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie
algebra, Invent. Math. 200:3 (2015), 671–760. (Preprint arXiv:1009.1654 [q-alg])
[22] T. Willwacher, M. Živković, Multiple edges in M. Kontsevich’s graph complexes and com-
putations of the dimensions and Euler characteristics, Adv. Math. 272 (2015), 553–578.
(Preprint arXiv:1401.4974 [q-alg])

Chapter 19
Universal cocycles and the graph
complex action on homogeneous
Poisson brackets by diffeomorphisms
This chapter is based on the peer-reviewed journal publication R. Buring and A.V. Kise-
lev, Physics of Particles and Nuclei Letters 17:5 Supersymmetry and Quantum Symme-
tries 2019, 707–713, 2020. (Preprint arXiv:1912.12664 [math.SG] – 8 p.) This result
was presented to M. Kontsevich at the IHÉS in December 2019.
Commentary. In reference to Part I of the dissertation, the material of this chapter
is used in Chapter 2 (especially §2.6), Chapter 7 (§7.1), and Chapter 8. The claim in
the end of this chapter (see Proposition 5 below) about the Poisson non-triviality—in
the class of differential polynomials—for the restriction of tetrahedral flow to the class
of rescaled Nambu–Poisson structures in 3D was false, as seen from the next chapter
(and from §7.1.5 in Part I). Besides, let us give a counterexample (in 2D) when a Poisson
structure (with non-polynomial coefficients) is homogeneous with respect to a vector field
(with polynomial coefficients) but its tetrahedral flow is not homogeneous in that way.
Counterexample. On R2 with coordinates x, y, let P = x2y2 exp(1/x) ∂x ∧ ∂y and
V = −x2 ∂x − y2 ∂y. Then [[V, P ]] = P but [[V,Qtetra(P )]] is not proportional to Qtetra(P ).
Indeed, we have
Qtetra(P ) = −32(6x− 1)x2y5 exp(4/x) ∂x ∧ ∂y
and
[[V,Q 2tetra(P )]] = 32(6x + 18xy − 24x− 3y + 4)x2y5 exp(4/x) ∂x ∧ ∂y,
where 6x2 + 18xy − 24x− 3y + 4 is not divisible by 6x− 1.
435
Universal cocycles and the graph complex action on
homogeneous Poisson brackets by diffeomorphisms
R. Buring,∗,¶ A.V.Kiselev† ‡ §,£
E-mail: ¶rburing@uni-mainz.de, £A.V.Kiselev@rug.nl
Abstract
The graph complex acts on the spaces of Poisson bi-vectors P by infinitesimal sym-
metries. We prove that whenever a Poisson structure is homogeneous, i.e. P = LV⃗ (P)
w.r.t. the Lie derivative along some vector field V⃗ , but not quadratic (the coefficients of
P are not degree-two homogeneous polynomials), and whenever its velocity bi-vector
Ṗ = Q(P), also homogeneous w.r.t. V⃗ by LV⃗ (Q) = nQ whenever Q(P) = O⃗r(γ)(P
⊗n)
is obtained using the orientation morphism O⃗r from a graph cocycle γ on n vertices
and 2n − 2 edges, then the 1-vector n−1X⃗ = O⃗r(γ)(V⃗ ⊗ P⊗ ) is a Poisson cocycle.
Its construction is uniform for all Poisson bi-vectors P satisfying the above assump-
tions, on all finite-dimensional affine manifolds M . Still, if the bi-vector Q 6≡ 0 is
exact in the respective Poisson cohomology, so there exists a vector field Y⃗ such that
Q(P) = [[Y⃗,P]], then the universal cocycle X⃗ does not belong to the coset of Y⃗ mod
ker[[P, ·]]. We illustrate the construction using two examples of cubic-coefficient Poisson
brackets associated with the R-matrices for the Lie algebra gl(2).
Introduction. Bi-vector cocycles Q n(P) = O⃗r(γ)(P⊗ ) ∈ ker[[P , ·]] are obtained by Kon-
tsevich’s graph orientation morphism O⃗r from graph cocycles γ on n vertices and 2n − 2
edges in a way which is uniform for all finite-dimensional affine Poisson manifolds (M r,
P). The (non)triviality of cocycles Q(P) in the second Poisson cohomology w.r.t. the dif-
ferential ∂P = [[P , ·]] remains an open problem, twenty-five years after the discovery of the
graph complex and orientation morphism (see [11]). In all the Poisson geometries probed
so far, the known infinitesimal symmetries Ṗ = Q(P) of the Jacobi identity 1 [[P ,P ]] = 0
2
are ∂P-exact: there always exists a vector field Y⃗ such that Q(P) = [[Y⃗,P ]]. The evolution
P(ε = 0) 7−→ P(ε > 0) of the tensor P then amounts to its reparametrisations under the
diffeomorphisms of Poisson manifold which are induced by the shifts along the integral tra-
jectories of the vector field Y⃗. This is why, instead of producing new Poisson brackets from a
given one, the Kontsevich graph flows on the spaces of Poisson bi-vectors induce (non)linear
∗Mathematical Institute, Johannes Gutenberg University of Mainz, Staudingerweg 9, D-55128 Germany.
A part of this research was done while R.B. was visiting at the IHÉS, supported by MI JGU project 5020.
†Bernoulli Institute for Mathematics, Computer Science & Artificial Intelligence, University of Groningen,
P.O. Box 407, 9700 AK Groningen, The Netherlands.
‡Current address: Institut des Hautes Études Scientifiques (IHÉS), Le Bois–Marie, 35 route de Chartres,
Bures-sur-Yvette, 91440 France.
§Supported by BI RUG project 135110 (Groningen) and the IHÉS (in part, by the Nokia Fund).
436
diffeomorphisms of the base manifold M , although no more than its affine structure was the
initial assumption and no possibility of smooth coordinate reparametrizations was presumed.
For a much used class of (scaling-)homogeneous Poisson bi-vectors P = LV⃗ (P), we obtain
an explicit formula, n−1X⃗ = O⃗r(γ)(V⃗ ⊗P⊗ ), of a 1-vector cocycle X⃗(γ, V⃗ ,P) ∈ ker[[P , ·]] which
is built from the graph cocycles γ uniformly for all homogeneous Poisson bi-vectors P on
affine manifolds M r<∞. The cocycle X⃗ is however not necessarily a 1-vector representative
of the coset Y⃗ mod {Z⃗ ∈ ker[[P , ·]]} which would trivialise the value Q(P) = [[Y⃗,P ]] of
Kontsevich’s symmetries at homogeneous Poisson structures. Indeed, the Poisson cocycle
Q(P) can be, we show, a nonzero bi-vector on M r, whereas the bi-vector [[X⃗,P ]] is identically
zero on M r by construction. We contrast the formulas of universal cocycles X⃗(γ, V⃗ ,P) and
trivialising vector fields Y⃗ for nonzero symmetries Ṗ = O⃗r(γ)(P) by two examples, namely,
using cubic-coefficient Poisson brackets associated with the R-matrices for gl(2).
This paper is organized as follows. In §1 we recall elements of Poisson cohomology theory
in the context of Kontsevich’s universal deformations of bi-vectors by using the unoriented
graph cocycles. In §2 we phrase the notion of structures which are homogeneous w.r.t. a
1-vector field, and we prove the main theorem. Finally, we illustrate the result (cf. [10]).
1. Poisson cohomology and the graph complex. A Poisson bracket {·, ·}P on a real
manifold M is a bi-linear skew-symmetric∑bi-derivation which takes C∞(M) × C∞(M) →
C∞(M) and satisfies the Jacobi identity 1 σ∈S {{σ(f), σ(g)}P , σ(h)}P = 0 for any f, g, h ∈2 3
C∞∑(M). The fact that both the arguments f, g and their bracket {f, g}P∑are scalars dictatesthe tensor transformation law of the c∑omponents P ij of a bi-vector P = P iji,j (x)∂i⊗∂j =
1 ij
i,j P (x)(∂i ⊗ ∂j − ∂j ⊗ ∂ ) =
1 ij
i i,j P ∂i ∧ ∂j whenever the structure is referred to a2 2
system of coordinates x = (x1, . . . , xr) and ∂i = ∂∧/∂xi is a shorthand notation.
The calculus on the space of multivectors Γ( • TM) ∼= C∞(ΠT ∗M) is simplified if one
uses the parity-odd coordinates ξi along the directions dxi in the fibres of the cotangent
bundle T ∗M over points a ∈ M (which are parametriz∑ed by xi). The symbol ξi thus
corresponds to ∂/∂xi dual to dxi, and bi-vectors are P = 1 i,j P ijξiξj, so that {f, g}P(a) =2
(f)∂/⃗∂xµ · ∂⃗/∂ξµ (P) ∂/⃗∂ξν · ∂⃗/∂xν(g); here, both the coefficients P ij and derivatives ∂/∂xk
are evaluated at the point a ∈ M as in the left-hand side.1
The space of multivectors is endowed with the parity-odd Poisson bracket [[·, ·]] (the
Schouten bracket, or antibracket) of own degree −1. For arbitrary multivectors P ,Q, the
formula is [[P ,Q]] = (P)∂/⃗∂ξi·∂⃗/∂xi(Q)−(P)∂/⃗∂xi·∂⃗/∂ξi(Q); in particular, [[X⃗, Y⃗]] = [X⃗, Y⃗] is
the usual commutator of vector fields X⃗, Y⃗ onM . The Schouten bracket [[·, ·]] is shifted-graded
skew-symmetric: [[Q,P ]] = −(−)(|P|−1)·(|Q|−1)[[P ,Q]] for P and Q grading-homogeneous. This
is why, unlike the tautology [[X⃗, X⃗]] ≡ 0, the equation [[P ,P ]] = 0 is a nontrivial restriction for
bi-vectors P , containing the tri-vector in the l.-h.s. of the Jacobi identity 1 [[P ,P ]](f, g, h) = 0
2
for the bracket {f, g}P = [[[[f,P ]], g]]. The Schouten bracket itself satisfies the graded Jacobi
identity [[P , [[Q,R]]]] − (−)(|P|−1)·(|Q|−1)[[Q, [[P ,R]]]] = [[[[P ,Q]],R]] with P and Q grading-
homogeneous. This identity implies that for Poisson bi-vectors P , their adjoint action by
∂P = [[P , ·]] is a differential of degree +1 on the space of multivectors on M . The Poisson
differential ∂P gives rise to the Poisson cohomology H iP(M) of the manifold M (see [13]).2
1The dot · denotes the coupling of iterated variations of the objects f , P, and g with respect to the
canonically conjugate variables xi and ξi, see [9] and references therein.
2The group H0 ∞P(M) spans the Casimirs, i.e. the functions which Poisson-commute with any f ∈ C (M);
the group H1P(M) consists of vector fields which preserve the Poisson structure but do not amount to the
Hamiltonian vector fields X⃗h = [[P, h]]; the second group H2P(M) 3 Q contains infinitesimal symmetries
P 7→ P + εQ+ ō(ε) of Poisson bi-vectors, whereas the next group H3P(M) stores the obstructions to formal
437
If a bi-vector Q = [[X⃗,P ]] is a trivial Poisson cocycle, then it certainly is an infinitesimal
symmetry of the Jacobi identity 1 [[P ,P ]] = 0. But the infinitesimal change [[X⃗,P ]] of the
2
tensor P then amounts to its reparametrisation under the infinitesimal change of coordinates
x′(x) ⇄ x(x′) along the integral trajectories of the vector field X⃗ on the manifold M . The
following fact is true for all multivectors (regardless of the concept of Poisson cohomology).
Proposition 1. Let a ∈ M be a point of an r-dimensional manifold and X⃗ ∈ Γ(TM)
be a vector field on it. For every ε ∈ I ⊆ R such that there is the integral trajectory
bringing b(−ε) := exp(−εX⃗)(a) to a by the (+ε)-shift, and for any choice of the r-tuple
x = (x1, . . . , xr) of local coordinates in a chart Uα around a ∈ M (and for |ε| small enough
for the points b(−ε) to not yet run out of the chart U 3α), introduce a new parametrization
for the point a by using the new r-tuple x′. By definition, put x′(a) :∣= x(b(−ε)). Let Ω beany multi-vector field near a on M . Under the reparametrization x′(x), the speed at which
the components o∣f Ω at the point a change in ε, as ε → 0, equals
d ∣ Ω(a) = [[X⃗,Ω]](a).
dε ε=0
In particular, a 1-∣vector field Y⃗ near a would change at a as fast as its commutator with thevector field X⃗: d Y⃗(a) = [X⃗, Y⃗](a).
dε ε=0
The geography of the set of Poisson structures near a given bracket {·, ·}P on a given manifold
M r is, generally speaking, unknown. All the more it was a priori unclear whether Poisson bi-
vectors P , irrespective of the dimension r ⩾ 3, topology of M r, etc., can be infinitesimally
shifted by Poisson 2-cocycles Q(P), the construction of which would be universal for all
P . The discovery of the graph complex in 1993–94 allowed Kontsevich to state (in [11])
the affirmative answer to the above question. Namely, the graph orientation morphism
O⃗r(·)(P) : ker d 3 γ 7→ Q(P) ∈ ker ∂P takes graph cocycles on n vertices and 2n−2 edges in
each term (e.g., the tetrahedron, cf. [1, 3, 5, 6]) to Poisson cocycles whenever the bi-vector
P itself is Poisson. Willwacher [15] revealed that the generators of Drinfeld’s Grothendieck--
Teichmüller Lie algebra grt are source of at least countably many such cocycles in the vertex-
edge bi-grading (n, 2n− 2); these cocycles are marked by the (2ℓ+1)-wheel graphs (e.g., see
[6, 7]). Brown proved in [2] that, under the Willwacher isomorphism grt ∼= H0(Gra) these
graph cocycles with wheels generate a free Lie subalgebra in grt, which means effectively that
the iterated commutators of already known cocycles – under the bracket in the differential
graded Lie algebra Gra of graphs – would never vanish. The commutator of two cocycles
is a cocycle by the Jacobi identity. All of them again being of the bi-grading (n, 2n − 2),
these graph cocycles determine countably many infinitesimal symmetries of a given Poisson
bi-vector P ; the construction is uniform for all the geometries (M r,P).
Lemma 2. For a given Poisson bi-vector P, the graph orientation mapping O⃗r(·)(P) : ker d 3
γ 7→ Q(P) ∈ ker ∂P is a Lie algebra morphism that takes the bracket of two cocycles in bi-
grading (n, 2n− 2) to the commutator [ d , d ](P) of two symmetries d (P) = Q (P).4
dε1 dε i2 dεi
By construction, the components of universal symmetry bi-vectors Q(P) are differential
polynomials w.r.t. the components P ij of the Poisson bi-vector P that evolves. It can
of course be that a grap∑h flow Ṗ = O⃗r(γ)(P) vanishes identically over the manifold M
r
integration P 7→ P(ε) = P + kk⩾1 ε Q(k) of infinitesimal symmetries Q = Q(1) to Poisson bi-vector formal
power series satisfying [[P(ε),P(ε)]] = 0.
3Actually, this is a way to construct new coordinates for all points of M near a in Uα, i.e. not only those
which lie on a piece of the integral trajectory of X⃗ passing through a.
4By Brown [2], the commutator does in general not vanish for Willwacher’s odd-sided wheel cocycles.
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whenever Q is evaluated at a particular class of Poisson structures P .5 Nevertheless, there
is no mechanism which would force a given Kontsevich’s graph flow to vanish at all Poisson
structures on all manifolds of all dimensions.6 Independently, it remains an open problem
(cf. [10]) whether there is a Poisson manifold (M r,P) and a graph cocycle γ such that the
Poisson cohomology class of Q(P) := O⃗r(γ)(P) would be nontrivial in H2P(M). In other
words, for all the shifts Q = O⃗r(γ) and all Poisson bi-vectors tried so far, the Poisson
coboundary equation Q(P) = [[X⃗,P ]] did have vector field solutions X⃗ on the manifolds M .
Remark 1. Obtained from the graphs γ ∈ ker d, the symmetriesQ(P) = O⃗r(γ)(P) ∈ ker[[P , ·]]
are independent of a choice of local coordinates xi (hence ξi) on a chart if, the Kontsevich
construction requires, the manifoldM r is endowed with an affine structure: all the coordinate
transformations amount to x′ = Ax + b⃗ with a constant (over the intersection of charts)
Jacobian matrix A. The parity-odd fibre variables are transformed using the inverse Jacobian
matrix, i′ξ ′i = Ai ξi′ , making sense of the couplings ∂/⃗∂ξi · ∂⃗/∂xi which decorate the oriented
edges of Kontsevich’s graphs after the morphism O⃗r works (see [3, 11]). The problem of
Poisson cohomology class (non)triviality for the Kontsevich infinitesimal symmetries Ṗ =
Q(P) ∈ ker[[P , ·]] thus acquires two diametrally opposite interpretations:
1 (as in [11]). The Poisson manifold M r<∞ is equipped with both the smooth and affine
structures.7 By definition, two Poisson bi-vectors are equivalent, P1 ∼ P2, if they are related
by a diffeomorphism of the manifold M : using its smooth structure, the diffeomorphism
identifies points in two copies of M , then relating the Poisson tensors by local coordinate
reparametrizations near the respective points. The affine structure on M is now used to run
the Kontsevich flows in two initial value problems Ṗi(ε) = Q(Pi(ε)), Pi(ε = 0) = Pi. The
Poisson triviality Q(P(ε)) = [[X⃗(ε),P(ε)]] would relate either of bi-vectors Pi(ε) back to the
Cauchy datum Pi by diffeomorphisms (as long as |ε| is small enough). Consequently, the
Poisson bi-vectors P1(ε) ∼ P2(ε) do not run out of the old equivalence class. In conclusion,
the goal is to produce essentially new Poisson brackets by using a nontrivial cocycle Q,
two given structures on the manifold M r, and its diffeomorphism. No examples of nontrivial
action, so that P2(ε) 6∼ Pi 6∼ P1(ε) at ε > 0, have ever been produced since 1996 (see [7, 11]).
2 (as in [10]). The Poisson manifold M r<∞ is equipped only with an affine structure.
The countably many grt-related graph cocycles on n vertices and 2n− 2 edges in every term
(the tetrahedron, the pentagon-wheel cocycle, etc., see [6, 15]) generate a noncommutative
Lie algebra of infinitesimal symmetries Q(P) = O⃗r(γ)(P) for a given Poisson structure
P . Consider the extreme case when all the cocycles Q(P) ∈ ker[[P , ·]] are exact in the
5Example. So it is for the Kontsevich tetrahedral flow ([11] and [1]) evaluated at the Kirillov–Kostant
linear Poisson brackets on the duals g∗ of Lie algebras because in every term within the cocycle Q(P) under
study, at least one copy is P is differentiated at least twice with respect to the global coordinates on g∗.
6Example. The Poisson bi-vectors P = da1 ∧ . . . ∧ dam/ dvol(Rm+2) of Nambu type with arbitrary
Casimirs a , . . . , a ∈ C∞(Rm+21 m ) and an arbitrary density in the volume element can have polynomial
components Pij ∈ R[x1, . . . , xm+2] of degrees as high as need be w.r.t. the global Cartesian coordinates
xα on the vector space Rm+2. The universal symmetries Ṗ = O⃗r(γ)(P) obtained from Kontsevich’s graph
cocycles deform the symplectic foliation (which is given in Rm+2 by the intersections of the level sets for the
Casimirs a1, . . . , am) in a regular way on an open dense subset of Rm+2, so that the symmetries Ṗ = Q(P)
preserve this Nambu class of Poisson brackets: the flows force the evolution of the Casimirs and the volume
density. Its integrability is an open problem; by Lemma 2 and [2], the evolutions induced by different graph
cocycles do not commute.
7On the circle S1, the affine coordinate ‘angle’ is obvious whereas the smooth structure is used in the
realm of Poincaré topology. A smooth atlas is always available for the spheres Sr, but not for all r ∈ N
would the r-dimensional sphere admit an affine structure.
439
cohomology group H2P(M) w.r.t. the Poisson differential ∂P . This assumption gives rise to
the countable set of vector fields Y⃗(γ,P) on M such that Q(P) = [[Y⃗,P ]]. (Some of these
vector fields can be identically zero over M .) But if at least one such vector field is not
constant w.r.t. the affine structure on M , then the shifts along its integral trajectories are
nonlinear diffeomorphisms of M . The evolution of bi-vector P is Ṗ = Q(P) = [[Y⃗,P ]] or
similarly, Ω̇ = [[Y⃗,Ω]] for any multi-vector Ω on M (see Proposition 1); this evolution is
now seen as mutlivectors’ response to the diffeomorphism whose construction refers only to
the simple, affine local portrait of M . Summarizing, the store of flows O⃗r(γ)(P) from the
grt-related graph cocycles γ could be enough to approximate arbitrary smooth vector fields
on M r, that is, imitate its smooth structure. Whether this theoretical possibility is actually
realised in relevant Poisson models is an open problem.
The Kontsevich symmetry construction is, therefore, either a generator of new Poisson
brackets or the mechanism that provides diffeomorphisms of the underlying manifold.
2. Homogeneous Poisson structures. By definition, a bi-vector P on a manifold M is
called homogeneous (of scale λ) with respect to a vector field V⃗ on M if [[V⃗,P ]] = λ · P .
Example 1. Let M = Rr be a vector space (only linear reparametrizations x′ = Ax are
allowed, so that the polynomial degree∑s of monomials in the ring R[x1, . . . , xr] is well defined).
Introduce the Euler vector field = rE⃗ ii=1x ∂/∂xi, and let all the components P ij of a bi-
vector P be homogeneous polynomials of degree d in the variables xi. Then we have that
[[E⃗,P ]] = (d−2) ·P , which means that P is homogeneous of scale d−2 w.r.t. the Euler vector
field E⃗. In particular, if d 6= 2 (i.e. if the coefficients of bi-vector P are not quadratic), then
we set V⃗ = (d− 2)−1 · E⃗ and from the equality P = [[V⃗,P ]] we obtain that the same bi-vector
P has homogeneity scale λ = 1 w.r.t. the multiple V⃗ of the Euler vector∑field E⃗ on Rr.
Example 2. Under the same assumptions, suppose further that γ = a caγa is a graph
cocycle with n ve∑rtices and 2n − 2 edges in every term γa (e.g., take the tetrahedron).Orient the ordered (by First ≺ · · · ≺ Last) edges in every γa using the edge decoration
operators r (i) µ (j)∆⃗ µij = µ=1(∂⃗/∂ξµ ⊗ ∂⃗/∂x(j) + ∂⃗/∂ξµ ⊗ ∂⃗/∂x(i)). By placing a copy of bi-
vector P = 1Pkl(x)ξkξl in each vertex v(i) of γa and taking the sum (over the graph index2
a) of products of the content of vertices in γa after all the edge operators ∆⃗ij work, we
obtain8 the bi-vector Q(P) := O⃗r(γ)(P). Then the coefficients of the bi-vector Q(P) are
homogeneous polynomials of degree n · d − (2n − 2) with respect to x1, . . ., xr, so that
[[E⃗,Q(P)]] = n(d − 2)Q(P). In particular, if d 6= 2, then [[V⃗,Q(P)]] = n · Q(P), whereas
quadratic-coefficient bi-vectors P (with d = 2) are deformed within their subspace by the
quadratic bi-vectors Q(P) which are obtained from the Kontsevich graph cocycles.
Lemma 3. If a Poisson bi-vector P = [[V⃗,Q(P)]] is homogeneous and Q(P n) = O⃗r(γ)(P⊗ )
is built from a graph cocycle γ on n vertices, now containing a copy of P in each vertex,
then the bi-vector Q(P) is also homogeneous: [[V⃗,Q(P)]] = n · Q(P), so that its scale is n.9
Remark 2 ([14, Rem. 4.9]). Consider a Nambu-type Poisson bi-vector P = da/dxdydz on R3
with Cartesian coordinates x, y, z; here a ∈ R[x, y, z] is a weight-homogeneous polynomial
8We refer to the original paper [11] and to [3] for illustrations and discussion how the graph orientation
morphism works in practice.
9The proof amounts to the Leibniz rule: let us inspect how fast the bi-vector Q(P), which by construction
is a homogeneous differential polynomial of degree n in P, evolves along the vector field V⃗.
440
with an isolated singularity at the origin10, so that (w(x) · x∂/∂x + w(y) · y∂/∂y + w(z) ·
z∂/∂z)(a) = w(a) · a. Then a vector field V⃗ with polynomial components satisfying the first-
order PDE P = [[V⃗,P ]] exists if and only if 11 the weight degree w(a) of the polynomial a is
not equal to the sum w(x) + w(y) + w(z) of weight degrees for the variables x, y, z.12
Summarizing, the homogeneity assumption about bi-vectors P is restrictive; it is not
always satisfied in Poisson models.
Theorem 4. Let (M,P) be a∑n affine finite-dimensional real Poisson manifold with P =
[[V⃗,P ]] homogeneous. Let γ = a ca ·γa be a graph cocycle consisting of unoriented graphs γa
over n vertices and 2n−2 edges (with a fixed ordering of edges in each γa). Then the 1-vector
X⃗(γ, V⃗,P n−1) = O⃗r(γ)(V⃗⊗ P⊗ ), which is obtained by representing each edge i−−j with the
operator ∆⃗ij and by (graded-)symmetrizing over all the ways σ ∈ Sn to send the n-tuple
V⃗⊗ P⊗n−1 into the n vertices in each γa, is a Poisson cocycle: X⃗ ∈ ker[[P , ·]].13
The vector field X⃗ is defined up to adding arbitrary Poisson 1-cocycles Z⃗ ∈ ker[[P , ·]].
Proof. The expansion n0 = O⃗r(dγ)(V⃗⊗P⊗ ) for γ ∈ ker d goes along the lines of [11] and [3,
7, 8], b∧ut the (n+1)-tuple of multivectors now( contains one 1-vector and only n copies of thePoisson bi-vector P . By assumption, dγ = 0 ∈ Gra; recall that O⃗r(0)(any mu)ltivectors) =
0 ∈ Γ( • TM). This zero l.-h.s. equates 0 = π ◦⃗O⃗r(γ)−(−)(−1)·(−N) nS O⃗r(γ)◦⃗π ⊗ 14S (V⃗⊗P ).
The appointmen(t of graded (m)ulti)vectors into the vertices of dγ (hence, into the argu-ment slots of the endomorphism O⃗r(dγ)) is achieved by the graded symmetrization usingn
((n+1)!)−1 O⃗r(dγ) ±σ(V⃗⊗P⊗ ) . Fortunately, the field V⃗ is the only parity-odd object, so
its transpositions with the parity-even bi-vectors P produce no sign factor: these ± are all
+. Likewise, the n! permutations of n indistinguishable copies of P leave only n + 1 from
(n+1)! in the denominator; to get rid of it, let us multiply by n+1 both sides of the equality
n
0 = O⃗r(dγ)(V⃗ ⊗ P⊗ ). The symmetrization thus amounts, by the linearity of O⃗r(γ), to its
evaluation at the sum of arguments, V⃗ ·Pn+P · V⃗ ·Pn−1+ . . .+Pn · V⃗, in which the ordering
of (multi)vectors now matches an arbitrary fixed enumeration of the( vertices.
The rest of the proof is standard.15 There remains 0 = O⃗r(γ) π (V⃗,P) · Pn−1S ) + P ·
10The Milnor number is the dimension dimR R[x, y, z]/(∂a/∂x, ∂a/∂y, ∂a/∂z) – here, < ∞ by assumption.
11This means that not all Nambu-type Poisson bi-vectors P = da/dxdydz are homogeneous w.r.t. a vector
field V⃗ with polynomial components; the PDE P = [[V⃗,P]] with polynomial coefficients and unknown V⃗ can
in principle admit non-polynomial solutions.
12Example. If the weights of (x, y, z) are (1, 1, 1) and a = 13 (x3 + y3 + z3) is cubic-homogeneous, then
the components of Poisson bi-vector P are quadratic and (by the above and also by [12, Exerc. 4.5.7c]) not
of the form P = [[V⃗,P]] for any polynomial-coefficient vector field V⃗. The non-existence of a solution V⃗ with
smooth non-polynomial coefficients is a separate problem.
13Open problem (for P homogeneous and Poisson). Is the universal 1-vector field X⃗(γ, V⃗,P) ∈ ker ∂P
Hamiltonian, i.e. X⃗ = [[P, h]] for h ∈ C∞(M) or at least, X⃗ = P η for a maybe not exact 1-form η on M ?
14Here, πS is the graded-symmetric Schouten bracket (so πS(F,G) = (−)|F |−1[[F,G]]), the graph insertion
◦⃗ into vertices is now the endomorphism insertion into argument slots, |πS | = −1, and N = 2n − 2 is the
even number of edges in γ, hence minus the even number of ∂/∂ξµ in the edge operators ∆⃗ij making O⃗r(γ).
15We have 0 = O⃗r(γ)(πS(V⃗,P),Pn−1) + O⃗r(γ)(πS(P, V⃗),Pn−1) + O⃗r(γ)(πS(P,P), V⃗,Pn−2) +
. . . + O⃗r(γ)(π (P,P),Pn−2S , V⃗) + O⃗r(γ)(V⃗, πS(P,P),Pn−2) + O⃗r(γ)(P, πS(V⃗,P),Pn−2) +
O⃗r(γ)(P, πS(P, V⃗),Pn−2) + O⃗r(γ)(P, πS(P,P), V⃗,Pn−3) + . . . + O⃗r(γ)(P, πS(P,P),Pn−3, V⃗) +
. . . (the Schouten bracket πS pas)ses along the slots towards the end) + O⃗r(γ)(V⃗,P
n−[2, πS((P,P)) + . . . +
O⃗r(γ)(Pn−2, V⃗, πS(P,P))])+ O⃗r(γ)(P
n−1, πS(V⃗,P))+ O⃗r(γ)(Pn−1, π (P, V⃗)) − ((−)N · π O⃗r(γ)(V⃗ · Pn−1S S +
P · V⃗ · Pn−2 + . . .+ Pn−1 · V⃗),P + πS(O⃗r(γ)(Pn), V⃗) + πS(V⃗, O⃗r(γ)(Pn)) + πS P, O⃗r(γ)(V⃗ · Pn−1 + P · V⃗ ·
Pn−2 + . . . + Pn−1 · V⃗) . For P Poisson, πS(P,P) = 0, so we exclude all such terms ([4]). The remaining
graded-symmetric Schouten brackets πS contain a bi-vector as one of the arguments, hence those can be
swapped at no sign factor; all doubles, so let us divide by 2.
441
π)(V⃗,)P) ·Pn− ) [ ( (2)+ . . .+Pn−]1 ·π (V⃗,P) −(−)N π O⃗r(γ) V⃗ ·Pn−1 n−2 n−1S S S +P · V⃗ ·P + . . .+P ·
V⃗ ,P +πS(O⃗r(γ)(Pn), V⃗) . By the homogeneity assumption, πS(V⃗,P) = (−)1−1[[V⃗,P ]] = P ,
and by construction, O⃗r(γ)(Pn) = Q(P), whence the minuend equals n·Q(P). By Lemma 3,
the graph flow is also homogeneous: [[V⃗,Q(P)]] = λ · Q(P) with the vertex count λ = n. We
obtain the equality
(−)2n−
( )
2 · [[O⃗r(γ) V⃗ · Pn−1 + P · V⃗ · Pn−2 + . . .+ Pn−1 · V⃗ ,P ]] =
= n · Q(P)− (−)2n−2λ · Q(P) = (n− (−)even n) · Q(P) ≡ 0.
We conclude that the 1-vector n−1X⃗ := O⃗r(γ)(V⃗⊗ P⊗ ) lies in ker[[P , ·]].16
Example 3. Take the Lie algebra gl2(R) with its four-dimensional vector space stru(cture);
denote by x, y, z, v the Cartesian coordinates. Consider the R-matrix ( x y 0 yz v ) →7 −z 0
known from [12]; the standard construction then yeilds the Poisson bi-vector in the algebra
of coordinate functions, P = (x2y + y2z) ∂ ∧ ∂ + (x2z + yz2x y ) ∂x ∧ ∂z + (2 xyz + 2 yzv) ∂x ∧
∂v +(y
2z + yv2) ∂y ∧∂ 2 2v +(yz + zv ) ∂z ∧∂v. This bracket has cubic-nonlinear homogeneous
polynomial coefficients, hence d = 3. The vector field V⃗ = (d − 2)−1 · E⃗ is the (multiple of
the) Euler vector field on R4. As the graph cocycle γ, we take the tetrahedron (see [1, 11]);
then the symmetry flow is Ṗ = Q(P) = (−48x5y − 288x3y2z − 240xy3z2 + 192 y3z2v −
384xy2zv2−192 y2zv3)∂x∧∂ +(−48x5z−288x3yz2−240xy2y z3+192 y2z3v−384xyz2v2−
192 yz2v3)∂ ∧∂ +(−336x4yz−480x2y2z2−576x3yzv+480 y2z2v2x z +576xyzv3+336 yzv4)∂x∧
∂ +(192x3y2z−192xy3z2+288 y2zv3+48 yv5v +48 (8 x2y2z + 5 y3z2)v)∂y∧∂v+(192x3yz2−
192xy2z3 +288 yz2v3 +48 zv5 +48 (8 x2yz2 + 5 y2z3)v)∂z ∧ ∂v. We detect that this bi-vector
is a coboundary, Q(P) = [[Y⃗,P ]] with the vector Y⃗ = (−24x4 + 120 y2z2 − 96 yzv2)∂x +
(96x3y− 96 yv3)∂y + (96 x3z− 96 zv3)∂ + (96 x2yz− 120 y2z z2 +24 v4)∂v mod ker[[P , ·]]. The
vector field Y⃗ ∈/ ker ∂P cannot be Poisson-exact (clearly, Q(P) ≡6 0), hence Y⃗ does not mark
the Poisson cocycle of zero 1-vector.17 But the universal vector field X⃗(γ, V⃗,P) ∈ ker ∂P is
identically zero on R4. Indeed, the Euler field E⃗ = V⃗ is linear, yet it is readily seen from the
figures in [1] that in every orgraph from the 1-vector n−1O⃗r(γ)(V⃗⊗ P⊗ ), the vertex with V⃗
is differentiated at least twice (and at most thrice), so X⃗ ≡ 0.
Proposition 5. The flow Ṗ = O⃗(r(tetrahedron γ3)(P)) preserves the Nambu class of Poisson
brackets, {f, g}P = ϱ(x, y, z) · det ∂(a, f, g)/∂(x, y, z) with arbitrary ϱ and global Casimir a
on R3: the flow forces the nonlinear evolution ȧ, ϱ̇ with differential-polynomial r.-h.s.
• This flow Ṗ = Q(P) is not Poisson-exact in terms of any vector field Y⃗ with differential-
polynomial coefficients (cubic in both a and ϱ, of total differential order eight).
16Exercise. Extend the proof to the case n = 1, γ = •, dγ = −•−−• (so that the l.-h.s. was nonzero).
17Likewise, by using another -matrix for R , namely x y 7→ x yR gl2( ) ( z v ) (−z v ) also from [12], we obtain the
Poisson bi-vector P = 2x2y∂x ∧ ∂y + 2 yz2∂x ∧ ∂z + (2xyz + 2 yzv)∂x ∧ ∂v + (−2xyz + 2 yzv)∂y ∧ ∂z +
2 yv2∂y ∧ ∂v + 2 yz2∂z ∧ ∂v on R4 with Cartesian coordinates x, y, z, v. The tetrahedral flow then equals
Ṗ = Q(P) = (−384x5y−384x3y2z−1536xy2zv2+384 (x2y2z − 4 y3z2)v)∂ ∧∂ +(−384x3yz2−2688xy2z3x y +
1152xyz2v2 + 384 yz2v3 − 384 (3x2yz2 − 7 y2z3)v)∂ ∧ ∂ + (−384x4yz − 2688x2y2z2 − 1536x3x z yzv +
2688 y2z2v2 + 1536xyzv3 + 384 yzv4)∂ ∧ ∂ + (384x4yz + 384x2y2z2 + 1536 y3z3 − 384xyzv3x v + 384 yzv4 +
384 (x2yz + y2z2)v2 − 384 (x3yz − 2xy2z2)v)∂y ∧ ∂z +(1536xy3z2 +1536x2y2zv− 384xy2zv2 +384 y2zv3 +
384 yv5)∂y ∧∂v +(−384x3yz2−2688xy2z3+1152xyz2v2+384 yz2v3−384 (3x2yz2 − 7 y2z3)v)∂z ∧∂v. It is
Poisson-trivial: Q(P) = [[Y⃗,P]] with a representative Y⃗(= (−96x4 +)576 y2z2 − 384 yzv2)∂x + (−192xy2z +
192 y2zv−384 yv3)∂y+(−96x3z−96xzv2+96 zv3+96 x2z − 4 yz2 v)∂z+(−576 y2z2−384xyzv+96 v4)∂v.
These explicit examples of Poisson-exact bi-vector flows Ṗ = Q(P) = [[Y⃗,P]] will be useful in the future study
of the mechanism Y⃗ = Y⃗(γ, V⃗,P) of their observed ∂P -triviality.
442
The cocycle equation at hand, E(γ3, a, ϱ) = {Ṗ = [[Y⃗,P ]]}, is a first-order PDE with
differential-polynomial coefficients (their skew-symmetry under permutations of x, y, z is
inherited from the property of the Jacobian determinant and from the transformation law for
the density ϱ in P). Whether this equation E does not admit any non-polynomial solutions
Y⃗(a|σ|⩽3, ϱ|τ |⩽2) is an open problem.
Acknowledgements. A.V.K. thanks the organizers of international workshop SQS’19 (August
26–31, 2019 in Yerevan, Armenia) for a warm atmosphere during the event. A part of this research
was done at the IHÉS (Bures-sur-Yvette, France); the authors are grateful to the ratp and stif for
setting up stimulating working conditions. The authors thank G.H.E.Duchamp and M. Kontsevich
for helpful discussions.
References
[1] Bouisaghouane A., Buring R., Kiselev A. The Kontsevich tetrahedral flow revisited // J. Geom. Phys.
2017. V. 119. P. 272–285.
[2] Brown F. Mixed Tate motives over Z // Annals of Math. 2012. V. 175. P. 949–976.
[3] Buring R., Kiselev A.V. The orientation morphism: from graph cocycles to deformations of Poisson
structures // J. Phys.: Conf. Ser. 2019. V. 1194. Paper 012017 P. 1–10; Kiselev A.V., Buring R. The
Kontsevich graph orientation morphism revisited. 2019. Preprint arXiv:1904.13293 [math.CO]. P. 1–11.
[4] Buring R., Kiselev A.V. On the Kontsevich ⋆-product associativity mechanism // PEPAN Let-
ters. 2017. V. 14(2). P. 403–407; Buring R., Kiselev A.V. The expansion ⋆ mod ō(ℏ4) and com-
puter-assisted proof schemes in the Kontsevich deformation quantization // Experimental Math.
doi:10.1080/10586458.2019.1680463 (Preprint arXiv:1702.00681 [math.CO])
[5] Buring R., Kiselev A.V., Rutten N. J. Poisson brackets symmetry from the pentagon-wheel cocycle in
the graph complex // PEPAN Letters 2018. V. 49(5). P. 924–928.
[6] Buring R., Kiselev A.V., Rutten N. J. The heptagon-wheel cocycle in the Kontsevich graph complex
// J. Nonlin. Math. Phys. 2017. V. 24, Suppl. 1. P. 157–173.
[7] Dolgushev V. A., Rogers C. L., Willwacher T. H. Kontsevich’s graph complex, GRT, and the deforma-
tion complex of the sheaf of polyvector fields // Annals of Math. 2015. V. 182(3). P. 855–943.
[8] Jost C. Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fields // Differ.
Geom. Appl. 2013. V. 31(2). P. 239–247.
[9] Kiselev A.V. The calculus of multivectors on noncommutative jet spaces // J. Geom. Phys. 2018.
V. 130. P. 130–167.
[10] Kiselev A.V. Open problems in the Kontsevich graph construction of Poisson bracket symmetries //
J. Phys.: Conf. Ser. 2019. V. 1416. Paper 012018 P. 1–8.
[11] Kontsevich M. Formality conjecture // Proc. of Conf. “Deformation theory and symplectic geometry”.
Ascona, June 17–21, 1996. Dordrecht: Kluwer Acad. Publ., 1997. P. 139–156;
Kontsevich M. Derived Grothendieck–Teichmüller group and graph complexes [after T. Willwacher] //
Séminaire Bourbaki (69ème année, 2016–2017). Janvier 2017. No. 1126. P. 183–212.
[12] Laurent–Gengoux C., Pichereau A., Vanhaecke P. Poisson structures. Grundlehren der mathematischen
Wissenschaften 347. Springer-Verlag Berlin Heidelberg, 2013.
[13] Lichnerowicz A. Les variétés de Poisson et leurs algèbres de Lie associées // J. Differential Geom. 1977.
V. 12(2). P. 253–300.
[14] Pichereau A. Poisson (co)homology and isolated singularities // J. Algebra 2006. V. 299(2). P. 747–777.
[15] Willwacher T. M.Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra // Invent.
Math. 2015. V. 200(3). P. 671–760.
443

Chapter 20
The hidden symmetry of
Kontsevich’s graph flows on the
spaces of Nambu-determinant
Poisson brackets
This chapter is based on the preprint R. Buring, D. Lipper and A. V. Kiselev.
arXiv:2112.03897 [math.SG] – 23+iv p. (submitted).
Commentary. In reference to Part I of the dissertation, the material of this chapter is
used in Chapter 7. The final chapter concludes with a list of open problems.
445
THE HIDDEN SYMMETRY OF KONTSEVICH’S
GRAPH FLOWS ON THE SPACES OF
NAMBU-DETERMINANT POISSON BRACKETS
R. BURING∗,a, D. LIPPERb, AND A.V.KISELEV∗ §,b
This text involves graph theory, Poisson geometry, and cominatorics;
it concludes with 7 research problems about Kontsevich’s universal symmetries
of the spaces of Nambu-determinant Poisson brackets on R3 and R4.
Abstract. Kontsevich’s graph flows are – universally for all finite-dimensional affine
Poisson manifolds – infinitesimal symmetries of the spaces of Poisson brackets. We
show that the previously known tetrahedral flow and the recently obtained pen-
tagon-wheel flow preserve the class of Nambu-determinant Poisson bi-vectors P =
da/ dvol(x) = ϱ(x) · da/dx on Rd 3 x for d = 3 and 4, including the general case
ϱ 6≡ 1. We establish that the Poisson bracket evolution Ṗ = Q (P⊗#Vert(γ)γ ) is
trivial in the respective Poisson cohomology, Qγ = [[P, X⃗([ϱ], [a])]], for the Nambu-
determinant bi-vectors P (ϱ, [a]). For the global Casimirs a = (a1, . . . , ad−2) and
inverse density ϱ on Rd, we analyse the combinatorics of their evolution induced by
the Kontsevich graph flows, namely ϱ̇ = ϱ̇([ϱ], [a]) and ȧ = ȧ([ϱ], [a]) with differen-
tial-polynomial right-hand sides. Besides the anticipated collapse of these formulas
by using the Civita symbols (three for the tetrahedron γ3 and five for the pentagon-
wheel graph cocycle γ5), as dictated by the behaviour ϱ(x′) = ϱ(x) · det‖∂x′/∂x‖ of
the inverse density ϱ under reparametrizations x ⇄ x′, we discover another, so far
hidden discrete symmetry in the construction of these evolution equations.
Introduction. Kontsevich’s infinitesimal symmetries P 7→ P + εQ(P ) + ō(ε) of the
spaces of Poisson structures are universal for all finite-dimensional affine Poisson man-
ifolds (M daff, P ), preserving the property of the Cauchy datum P (ε = 0) to remain Pois-
son modulo ō(ε) at ε > 0. The goal of this paper is to explore the combinatorics that
arises for the restriction of these symmetry flows Ṗ = Q(P ) to the class of generalized
Nambu-determinant Poisson brack∥∥ets, ∥{f, g} = ϱ(x) · det ∂(a1, . . . , ad−2, f, g)/∂(x1, . . . , xd)∥,
Date: 7 December 2021.
2010 Mathematics Subject Classification. 05C22, 68R10, also 53D17.
Key words and phrases. Poisson geometry, Nambu-determinant Poisson bracket, Poisson cohomol-
ogy, symmetry, Kontsevich’s graph complex.
∗ A part of this research was done while RB and AVK were visiting at the IHÉS in December 2019.
§ Corresponding author. E-mail: A.V.Kiselev@rug.nl.
aAddress: Institut für Mathematik, Johannes Gutenberg–Universität, Staudingerweg 9, D-55128
Mainz, Germany.
bAddress: Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Uni-
versity of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands.
446
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 447
with d− 2 global Casimirs a = (a1, . . . , ad−2) and inverse density ϱ on Rd. Differential-
polynomial in the bi-vector compone∑nts,
ij { i j} · i1···i ij · ∂a1 · · · ∂ad−2 d−2P = x , x = ϱ(x) ε ,
i1,...,id−2 ∂xi1 ∂xid−2
the right-hand sides Q([P ]) of the flows are encoded by using the graph cocycles in the
Kontsevich undirected graph complex. The tetrahedral cocycle flow 4Ṗ = O⃗r(γ ⊗3)(P )
is the first example from the pioneering paper [14] (cf. [19] and [2]); graph cocycles
beyond the tetrahedron γ3 are discussed in [7] (see references therein); the next, higher
nonlinearity degree flows are constructed for the pentagon-wheel cocycle γ5 in [9] and
for the heptagon-wheel cocycle γ7 in [8]. We now study the restriction of this universal
construction to a particular class of Poisson brackets, so that their analytic properties
repercuss in the combinatorics of algebraic structures and in the Poisson-cohomological
(non)triviality of the infinitesimal deformations P →7 P+εQ(P )+ ō(ε) with the markers
Q ∈ ker[[P, ·]] of second Poisson cohomology classes [Q] ∈ H2 (M d = RdP aff ).
Linear in the functional parameters ϱ and a, the Nambu-determinant Poisson bi-
vectors P = ϱ(x)·da/dx are both special and generic in many situations within Poisson
geometry. Among the most well known examples of Poisson structures from this class
we recall, for instance,
• the Euler top bracket {xi, xj} = εijk · xk on E3 ' so(3)∗, that is {x, y} = z and
so on w.r.t. the signed permutations σ ∈ S3. This bracket is Nambu-class with
ϱ ≡ 1 and the global polynomial Casimir a(x, y, z) = 1(x2 + y2 + z2).
2
• the log-symplectic bracket {x, y} = xy (and so on, cyclically), given on R3
with ϱ ≡ 1 by the Casimir a = xyz. This bracket is important in deformation
quantization (on R2 ⊂ R3) since it is explicitly known that x⋆y = exp(ℏ)·y⋆x for
the associative noncommutative star-product with this Poisson bracket, {x, y} =
xy, in the leading deformation term (see [15, 16, 18] and [1]).
The generalized (ϱ 6≡ 1) Nambu{-determinant Poisson brackets P = ϱ(x) · da/dx are“generic” in the sense that as soon as a differ}ential-polynomial identity which holds by
force of the Jacobi identity E = 1 [[P, P ]] = 0 and its differential consequences (as well
2
as by force of other constraints: e.g., the cyclic weight relations and multiplicativity of
the Kontsevich graph weights in the ⋆-products, cf. [17]), for instance,
.
(f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) = 0 on E∞,
is achieved by splitting and solving an overdetermined system of homogeneous differ-
ential-polynomial equations in ϱ and a, then that identity in practice holds at once
for generic Poisson bi-vectors P , now by force of a nontrivial, explicitly constructed
operator ♢ in the identity’s right-hand side (see [4, Theor. 9 and 12]): respectively,
(f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) = ♢(P, [[P, P ]]).
On the other hand, for any choice of ϱ 6≡ 1, the Nambu-determinant Poisson brackets
P = ϱ(x) · da/dx are “special” in the sense that they admit the maximal set of d− 2
scalar Casimirs a = (a1, . . . , a dd−2). The space R is foliated by the intersections of the
level sets {ai = const} into symplectic leaves, which are generally two-dimensional: e.g.,
consider the concentric spheres {x | x2 + y2 + z2 = const > 0} for the Euler top.
448 R. BURING, D. LIPPER, AND A. V. KISELEV
But not every Poisson bracket on R3 admits a global polynomial Casimir a ≡6 const
if the coefficients P ij of the bi-vector P are polynomial. (Whereas for the Nambu class
this is achieved tautologically by taking ϱ, a ∈ R[x1i , . . . , xd] in any fixed system of
affine coordinates on Rd in any dimension d ⩾ 3.)
Counterexample 1 ([20]). On R∑d with Cartesian coordinates x = (x1, . . . , xd), con-
sider the Euler vector field E⃗∑= xi · ∂/∂xii and, for any k ⩾ 2, take another ho-
mogeneous vector field V⃗ = (xj)k · ∂/∂xjj . By definition, put P := V⃗ ∧ E⃗. Then
the bi-vector P is Poisson — yet it does not admit any non-constant global polynomial
Casimir a on Rd. (A proof is recalled in Appendix A, see p. 465 below.)
In the same context of competing “generic vs special”, Kontsevich’s graph flows
provide (markers of the) second Poisson cohomology classes Q([P ]) ∈ ker[[P, ·]] in an
extremely broad setting: indeed, universally for all finite-dimensional affine Poisson
manifolds (M daff, P ). This automatically poses the problem of (non)triviality for these
Poisson cohomology classes [Q] ∈ H2P (M daff). We recall from [2, 5] that for nontrivial
graph cocycles γ in the Kontsevich undirected graph complex, there does not exist
any mechanism that would trivialize the flows Ṗ = O⃗r(γ)(P⊗#Vert(γ)) at the level of
Kontsevich’s graphs, that is, by using a would-be universal trivializing vector field
X⃗ again determined within the graph language, and hence by a formula that would
work uniformly in all dimensions. For instance, such is manifestly the case for the
tetrahedron γ3, for the pentagon-wheel cocycle γ5, etc., provided the dimension d of
Poisson manifold is greater than two. In other words, the coboundary equation,
O⃗r(γ)(P⊗#Vert(γ))− [[P, X⃗(γ′)]] = ♢(P, [[P, P ]]),
has no solution (γ′,♢) for the main sequence of nontrivial graph cocycles γ3, γ5, γ7, . . .
and their iterated commutators (if d ⩾ 3 is not fixed a priori; if d = 2, the graph γ′
trivializing the tetrahedral γ3-flow is found in [2]). The present work serves to continue
– from [2, 5, 6] – the line of study on the Poisson (non)triviality of Kontsevich’s graph
flows.
The fact we discover is that, for rich classes of Poisson structures, the Kontsevich
graph flows are Poisson-trivial, so that the resulting shifts nQ([P ]) = O⃗r(γ)(P⊗ ) of
Poisson bi-vectors P are induced by highly nonlinear, non-affine reparametrizations
of the base coordinates – along the integral trajectories of the trivializing vector fields
X⃗ – on the affine Poisson manifolds M daff. Such is the case for the Nambu-determinant
class of brackets P (ϱ, [a]) on R3 and the graph flows preserving it. We establish the
fact of trivialization and we collapse the formula of the vector field X⃗([ϱ], [a]) by using
the features of the Nambu--Poisson geometry under study. (All these analytic and
combinatorial results are verified by direct calculation.) It remains to express the vector
field X⃗ through deeper invariants, that is, explain the work of the trivialization and
collapse mechanisms at the level of graphs and symplectic foliation.
Remark 1. For a chosen volume element dvol(x) = dx/ϱ(x) with smooth ϱ, needed for
construction of the Nambu-determinant bi-vectors P = da/ dvol(x), the zero locus of
the inverse density ϱ provides a tiling of the affine space Rd. Inside each cell bounded
by the walls {x | ϱ(x) = 0}, that is on every maximal subset where the restriction
of ϱ is nowhere vanishing, the inverse density can be brought to a constant ϱ′(x′) ≡ ±1
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 449
by a (non)linear, pointwise-dependent rescaling of local coordinates. The restriction of
the graph flows to the subclass of ‘genuine’ Nambu-determinant brackets P = da/dx
can either degenerate (e.g., for the tetrahedral flow over R3) or stay nonzero (e.g.,
for the tetrahedral flow over R4), see below. In all these cases, the trivializing vector
fields X⃗ behave in a usual way, as tensors do, in the course of such transformations to
the normal coordinates; note that the vector fields X⃗ can acquire arbitrary Poisson-
exact components [[P,H]]. Yet the construction of the normal coordinates satisfying
ϱ′(x′) = ±1 is a priori not correlated at all with the affine structure — which the graph
flows refer to.
This paper is structured as follows. In §1 we recall some facts about the Nambu-
determinant Poisson brackets P (ϱ, [a]) on Rd and about the Kontsevich graph flows over
affine Poisson manifolds (M daff, P ). Next, in §2 we detect that the Nambu class of Poisson
brackets on R3 and R4 is preserved by the graph flows for the tetrahedral cocycle γ3
and by the pentagon-wheel cocycle γ over R35 . The structure of induced evolution
ϱ̇([ϱ], [a]), ȧ([ϱ], [a]) is then put, in §3, in correspondence with the original graph cocycle,
and the formulas of induced velocities are collapsed by using the Civita symbols (one
per graph vertex minus one overall: e.g., three symbols for the tetrahedron); the affine
structure of Rd is crucial at that point. In §4 we analyze the algebra and combinatorics of
the marker-monomials under the sums with multiple Civita symbols. Here we discover
an extra symmetry of the Kontsevich graph flows’ restriction to the spaces of Nambu-
determinant Poisson structures. Finally, we establish in §5 that the tetrahedral flow
over R3 is Poisson-cohomology trivial, and we collapse the formula of the trivializing
vector field X⃗ by using the same mechanism of Civita symbols as before. The paper
concludes with a list of open problems about the graph flows and combinatorics of their
restrictions to the Nambu class of brackets.
1. Preliminaries
1.1. The generalized Nambu-determinant Poisson brackets. In the context of
quark dynamics and n-ary interactions, Nambu introduced ([22], cf. [10, 11]) a class
of Poisson brackets with global Casimirs a = (a1, . . ., ad−2) on Rd 3 x: the Poisson
bi-vectors are P = da/ dvol(x) = ϱ(x) · da1 ∧ . . . ∧ da 1 dd−2/dx ∧ . . . ∧ dx with a not
necessarily constant inverse of the volume density, ϱ(x). The coordinate expressions
are, for example,
∣∣∣∣ ∣∣∣∣ ∣
∣ ∣
a ∣
∂(a, f, g) ∣∣∣ x fx gx∣{f, g} = ϱ(x) · = ϱ(x, y, z) · ∣a ∣y fy gy∂(x, y, z) ∣a ∣z fz gz
on R3 3 x = (x, y, z), and likewise, ∣∣ ∣∣
{f, g} = ϱ(x, y, z, w) · ∣∣∂(a1, a2, f, g) ∣∂(x, y, z, w) ∣
on R4 with global (e.g., Cartesian) coordinates x = (x, y, z, w). It is obvious that
the given functions ai which show up in the construction of the bi-vector P Poisson-
commute with any argument f ∈ C∞(Rd). The scalars ai(x) = ai(x′(x)) do not change
450 R. BURING, D. LIPPER, AND A. V. KISELEV
under the coordinate transformations x(x′) ⇄ x′(x). Given two scalars f, g ∈ C∞(Rd),
their Poisson bracket is also a scalar. To counterbalance the behaviour of the Jacobian
determinant in the cou∣∣rse of coo∣∣∂(a, f, g) ∣∣
rdin∣∣ate transfor∣mations,∣∣ ∣∣ ∣ ∣∂(a, f, g) ∣∣ ∣∣∂(x′, y′, z′) ∣= ∣∂(x, y, z) ∂(x′, y′, z′ · ,)∣ ∣ ∂(x, y, z) ∣
the object ϱ(x) ⇄ ϱ′(x′) behaves a∣∣ccordingly,
· ∣∣ ∣∂(x′, y′, z′) ∣ϱ(x, y, z) ∣ = ϱ′(x′, y′, z′),∂(x, y, z) ∣
with an elementary general fact that∣∣dx/ϱ/(x) =∣ dx′/ϱ′(x′) andϱ(x) · ∂(x′) ∂x∣ = ϱ′(x′) (1)
for all dimensions d ⩾ 3. So, let us keep in mind that the inverse density ϱ(x) =
ϱ′(x′)·|∂x/∂x′| in the volume element dvol(x) = dx/ϱ(x) is nontrivially reparametrized
under the changes x(x′) ⇄ x′(x), whereas the scalars ai are not transformed. Let us
remember also that so far, the coordinate changes could be arbitrarily nonlinear, that
is, not necessarily linear or affine on Rd.
1.2. Kontsevich’s graph flows. In the seminal paper [14] (see also [19] and [2, 9,
5, 12, 13] for illustrations and discussion), Kontsevich designed a method to construct
infinitesimal symmetries P = Q([P ]) of the spaces of Poisson structures on affine fi-
nite-dimensional Poisson manifolds (M daff, P ). The construction is universal for all such
geometries (with x′ = Ax + b as the only admissible coordinate reparametrizations).
The right-hand side Q of the evolution Ṗ = Q([P ]), differential-polynomial in the
components of the bi-vector P , is described by using linear combinations (with real
coefficients) of directed graphs; these graphs are built of wedges ←•→ with prescribed
ordering Left ≺ Right of the outgoing arrows in every internal vertex. Each edge is dec-
orated with its own summation index which runs from 1 to the dimension d = dim Md;
each decorated edge −−i→ corresponds to the derivative ∂/∂xi w.r.t. a local coordinate in
an affine chart of M d; each internal vertex of the directed graph is inhabited by a copy
of the Poisson bi-vector P = (P ij(x)); each graph determines a differential-polynomial
expression (w.r.t. the structure P and the content of sink vertices) in a natural way:
take the product of the (differentiated) contents of the vertices and sum over all the in-
dices. Two factors, namely (i) the contraction of lower indices – from ∂/∂xi and ∂/∂xj
on the respective Left and Right outgoing edges – with the first and second indices i, j
in the skew-symmetric bi-vector components P ij(x) in the arrowtail vertex, and (ii)
the independence of the Jacobians A (in the affine changes x′ = Ax+ b) from a point
of two charts’ overlap, make the Kontsevich construction well defined for an arbitrary
choice of local affine coordinates on (M daff, P ).
The graph cocycles γ on n vertices and 2n − 2 edges in the Kontsevich undirected
graph complex (see [14] as well as [5, 13] and references therein), when directed (inher-
iting the edge ordering from γ) and evaluated at n copies of a given Poisson bi-vector,
yield a natural class of Kontsevich’s graph flows nṖ = O⃗r(γ)(P⊗ ) on the spaces of
Poisson structures. Willwacher’s construction of suitable graph cocycles γ from the
Grothendieck–Teichmüller Lie algebra grt gives us the main sequence to work with:
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 451
Kontsevich’s tetrahedron γ3 (which is the wheel graph with three spikes), the Kon-
tsevich–Willwacher pentagon-wheel cocycle γ5 (see [8, 7] and [9]), the heptagon-wheel
cocycle γ7 (see [7] and [5]), etc., and their iterated commutators (always on n vertices
and 2n−2 edges, for instance with 9 vertices and 16 edges in [γ3, γ5]). The construction
of Lie brackets on the vector space of graphs with wedge ordering of edges is explained
in [14] and [23, 7].
Example 2 ([14, 19] and [2, 13]). The tetrahedron γ3, when o(riented by the morph)ism
O⃗r to the balanced (by 8 : 24 = 1 : 3) sum Γ1(1 , 2 ) + 3 Γ′2(1 , 2 ) − Γ′2(1 , 2 ) of
two skew-symmetrized bi-vector graphs built of wedges (see Fig. 1) now encodes the
m′

 RC}b 
bb }C b
R
 C > k′ 
 C


 R C  

′
ℓ CL L ℓ j
Γ = P


 C  Γ′ = P
=

 C 
1 PPPqCCW 2 Pk PPqCCWL m

A ? i
L
 AR ?

 UA
Figure 1. T(he components of K)ontsevich’s tetrahedral flow Ṗ (1 , 2 ) =
Γ ′ ′1(1 , 2 ) + 3 Γ2(1 , 2 ) − Γ2(1 , 2 ) on the on the space of Poisson bi-
vectors P on Rd in any dimension d ⩾ 3.
differential-polyn(omial velocity of Poisson bi-vect)ors:′ ′ ′
· ∂
3P ij ∂P kk ∂P ℓℓ ∂Pmm ∂ ∧ ∂Qtetra(P ) = 1
∂xk∂xℓ∂xm ∂xℓ′ ∂xm(′ ∂xk′ ∂xi ∂xj )
∂2P ij
′
∂2P km ∂P k ℓ ∂Pm
′ℓ′ ∂ ∂
+ 3 · ′ ′ ∧ .∂xk∂xℓ ∂xk ∂xℓ ∂xm′ ∂xj ∂xi ∂xm
Indeed, we place copies of a given bi-vector P into the internal vertices, match their
first and second indices with the summation indices that decorate the arrows (note
that the ordering, available in the digraph encoding in loc. cit., is not everywhere
displayed in Fig. 1, but it is easily retrieved from the differential-polynomial formula),
and for all values of all the indices, we take the sum of products of all the differentiated
contents of the vertices. It is clear that for an arbitrary affine Poisson manifold, the
flow 4Ṗ = O⃗r(γ3)(P⊗ ) is coordinate-free.
Other examples of nonlinear proper ( 6≡ 0 if P is Poisson) Kontsevich’s graph flows are
constructed in [9] for the pentagon-wheel cocycle γ5 and in [5] for the heptagon-wheel
cocycle γ7 (see also [13]).
2. Structural stability of the Nambu-determinant brackets under
Kontsevich’s flows
The first main question is which we explore in this note is how, in precisely which
way the class P = da/ dvol(x) of generalized (ϱ 6≡ 1) Nambu-determinant bi-vectors
P (ϱ, [a]) on Rd is stable under Kontsevich’s universal deformations Ṗ = Qγ([P ]) given
by the graph cocyles γ. The other question for us to explore is the combinatorial
452 R. BURING, D. LIPPER, AND A. V. KISELEV
mechanism of this stability. So, let us first inspect how the infinitesimal symmetries Ṗ =
#V (γ)
O⃗r(γ)(P⊗ ) of the – actually, unknown – space of all Poisson brackets P on Rd (where
Rd is viewed as an affine manifold) restrict to the subspace of Nambu-determinant
Poisson brackets.
Because the Nambu-determinant bi-vectors P (ϱ, [a]) = ϱ(x) · da/dx are linear in
both the inverse density ϱ and Casimirs a = (a1, . . ., ad−2), the class {P (ϱ, [a])} is
stable if, by definition, there exist the velocities ϱ̇ and ȧ (depending on the point (ϱ,a)
in the functional parameter space) su∑ch thatd−2
Ṗ (ϱ, [a]) = P (ϱ̇, [a]) + P (ϱ, [a1], . . . , [ȧi], . . . , [ad−2]). (2)
i=1
In particular, the stability of the class is achieved if the evolution ϱ̇ and ȧ is differential-
polynomial (of finite degrees and differential orders) in the parameters that evolve,
ϱ̇ = ϱ̇([ϱ], [a]), ȧ = ȧ([ϱ], [a]). (3)
The construction of the Kontsevich flow ⊗nṖ = O⃗r(γ)(P ) from a graph cocycle γ on n
vertices and the count of homogeneities always allow us to estimate both the order and
polynomial degrees of such (non)linear PDE evolution — provided it exists.
Example 3 (γ3-flow over R3). First, if ϱ ≡ 1 and the Poisson bi-vector is P =
da(x, y, z)/dxdydz, then the Kontsevich tetrahedral flow 4Ṗ = O⃗r(γ3)(P⊗ [a]) vanishes
identically. In retrospect, this is true because every term in ȧ contains a derivative of ϱ,
and all the more each term in ϱ̇ does so, whence the Cauchy datum ϱ = const makes
the flow well defined but identically zero.
Let the inverse density ϱ(x, y, z) be not necessarily constant over R3. A simple a
priori estimate of homogeneities suggests that the terms in the differential-polynomial
right-hand side of ϱ̇ and ȧ are constrained by the ansatz
ȧ ∼ a4ϱ3, with 9 derivatives in each monomial,
at most 3rd order derivatives of a and of ϱ;
(4)
ϱ̇ ∼ a3ϱ4, with 9 derivatives in each monomial,
at most 3rd order derivatives of a and of ϱ.
By using the method of undetermined coefficients, implementing the problem in soft-
ware for differential calculus on jet spaces (e.g., Jets by M.Marvan [21] or gcaops by
R.Buring [3]), we obtain the nontrivial solution (see also Example 6 on p. 454 below).
The differential polynomial ȧ([ϱ], [a]) consists of 228 monomials with nonzero coeffi-
cients, and ϱ̇([ϱ], [a]) is 426 monomial long. It is seen that the actual dependence of ȧ
and ϱ̇ on the jet variables aσ and ϱτ is such that the lengths of multi-indices σ and τ
are bounded by 1 ⩽ |σ| ⩽ 3 and 0 ⩽ |τ | ⩽ 1 for ȧ and by 1 ⩽ |σ| ⩽ 2 and 0 ⩽ |τ | ⩽ 3
for ϱ̇. Apparent is also that in each monomial, there are exactly three derivatives w.r.t.
x, exactly three w.r.t. y, and exactly three w.r.t. z. Here is a sample how these formulas
read:
ȧ = −12ϱ2axϱyaxya 2 2zzaxyz + 12ϱ axϱyaxyaxzayzz + 12ϱ axϱyaxyaxzzayz + . . . ,
ϱ̇ = −12ϱϱxϱyaxazϱxxyazz − 12ϱϱxϱyaxazϱxzzayy + 24ϱϱxϱyaxazϱxyzayz + . . . ;
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 453
both formulas are given in full in Appendix B. In what follows, we shall explain these
empiric facts; by understanding the combinatorics in these formulas, we collapse them
to tiny Eqs. (7) on p. 457.
Example 4 (γ3-flow over R4). In contrast with 3D, the tetrahedral flow Ṗ = O⃗r(γ3)
4
(P⊗ ) is nonzero for the “authentic” Nambu-determinant Poisson bi-vector P = da1 ∧
da2/dxdydzdw with pre-factor ϱ ≡ 1 on R4 3 x = (x, y, z, w). With this Cauchy datum
ϱ ≡ 1 implying ϱ̇ ≡ 0, we obtain that the differential polynomial velocities ȧ1([a1], [a2])
and ȧ2([a1], [a2]) each contain 9024 monomials (of two unequal differential profiles, 4512
and 4512 each, see Example 9).
Now the full case on R4: take the generalized Nambu--Poisson bi-vector P (ϱ, [a1], [a2])
= ϱ(x) ·da1∧da2/dxdydzdw. The tetrahedral flow
4
Ṗ = O⃗r(γ3)(P
⊗ ) does preserve this
class of Poisson brackets: there exist differential-polynomial velocities of the inverse
density ϱ(x) and of the two Casimirs a1, a2 such that
ϱ̇ = ϱ̇([ϱ], [a1], [a2]) with 90,024 terms,
ȧ1, ȧ2([ϱ], [a1], [a2]) with 33,084 terms each.
The combinatorial structure of these right-hand sides in the general case (ϱ 6≡ 1) can
be analysed by the technique which we develop in what follows: each of the three
expressions is collapsed by using the marker-monomials for the triple summation with
the Civita symbols on R4. For instance, the differential monomials in either ȧ1 or ȧ2 are
partitioned according to the homogeneity profiles of derivatives of ϱ, a1, and a2 with
respect to the four coordinates on R4 (see Table 2 on p. 462 below). And all the 33,048
terms in ȧ1 and ȧ2 are expressed by formulas (9) on p. 458.
Example 5 (γ5-flow over R3). The pentagon-wheel flow
6
P = O⃗r(γ ⊗5)(P ) on the space
of all Poisson structures on R3 restricts to the Nambu-determinant class of brackets
{P (ϱ, [a])}. In the differential-polynomial formulas of evolution ϱ̇([ϱ], [a]) and ȧ([ϱ], [a]),
the right-hand side of ȧ contains 79,212 monomials, and there are as many as 146,340
in ϱ̇ (before either formula is collapsed by using five Civita symbols). In the meantime,
one can estimate the homogeneity degrees and orders, that is the polynomial degrees
of each term in ȧ and ϱ̇ with respect to the jet variables aσ and ϱτ , as well as the
bounds on the possible (but not necessarily attained) lengths of the multi-indices σ and
τ counting the derivatives. We note that in every monomial, there are 5 subscripts x
(for derivatives, which is the usual notation), 5 subscripts y, and 5 subscripts z; both
ȧ and ϱ̇ are manifestly skew-symmetric w.r.t. permutations of the base variables x, y, z
(meaning that the right-hand sides contain at least one Civita symbol εi1i2i3).
3. The structure of induced evolution ϱ̇, ȧ
3.1. Encoding ȧ, ϱ̇ by the Kontsevich graphs. The Kontsevich flow Ṗ = O⃗r(γ)
n
(P⊗ ) on the class of generalized (ϱ ≡6 1) Nambu-determinant Poisson brackets P (ϱ, [a])
preserves their structure. Let us interpret this fact back in the language of Kontsevich’s
directed graphs.
454 R. BURING, D. LIPPER, AND A. V. KISELEV
Proposition 1 ([20]). The evolution ȧi of each Casimir in the Jacobian determinant
within the Nambu–Poisson bracket, ∥ ∥
{f, h} ∥ 1 d ∥P (ϱ,[a]) = ϱ(x) · det ∂(a1, . . . , ad−2, f, g)/∂(x , . . . , x ) ,
is equal to the value of the graph orientation morphism O⃗r at the n-tuple P⊗n−1 ⊗ ai
(here n = #Vert(γ)):
⊗n−1ȧi = O⃗r(γ)(P , ai), (5)
where the right-hand side represents the sum of n-linear polydifferential operators which
are encoded by the directed graph cocycle O⃗r(γ) and which are evaluated at ai placed
consecutively in one of the vertices and the other vertices filled in by copies of the
bi-vector P (ϱ, [a]).
Commentary. Indeed, the Kontsevich graph flows nṖ = O⃗r(γ)(P⊗ ) are such that no
arrows fall on the checked factor ϱ̌ in∑the Leibniz formula for Ṗ ,d−2
Ṗ ([ϱ], [a]) = P (ϱ̇, [a]) + P (ϱ̌, [a1], . . . , [ȧi], . . . , [ad−2]). (2)
i=1
More specifically, to let exist the restriction of Kontsevich’s graph flow to the class
of Nambu-determinant Poisson brackets P (ϱ, [a]) = ϱ(x) · da/dx, the directed graph
formula, working over the content of each internal vertex by the Leibniz rule for each in-
coming arrow, automatically singles out the terms in which (i) the pre-factor ϱ remains
intact and (ii) the in-coming derivatives are not spread over several Casimirs in the
Jacobian inside that vertex. Nontrivial in this claim is that precisely all – without
exception – terms of such structure do form the well defined tuple of velocities ȧ. □
Corollary 2. As soon as the evolution ȧ of the Casimirs is obtained according to
formula (5), from the structure of P = ϱ(x) · da/dx of the Nambu bracket and from
the Leibniz rule in Eq. (2) we deduce the speed of evolution for the inverse density ϱ.
Nam∣∣ely, we have that∣∣ ∣ ( d−2 )· ∂(a1, . . . , ad−2, f, g) ∣ϱ̇ ∣ ∑= Ṗ ([ϱ], [a])− P (ϱ, [a1], . . . , [ȧi], . . . , [ad−2]) (f, g), (6)∂(x1, . . . , xd) ∣
i=1
where f, g ∈ C∞(Rd), the right-hand side with the known flow nṖ = O⃗r(γ)(P⊗ ) is
the value of the linear combination of bi-vectors at f ⊗ g, and ϱ̇ is extracted from the
left-hand side by division.
Commentary. Indeed, by the above, the right-hand side is a whole multiple of the
Jacobian, which itself is equal to P (ϱ ≡ 1, [a])(f, g). □
Example 6. The above proposition and corollary, resulting in the explicit differential-
polynomial expressions for the velocities ϱ̇ and ȧ that induce a given graph flow Ṗ =
n
O⃗r(γ)(P⊗ ) for the Nambu structures P = ϱ(x) · da/dx on Rd, are illustrated in [3,
Ch. 6] by using the gcaops software for differential calculus on jet spaces. So far, the
graph formulas are explicitly verified for
• the tetrahedral flow (γ = γ3) on R3;
• the tetrahedral flow (γ = γ3) on R4 3 x with ϱ ≡ 1 (special case) and generic
ϱ(x) which implies ϱ̇ 6≡ 0;
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 455
• the pentagon-wheel flow (γ = γ ) on R35 with generic ϱ.
For the tetrahedral γ3-flow on R3, the findings from Example 3 are reproduced iden-
tically. A naive attempt to use the method of undetermined coefficients would be
practically unfeasible in the other three cases, yet Eqs. (5) and (6) serve the correct
formulas of ȧ and ϱ̇ without any need to solve a linear algebraic system.
3.2. Civita symbols in ϱ̇, ȧ: collapsing the formulas. Our present task is to
analyze the combinatorial structure of the differential-polynomial expressions for ȧ and
ϱ̇ in formulas (5) and (6), respectively, and collapse them as much as possible by using
this new knowledge.
3.2.1. The determinant provides one Civita symbol. One simple fact is immediate from
the presence of Jacobian determinant in the Nambu brackets.
Proposition 3. The differential polynomials ȧ([ϱ], [a]) and ϱ̇([ϱ], [a]) are shifted-graded
skew-symmetric w.r.t. permutations of the base variables x1, . . ., xd (i.e. the coordinates
on the Poisson manifold Rd): for a graph cocycle γ on n vertices in each term, the ve-
locities ϱ̇ and ȧi are skew-symmetric in x1, . . ., xd if n is even (e.g., as for γ3, γ5, γ7, . . .,
γ 1 d2ℓ+1, . . .) and symmetric in x , . . ., x if n is odd (e.g., such is the case for the cocycle
[γ3, γ5] on 9 vertices and 16 edges).
Proof. Every Poisson bracket of two scalar functions itself is a scalar. For the Nambu
bracket in particular, ∥ ∥
{f, g}P (ϱ,[a]) = ϱ(x) · det∥∂(a1, . . . , ad−2, f, g)/∂(x1, . . . , xd)∥,
this invariance is provided by the response (1) of the inverse density ϱ to a permutation
σ of rows in the Jacobian determinant: ϱ(x) = (−)σ · ϱ′(x′(x)) if x = σ(x′). A simple
count shows that for a graph cocycle γ on n vertices (and 2n− 2 edges), we have that
ϱ̇ ∼ ϱn · an−1 ·∏. . . · an−11 d−1 with (n− 1)× d base variables x1, . . . , xd;′
ȧ ∼ ϱn−1i · an · an−1i j with (n− 1)× d base variables x1, . . . , xd.
j ̸=i
For th∣∣e velocities ∣∣ȧi to be scalars and for the objects ϱ̇ to behave according to the samelaw, ϱ̇ = (−)σϱ̇′ ′ , as the inverse density ϱ satisfies, both the right-hand sides havex x (x)
the parity ((−)σ)n−1 whenever the base variables are permuted: x = σ(x′). □
Example 7 (γ3-flow over R3). Indeed, for the tetrahedral γ3-flow on the space of
Nambu–Poisson structures P (ϱ, [a]) on R3, with 228 terms in ȧ and 426 terms in ϱ̇, we
verify that ∑
ȧ([ϱ], [a])(x, y, z) = ∑ (−)σσ(x, y, z) acts on (sum of 38 terms),σ∈S3
ϱ̇([ϱ], [a])(x, y, z) = (−)σσ(x, y, z) acts on (sum of 71 terms).
σ∈S3
The differential monomials in the right-hand sides are obtained by the greedy algorithm:
for a monomial that still remains in the expression to be represented as an alternating
sum, take its skew-symmetrization w.r.t. σ ∈ S3 acting on x, y, z, subtract it from the
expression, collect similar terms and reduce, then proceed recursively until the list of
monomials, initially met in the velocity, is empty.
456 R. BURING, D. LIPPER, AND A. V. KISELEV
We shall presently recognize such one-time skew-symmetrizations (when n is even)
within ϱ̇ and ȧi as a consequence of a much stronger claim about the independent
action of n − 1 copies of the permutation group S 1 dd on the d-tuples {x , . . ., x }k for
1 ⩽ k ⩽ n−1 in the right-hand sides ϱ̇ and ȧ. For instance, in the above example (here
n = 4 and d = 3) the sign factor (−)σ is produced by the restriction on the diagonal,
∑ ⊗n−1 ∑ ⊗n−1 ∣
(−)σσ( {x, y, z} ) = (−)σ1 · · · (−)σn−1 ∣k σk({x, y, z}k)∣ ,
σ =σ
σ∈S k3 k=1 σ1,...,σn−1∈S3 k=1
in the set of n − 1 = 3 permutations σk ∈ Sd acting on the n − 1 non-intersecting
d-tuples {x1, . . ., xd}k that partition the set of (n− 1)× d derivatives occurring in the
right-hand sides of ϱ̇ and ȧ.
3.2.2. How ϱk yields k Civita symbols, or: Jacobians generalized. Let us recall three
facts from analysis:
• the Casimirs a = (a1, . . . , ad−2) of the Nambu–Poisson brackets are scalars; ∣
• the inverse density ϱ obeys the transformation law ϱ(x) = ϱ′(x′)·det‖∂x/∂x′‖∣
x′(x)
under a change x(x′) ⇄ x′(x);
• the objects’ velocities inherit the behaviour of those objects under the coordinate
transformations.
Consider a Kontsevich flow nṖ = O⃗r(γ)(P⊗ ) associated with a graph cocycle γ on n
vertices. These three facts, put together, reveal that the reparametrization of derivatives
of a and ϱ in the differential monomials within ϱ̇([ϱ], [a]) and ȧ([ϱ], [a]) match the
nontrivial reparametrization of n− 1 copies of ϱ therein.
More specifically, the (n− 1)× d derivations ∂⊗n−1x1 ⊗ · · · ⊗
⊗n−1∂ d arrange into n− 1x
totally skew-symmetric d-tuples εi11···i1 n−1 n−1d ∂x1⊗· · ·⊗∂ d , . . ., εi1 ···ix d ∂x1⊗· · ·⊗∂xd , where
ı⃗αε is the Civita symbol on Rd. The derivatives from each d-tuple act on different
comultiples of a marker-monomial (which stands under the sum over the n− 1 tuples
ı⃗ 1, . . . , ı⃗ n−1 with d indices in each tuple and which thus marks, generally speaking,
many monomials in the polynomial expressions ϱ̇ and ȧℓ when the sums over ı⃗ α are
expanded). In effect, each of the d-tuples ∂x1 ∧ . . . ∧ ∂xd provides its own Jacobian
determinant det‖∂x′/∂x‖ when the coordinates are reparametrized on the affine base
manifold Rd. These n−1 Jacobians |∂x′/∂x| cancel against the n−1 Jacobians |∂x/∂x′|
from the reparametrizations of the inverse density ϱ in either ϱ̇ or ȧϱ.
Theorem 4. For a graph cocycle γ on n vertices, the Kontsevich flow nṖ = O⃗r(γ)(P⊗ )
restricts to the Nambu class P = ϱ(x) ·da/dx of Poisson brackets on the affine space Rd
in such a way th(at∑ n∏−1 ) ∏
ȧ = (−)σiσ 1 d n−
′
1 n n−1
ℓ i((x , . . . , x )i) (marker-monomials ∼ ϱ aℓ · ak ),
σ1,...,∑σn−1∈Sd(∏i=1 ) k≠ ℓn−1
ϱ̇ = (−)σiσ ((x1, . . . , xd) ) (marker-monomials ∼ ϱnan−1i i 1 · · · an−1d−2).
σ1,...,σn−1∈Sd i=1
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 457
⊔The permutations σi ∈ Sd act on the partitioned set of subscripts (for derivatives)n−1
i=1 ((x
1
∑, . . . ,(x
d
⊗)i) in each)marker-monomial. Equivalently, we have thatn−1 ∏′
ȧ = εı⃗
α
ℓ · ∂⃗ıα (comultiples in marker-monomials ∼ ϱn−1an · an−1ℓ k ),
ı⃗ 1,.∑..,⃗ın−1(⊗α=1 ) k≠ ℓn−1
α
ϱ̇ = εı⃗ · ∂⃗ n n−1 n−1ıα (comultiples in marker-monomials ∼ ϱ a1 · . . . · ad−2),
ı⃗ 1,...,⃗ın−1 α=1
with d-component multi-indices ı⃗ α = (iα α1 , . . . , id ) in the Civita symbols
α
εı⃗ on Rd.
Commentary. The right-hand side of the velocity ȧℓ or ϱ̇ is a very interesting object
from analytic and combinatorial viewpoint: with all the first- and higher-order deriva-
tives in the velocity, it looks like the Jacobian determinant w.r.t. each tuple (x1, . . .,
xd) ⇄ (∂x1 , . . ., ∂xd), although the derivatives from different tuples can act on the same
comultiple. It remains therefore to control the non-tensorial behaviour of the higher-
order derivatives (which do actually occur in the expressions under study, as seen from
examples). Fortunately, this is where our initial assumption works: Kontsevich’s graph
flow is defined over an affine manifold so that the second- and higher-order derivatives
of the coordinate changes vanish identically for x = Ax′ + b (with a constant Jacobian
matrix A). Thus, higher derivatives of aℓ and ϱ are transformed by using only the first
derivatives of the coordinate changes, whence the assertion. □
Example 8 (γ3-flow over R3). For the tetrahedral flow
4
Ṗ = O⃗r(γ3)(P
⊗ ) over R3 we
recall from Eq. (4) in Example 3 that
ȧ ∼ ϱ3a4 with xxxyyyzzz in each monomial,
ϱ̇ ∼ ϱ4a3 with xxxyyyzzz in each monomial.
Now Theorem 4 works: the (4 − 1) × 3 base variables are partitioned in three triples
(x, y, z) in each term, with a skew-symmetrization over each triple. Indeed, by a brute
force calcu∑lation we verify (that for the γ -flow over R33 ,
ȧ = (−)σ(−)τ (−)ζ 2au a1 u a2 u ϱ3 w ϱ1 w ϱ2 w a − 6ϱa a3 v1v2v3 ) u1v2 u au ϱ ϱ a2 3 w1 w3 v1v3w2σ,τ∑,ζ∈S3 (− 6ϱ2au au u av v ϱw av w w ,1 2 3 1 2 3 3 1 2
ϱ̇ = (−)σ(−)τ (−)ζ − 2a a a ϱ (7)u1 u2 u3 v ϱ1 v ϱv ϱw + 6a a a ϱ2 3 1w2w3 u1v2 u2 u3 v ϱv ϱ ϱ1 3 w2 w1w3
σ,τ,ζ∈S3 − 12ϱau1au2u3av1v2ϱv3ϱw1ϱw2w)3 − 6ϱau1v2au2au3ϱv1ϱv3ϱw1w2w3
+ 6ϱ2au au u av v ϱ ϱ1 2 3 1 2 v3 w1w ,2w3
where each summation runs over three permutations σ, τ, ζ ∈ S3 giving three triples
(u1, v1, w1) = (σ(x), σ(y), σ(z)), also (u2, v2, w2) = (τ(x), τ(y), τ(z)), and (u3, v3, w3) =
(ζ(x), ζ(y), ζ(z)).
We conclude that the 228 monomials in ȧ and 426 monomials in ϱ̇ which we started
with are completely determined by only three marker-monomials for ȧ and five marker-
monomials for ϱ̇ by using three Civita symbols in either formula.1
1 Not only this: the three and five respective marker-monomials in both the velocities ȧ and ϱ̇ and
the 1,504 differential monomials in each component of the bi-vector flow 4Ṗ = O⃗r(γ ⊗3)(P ) for P (ϱ, [a])
458 R. BURING, D. LIPPER, AND A. V. KISELEV
The natural question is how the nine symbols xxxyyyzzz in each term were dis-
tributed among the disjoint triples xyz, xyz, xyz (to be permuted by σ, τ and ζ respec-
tively); we shall analyze this in the next section.
Example 9 (γ3-flow of P (ϱ ≡ 1, [a1], [a2]) on R4). Consider the “authentic” Nambu-
determinant bracket P (ϱ ≡ 1, [a1], [a2]) and induce the γ3-flow of the Casimirs a1 and a2,
see Example 4. Owing to Theorem 4 we collapse the 9,024 terms in either ȧ1 and ȧ2 to
the thrice altern∑ating formulas, namely(
ȧ1 = (−)σ(−)τ (−)ζ 3a1;s1u2u3a1;t1t2a2;s2a2;s3v1a2;t3u1a1;v2a1;v3
σ,τ∑,ζ∈S4 ( )+ 6a1;s1u2a1;t1a1;t2v3a1;u3v1v2a2;t3u1a2;s2a2;s3 , (8a)
ȧ = (−)σ2 (−)τ (−)ζ 3a1;s1a2;t1u2a2;u1u3v2a1;s2t3a1;s3t2a2;v1a2;v3
σ,τ,ζ∈S4 )
− 3a1;s1t2a2;u1a2;u2u3v1a1;t1a1;t3a2;s2v3a2;s3v2 , (8b)
where each summation runs over three permutations σ, τ, ζ ∈ S4 giving three 4-tuples
(s1, t1, u1, v1) = (σ(x), σ(y), σ(z), σ(w)), also (s2, t2, u2, v2) = (τ(x), τ(y), τ(z), τ(w)),
and (s3, t3, u3, v3) = (ζ(x), ζ(y), ζ(z), ζ(w)).
Again, our task is to explain how these formulas are obtained, i.e. how one can
guess the right partitionings of xxxyyyzzzwww in each monomial into three 4-tuples
(x, y, z, w).
Remark 2. The partitioning xxxyyyzzzwww = xyzwtxyzwtxyzw within the second
marker-monomial in the polynomial under the sum for the velocity ȧ1 in (8a) is different
from the analogous partitioning in the second marker-monomial (with coefficient −3)
in the mirror-symmetric formula (8b) of the velocity ȧ2. The structural inequivalence of
the two partitionings does occur modulo the relabelling a1 ⇄ a2 and modulo arbitrary
reshuffles of the three 4-tuples {s, t, u, v}k indexed by k ∈ {1, 2, 3} and arbitrary
permutations of s, t, u, v in any of the 4-tuples. Indeed, the last marker-monomial
in (8a) for ȧ1 contains the product of second derivatives a1;s1u2 · a1;t2v3 in which all the
three 4-tuples are mixed, whereas one of the 4-tuples is not present at all in the product
of second derivatives a2;s2v3 · a2;s3v2 in the last marker-monomial in (8b) for ȧ2.
Yet both the marker-monomials yield the sums (over σ, τ, ζ ∈ S4) which are mirror-
reflections of each other under the swap a1 ⇄ a2. This is an example of marker-
monomials’ hidden symmetry which we discuss in the next section.
Example 10 (γ3-flow over R4 with ϱ 6≡ 1). The 33,084 terms in ȧ1 or in its mirror-
reflection ȧ2 are captured – for the tetrahedral γ3-flow on the space of generalized
Nambu–Poisson brackets P (ϱ, [a1], [a2]) on R4 3 x = (x, y, z, w) – by three Civita sym-
bols (or equivalently, by three permutations) using the formulas
on R3 are completely determined by the eleven marker-monomials in the trivializing vector field X⃗,
which we obtain for the γ3-flow over R3 in section 5.1.
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 459
∑ (
ȧ = (−)σ1 (−)σ2 (−)σ31 · 3a 31;s1u2u3a1;t1t2a2;s2a2;s3v1a2;t3u1a1;v2a1;v3ϱ
σ1,σ2,σ3∈S4
+ 6a 3 21;s1u2a1;t1a1;t2v3a1;u3v1v2a2;t3u1a2;s2a2;s3ϱ + 3a1;v2a1;t1u2v3a2;v1ϱs1a1;u1a1;u3a2;s2t3a2;s3t2ϱ
− 6a1;s1v3a2;s2t1ϱv1a
2
1;s3u1v2a1;u2a1;u3a2;t2a2;t3ϱ − 6a1;s1v2v3a1;t1a1;u2a1;u3v1a2;s2a2;s3u1a2;t3ϱt2ϱ
2
+ 6a 21;s1a1;s2v3a1;s3u1a1;t1t2t3a2;v1ϱv2a2;u2a2;u3ϱ − 6a1;s1a1;t1u2u3a2;u1v2a1;t2a1;t3a2;s2a2;s3ϱv1ϱv3ϱ )
+ 6a∑1;v2a1;s1a1;s2s3t1a1;t2t3ϱv(1ϱv3a2;u1a2;u2a2;u3ϱ− 2a1;s1a1;t2a1;t3u1u2a1;u3a2;t1a2;s2a2;s3ϱv1ϱv2ϱv3 , (9a)
ȧ = (−)σ1 (−)σ2 (−)σ3 · 3a 32 1;s1a2;t1u2a2;u1u3v2a1;s2t3a1;s3t2a2;v1a2;v3ϱ
σ1,σ2,σ3∈S4
− 3a1;s1t2a2;u1a2;u2u3v1a1;t1a1;t3a
3
2;s2v3a2;s3v2ϱ − 6a1;u1a2;t1t2v3ϱt3a1;u2v1a
2
1;u3v2a2;s1a2;s2a2;s3ϱ
+ 6a 2 21;s1a1;t1t3a1;u2a2;v2a2;s2v1v3a2;s3t2a2;u1ϱu3ϱ + 6a1;t1u2a2;u1v2ϱu3a2;s1s2s3a1;v1a1;v3a2;t2a2;t3ϱ
+ 3a 22;v1a2;s1s2s3ϱt1a2;t2v3a2;t3v2a1;u1a1;u2a1;u3ϱ − 6a1;t1u2a2;u1u3v2a1;v1a1;v3ϱt2ϱt3a2;s1a2;s2a2;s3ϱ )
− 6a2;t1u2v3a2;u1u3a2;v1a2;v2ϱt2ϱt3a1;s1a1;s2a1;s3ϱ+ 2a2;s1a2;s2s3t1ϱv1a2;v2a2;v3ϱt2ϱt3a1;u1a1;u2a1;u3 , (9b)
here {si, ti, ui, vi} = σi(x, y, z, w) for σi ∈ S4. Finding a compact expression of ϱ̇ 6≡ 0
with 90,024 differential monomials in it, now using three Civita symbols, is a compu-
tationally much larger task than collapsing the velocities of the Casimirs.
4. Marker-monomials and their hidden symmetry
Definition 1. A marker-monomial in the fibre variables ϱ, a1, . . ., ad−2 over the base
variables x1∑ , . . ., x
d is a differential monomial in the jet variables ϱκ, a1;λ1 , . . ., ad−2;λd−2
(here the multi-indices κ, λi for the derivatives satisfy⋃0 ⩽ |κ|, |λ⋃i| < ∞) such that
|κ| + d−2i=1 |λi| = µ · d with µ ∈ N⩾1, such that κ ∪
d−2 µ 1 d
i=1 λi = k=1{x , . . ., x }k,
and such that all the base variables xℓ∑in the multi-indices (denoting the respectivederivatives) are partitioned into µ disjoint d-tuples x1, . . ., xd. E∑very such tuple thencorresponds to its own alternating sum σ∈S (−)σσ(x1, . . ., xd) acting on the marker-d
monomial, or equivalently, corresponds to the Civita summation ı⃗ εı⃗ chosen such that
the marker-monomial “as is” occurs with the plus sign when ı⃗ = (1, 2, . . ., d), so the
base variables x1, . . ., xd in the d-tuple are then represented by the variables xi1 , . . .,
xid in the subscripts, respectively.
Definition 2. A marker-monomial is called zero if the alternating sum over all permu-
tations of all the d-tuples (x1, . . ., xd) in it is identically equal to zero.
Example 11. Let x, y be the base variables and ϱ be the fibre v⊔ariable. Consider the
marker-monomial M1 = ϱx1ϱy1ϱx2y2 with the two-tuples {x1y1} {x2y2}. Taking the
alternatin∑g su∑m,
(−)σ(−)τϱσ(x)ϱσ(y) · ϱτ(x)τ(y) = (ϱxϱy − ϱyϱx) · (ϱxy − ϱyx) ≡ 0,
σ∈S2 τ∈S2
we establish that the marker-monomial M1 is a zero marker.
• But let us instead take the marker-monomial M2 = ϱx1ϱy2ϱx2y1 with a different
partitioning of the letters xxyy as they are seen in M1. We now have that
ϱxϱyϱxy − ϱyϱyϱxx − ϱxϱxϱyy + ϱyϱxϱyx 6≡ 0. (10)
460 R. BURING, D. LIPPER, AND A. V. KISELEV
In other words, the new marker-monomial M2 is not zero any longer, even though the
profile |σ1| = 1 = |σ2|, |σ3| = 2 of the comultiples in the product ϱσ1 · ϱσ2 · ϱσ3 is the
same as in M1.
Definition 3. The differential profile of orders of the derivatives in a marker-monomial
M = ϱκ1ϱκ2 . . . a1;λ1 . . . ad−2;µ1 . . . is the set of pairs {ϱ|κ1|, ϱ|κ2|, . . ., a1|λ1|, . . ., ad−2|µ1|,
} def. . . = {ϱ|κ1||κ2| . . . , a1|λ1| . . . , . . ., ad−2|µ1| . . . }: each (instance of a) fibre variable is
followed by the nonnegative order(s) of its derivative(s).2
Example 12. Both marker-monomials in Example 11 have the same differential profile
ϱ1ϱ1ϱ2 (equivalently, ϱ112), yet M1 is zero whereas M2 is not zero as a marker.
The differential profile of a marker-monomial is thus a coarse invariant (w.r.t. per-
mutations of all the base variable in it, or w.r.t. a permutation of the base variables
within one of the d-tuples x1, . . ., xd). It is clear also that marker-monomials of unequal
differential profiles cannot be obtained one from another by permuting the comultiples
or by permuting the base variables (what the alternating sum does by definition). This
implies that to represent a differential polynomial by an alternating sum over the per-
mutations which act on the base variable in the marker-monomials, the sums of terms
of unequal differential profiles can be processed independently one from another.
Remark 3. Representations of a differential polynomial by using marker-monomials are
not unique. Indeed, the marker can be picked for any value of the permutation(s). For
inst∑ance, we have that ∑
(−)σ(−)τϱ σ τ
∑σ(x)
ϱτ(x)ϱσ(y)τ(y) = (−) (−) ϱσ(y ϱ∑) τ(y)
ϱσ(x)τ(x) =
σ,τ∈S2 σ,τ∈S2
= − (−)σ(−)τϱ σ τσ(x)ϱτ(y)ϱσ(y)τ(x) = − (−) (−) ϱσ(y)ϱτ(x)ϱσ(x)τ(y).
σ,τ∈S2 σ,τ∈S2
Indeed, each of the four choices of the monomial marks the same expression, ϱ2xϱyy −
2ϱxϱ
2
yϱxy + ϱyϱxx 6≡ 0.
At the same time, for two nonzero marker-monomials of equal differential profiles it
can be that their alternating sums are neither equal nor proportional to each other but
intersect, that is, the two resulting differential polynomials have common term(s).
Counterexample 13. The monomial axϱxayyϱxy is a term in the alternating sums for
the markers
M3 = aσ(x)ϱτ(x)aσ(y)ζ(y)ϱζ(x)τ(y) and M4 = aσ(x)ϱτ(x)aτ(y)ζ(y)ϱζ(x)σ(y),
indeed showing up when σ = τ = ζ = id, but the two fully alternating sums are not
equal, ∑ ∑
(−)σ(−)τ (−)ζσ ⊗ τ ⊗ ζ(M3) =6 (−)σ(−)τ (−)ζσ ⊗ τ ⊗ ζ(M4),
σ,τ,ζ∈S2 σ,τ,ζ∈S2
which can be seen by straightforward expansion. The two differential polynomials are
not even multiples of one another.
2The first variant of notation is inevitable if some of the orders is at least 10; in this note, the other
variant of notation is enough (see Tables 1–2 in the next section).
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 461
This implies that to represent a given sum, the marker-monomial can be unique (up
to a given permutation of the base variables in a d-tuple) but the choice of the base
variables’ partitioning (into the disjoint d-tuples) can be not unique, and only the right
choice does the job. This ambiguity yields a nontrivial problem of finding the “true”
partitioning of the µ · d derivatives into µ tuples {x1, . . ., xd} in each term of the right-
hand sides ϱ̇([ϱ], [a]), ȧℓ([ϱ], [a]) for a given Kontsevich’s graph flow on the space of
Nambu–Poisson brackets P (ϱ, [a]).
We discover that this anticipated ambiguity is heavily suppressed by an extra, so far
hidden symmetry of these graph flows on this particular class of Poisson brackets on Rd.
Proposition 5 (ȧ, ϱ̇ for γ3-flow over R3). In the evolution ϱ̇, ȧ which is induced by the
tetrahedral flow on the class of generalized (ϱ 6≡ 1) Nambu–Poisson brackets P (ϱ, [a])
on R3, the count of differential monomials of unequal differential profiles is presented
in Table 1.3
Table 1. The number of monomials and their differential profiles in ȧ
and ϱ̇ for the tetrahedral γ3-flow over R3.
In ȧ In ϱ̇
54: a1113ϱ111 54: a111ϱ1113
102: a112ϱ1112
102: a1123ϱ011 102: a112ϱ0113
96: a122ϱ0112
72: a1223ϱ001 72: a122ϱ0013
• For each of the three differential profiles of monomials in ȧ and five in ϱ̇, we dis-
cover that for any choice of nonzero marker-monomial with that profile, its total skew-
symmetrization (using three permutations, each acting on its own tuple xyz), taken
with a suitable nonzero coefficient, exactly equals the entire sum of all the terms with
that differential profile. In other words, for each of the 3+5 differential profiles of mono-
mials in ȧ and ϱ̇ respectively, the total skew-symmetrizations of all nonzero markers of
a fixed profile are multiples of each other.
This reveals a previously hidden, extra symmetry of the objects in Kontsevich’s flow
under study.
The case of Nambu–Poisson structures (with ϱ 6≡ 1) on R4, when the tetrahedral
γ3-flow induces the evolution ȧ1, ȧ2 and ϱ̇ ≡6 0, is even more interesting: we observe
the exact same extra symmetry for all but one differential profiles, and one profile
exceptionally requires the use of two marker-monomials.
3 From Proposition 1 we recall that the velocity ȧ is encoded using the Kontsevich graphs by
formula (5). Because the entire flow 4Ṗ = O⃗r(γ3)(P⊗ ) is specified by the directed graph cocycle
O⃗r(γ3), the velocity ϱ̇ is deduced from Eq. (6). One can inspect in full detail how the arrows, targeted
on a copy of a in the construction of ȧ, spread over copies of ϱ and a to form ϱ̇ in Eq. (2). This is why
there is much similarity in the differential profiles of terms in the two velocities (as seen from Table 1).
462 R. BURING, D. LIPPER, AND A. V. KISELEV
Proposition 6 (ȧ1, ȧ2 for γ3-flow with ϱ 6≡ 1 over R4). The count of differential mono-
mials of unequal profiles in the velocities ȧ1 and ȧ2 (see Example 9) is summarized in Ta-
ble 2. (The symmetry in how the Casimirs a1 and a2 appear in the Nambu-determinant
Poisson bracket is naturally reflected in their evolution under the tetrahedral γ3-flow).
Table 2. The count of monomials w.r.t. their differential profiles in ȧ1
and ȧ2 for the tetrahedral γ3-flow on the space of generalized (ϱ 6≡ 1)
Nambu–Poisson brackets on R4.
In ȧ1 In ȧ2
4512: a11123a2122ϱ000 4512: a1122a21123ϱ000
4512: a11223a2112ϱ000 4512: a1112a21223ϱ000
3168: a11113a2122ϱ001 3168: a1122a21113ϱ001
7872: a11123a2112ϱ001 7872: a1112a21123ϱ001
3168: a11223a2111ϱ001 3168: a1111a21223ϱ001
3984: a11113a2112ϱ011 3984: a1112a21113ϱ011
3984: a11123a2111ϱ011 3984: a1111a21123ϱ011
1848: a11113a2111ϱ111 1848: a1111a21113ϱ111
• The homogeneous differential polynomial components of all profiles except the 7872
terms with a11123a2112ϱ001 and the 7872 terms with a1112a21123ϱ001 enjoy the same
extra symmetry as at d = 3: just one, arbitrarily chosen nonzero marker-monomial
suffices to express the entire sum. In particular, this is always so in the restricted case
ϱ ≡ 1 when ϱ̇ ≡ 0 and the nontrivial velocities ȧ1, ȧ2 realize the entire evolution of the
class {P (ϱ ≡ 1, [a1], [a2])}.
In either of the two exceptional cases (one in ȧ1 and the other in ȧ2, with necessarily
ϱ 6≡ 1), when two marker-monomials are needed, the first choice is still arbitrary but
the next choice is constrained by the former.4
The marker-monomial expression of ϱ̇ in the generic case ϱ 6≡ 1 on R4 carrying the
tetrahedral γ3-flow – and a simultaneous study of the presence or absence of the new
extra symmetry in it – is a computationally challenging problem; the same applies to
the pentagon-wheel γ5-flow on R3 (to collapse the known evolution ϱ̇, ȧ 6≡ 0 by using
five Civita symbols and to check the extra symmetry in the course of building the
hypotheses about the µ · d = 5 · 3 base variables’ partitioning into µ · {xyz}).
5. Vector fields which trivialize the flows of Nambu brackets
Finally, we examine the Poisson triviality of the restriction of Kontsevich’s graph flow
n
Ṗ = O⃗r(γ)(P⊗ ) to the space of Nambu–Poisson structures P (ϱ, [a]) on Rd. (There is
no known mechanism for Kontsevich’s graph flows to be trivial in the second Poisson
cohomology of P for nontrivial graph cocycles γ and generic Poisson structures.)
4 This looks similar to the construction of a basis in E2 by using a root system with the Coxeter
graph •—•: selecting the first vector is free but as one proceeds, the remaining direction is constrained.
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 463
5.1. The tri∑vializing vector field X⃗(γ3, ϱ, a) and Civita symbols in it. Let usmake a few estimates of differential polynomial degrees and orders. For every graph
cocycle γ = ℓ cℓ · γℓ with graphs γℓ on n vertices and 2n − 2 edges, the restriction
of Kontsevich’s flow nṖ = O⃗r(γ)(P⊗ ) to the space of generalized (ϱ 6≡ 1) Nambu-
determinant Poisson bi-vectors P (ϱ, [a]) with d− 2 global Casimirs a = (a1, . . . , ad−2)
on Rd contains, in each term of the differential-polynomial coefficient of the bi-vector
P (ϱ, [a]), n·(d−2)+2n−2 = nd−2 derivatives spread over ϱn ·∏an1 . . . and−2. (Hence there
are (nd−2)−(d−2) = (n−1)d derivatives spread over ϱn−1 ·∑an · ′ n−1k j ̸=k aj in ȧk and over
ϱn · an−1 n−11 · · · ad−2 in ϱ̇.) The trivializing vector field X⃗ =
d X ii=1 ([ϱ], [a]) ∂/∂x
i with
differential-polynomial coefficients satisfying the coboundary equation O⃗r(γ)(P⊗n) =
[[P, X⃗]] for P (ϱ, [a]) would therefore have (nd−2)−(d−2)−1 = (n−1)d−1 derivatives
spread over ϱn−1 · an−11 · · · an−1 in every term of each coefficient X id−2 . The Civita mech-
anism of base coordinates’ partitioning now applies to the trivializing vector field. For
the object X⃗ to be a vector field under coordinate reparametrizations x(x′) ⇄ x′(x),
the behaviour of n − 1 comultiples ϱ dictates that there are n − 1 Civita symbols in
each X i:
∑
1 n−1
X⃗ = εı⃗ · · · εı⃗n−1 ·X i1
ı⃗ 1,...,⃗ın−1 ı⃗ ,...,⃗ı
n−2;in−11 ···i
n−1 · ∂/∂x d . (11)
d−1
In other words, the vector field coefficients X i collapse by using all the indices of n− 2
Civita symbols ı⃗αε on Rd and by using all but one last index of the (n − 1)th Civita
symbol.
It is readily seen that if the trivializing vector field exists, the velocities of the scalar
Casimirs are ȧk = −X⃗(ak); the proof is standard. N(o∑ntrivial here is that ze)ro marker-
monomials can be produced in the velocity ȧ = − d X ik i=1 ([ϱ], [a]) ∂/∂xi (ak) from
nonzero marker-monomials in the right-hand side of (11). This prompts that the ve-
locity ȧk, which was obtained directly from the graph cocycle γ by using formula (5),
can involve fewer marker-monomials than there are terms to express the coefficient X i
by (11). We observe this effect already in the simplest case, namely for the Kontsevich
tetrahedral flow (so n = 4) and the generalized (ϱ 6≡ 1) Nambu-determinant Poisson
structures P (ϱ, [a]) on R3 (so d = 3).
Theorem 7. The Kontsevich tetrahedral flow 4Ṗ = O⃗r(γ ⊗3)(P ) for the Nambu–Poisson
brackets P (ϱ, [a]) on R3 is Poisson-cohomology trivial.
• The equivalence class X⃗ mod [[P,H]] of trivializing vector fields X⃗ satisfying the
coboundary condition 4O⃗r(γ3)(P⊗ ) = [[P, X⃗]] is represented by the following vector field
with differential-polynomial coefficients X i([ϱ], [a]):
∑
X⃗ = εı⃗εȷ⃗εk⃗ ·X
ı⃗,⃗ȷ,⃗k ı⃗ ȷ⃗ k⃗
,
where
464 R. BURING, D. LIPPER, AND A. V. KISELEV
Xı⃗ ȷ⃗ k⃗ =+ 12ϱϱ
k1
xk2ϱxi1xj1axk3axi2xj2axi3xj3 · ∂/∂x + 48ϱϱxj3ϱ k2xi1xj1axk3axi2xj2axi3xk1 · ∂/∂x
+ 8ϱxj2ϱ
j1 j1
xi1xk1ϱxi2xk2axi3axj3axk3 · ∂/∂x − 40ϱxi3ϱxj2ϱxi1xk1axj3axk3axi2xk2 · ∂/∂x
+ 8ϱ j1 i2xi3ϱxj2ϱxk3axj3axi1xk1axi2xk2 · ∂/∂x + 24ϱxj2ϱxk3ϱxi1xk1axi3axj3axj1xk2 · ∂/∂x
− 12ϱ2ϱxk2axi1xj1axi2xj2axi3xj3xk3 · ∂/∂xk1 + 24ϱϱxj2ϱxk1axk2axi1xj1a i2xi3xj3xk3 · ∂/∂x
− 36ϱϱxi2ϱxj2a a k1 i1xk2 xi1xj1axi3xj3xk3 · ∂/∂x + 8ϱxi2ϱxj1ϱxk1axj2axk2axi3xj3xk3 · ∂/∂x
− 8ϱxj1ϱxk1ϱxi3xj3xk3axi2axj2axk2 · ∂/∂xi1 .
There are eleven terms in the marker-polynomial for Xı⃗ ȷ⃗ k⃗ but only the three underlined
terms survive when the vector field X⃗ acts on the Casimir a; the rest contributes to the
velocity ȧ with zero markers. The evolution ȧ in Eq. (5) is thus reproduced: we verify
that ȧ = −X⃗(a).
5.2. Open problems about the graph flows and their trivializing vector fields
X⃗([ϱ], [a]). The study of Kontsevich flows – for the tetrahedral and pentagon-wheel
graph cocycles (or higher vertex number cocycles γ7, [γ3, γ5], γ9, etc.) – restricted to the
spaces of generalized (ρ 6≡ 1) Nambu-determinant Poisson brackets P ([ϱ], [a]) on R3
and R4 (or higher-dimensional affine spaces Rd) is, first of all, a source of combinatorial
and algorithmic problems about finding the explicit shape of the objects. In particular,
such is the task to collapse formulae, originally derived within the graph language, by
using the Civita symbols. The other set of problems concerns the geometric nature and
properties of the objects; such are the construction of the trivializing vector fields and
explanation of the deeper symmetry in the choice of marker-monomials under the sums
with Civita symbols. Let us summarize these problems in the order how they naturally
emerge.
Open problem 1 (ϱ̇ in γ3-flow over R4). Represent the known velocity ϱ̇([ϱ], [a]) for
the tetrahedral γ3-flow over R4 by using three Civita symbols. Does the choice of
marker-monomials enjoy the extra symmetry which is revealed in Proposition 5 for the
γ3-flow over R3 and in Proposition 6 for ȧ over R4?
Open problem 2 (ϱ̇, ȧ in γ5-flow over R3). Represent the known velocities ϱ̇([ϱ], [a])
and ȧ([ϱ], [a]) for the pentagon-wheel γ5-flow over R3 by using five Civita symbols. Does
the extra symmetry persist for the marker-monomials in either velocity?
Open problem 3 (X⃗ for γ3-flow on R4 with ϱ ≡ 1). Inspect whether the restriction of
the tetrahedral γ3-flow to the space of Nambu-determinant Poisson structures P ([a])
on R4 with ϱ ≡ 1 is trivial in the second Poisson cohomology. — Let us presume that
there exists a trivializing vector field X⃗([a]) with differential-polynomial coefficients.
If it actually does, represent the coefficients – of possibly another vector field from the
coset X⃗ mod [[P, ·]] – by using three Civita symbols on R4. Do the marker-monomials
in Y⃗ ([a]) enjoy the extra symmetry?
Open problem 4 (X⃗ for γ3-flow on R4 with ϱ 6≡ 1). Extend and solve Problem 3 in
the general case ϱ 6≡ 1 on R4, now for the trivializing vector field X⃗([ϱ], [a]).
Open problem 5 (X⃗ for γ -flow on R35 ). Solve the trivialization problem – fully anal-
ogous to the above Problems 3–4 – for the pentagon-wheel γ5-flow over R3. If it exists,
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 465
the trivializing vector field Y⃗ ([ϱ], [a]) from the coset X⃗ mod [[P, ·]] (defined modulo
Hamiltonian vector fields) will again be realizable by using five Civita symbols on R3.
Open problem 6. Can the trivializing vector fields X⃗([ϱ], [a]) be constructed – for
nontrivial graph cocycles γ – and induce the graph flows Ṗ = O⃗r(γ)(P⊗n) on the spaces
of Nambu-determinant Poisson brackets P (ϱ, [a]) directly from the graph cocycles γ on
n vertices and from the properties of the particular Poisson geometry of the Nambu
brackets with global Casimirs? In other words, what are the marker-monomials for
Y⃗ ∈ X⃗ mod [[P, ·]] as differential-geometric objects? (Note that the knowledge of the
vector field X⃗([ϱ], [a]) as the parent object for the Lie derivative LX⃗ and for Ṗ = [[P, X⃗]]
is enough to calculate ȧ and ϱ̇.)
Open problem 7. Is there a relation between the (pseudo)group of diffeomorphisms
generated by the highly nonlinear vector fields X⃗([ϱ], [a]) from the graph cocycle flows
and, on the other hand, the (local) diffeomorphisms x ⇄ x′ that map (by blowing up
the local coordinates) the cells bounded by ϱ(x) = 0 in Rd to the domains on which
ϱ′(x′) ≡ ±1?
In conclusion, we note that whenever they are Poisson-cohomology trivial (as we
observe so far in all the cases), the nontrivial graph cocycle flows on the spaces of
generalized Nambu-determinant Poisson brackets P (ρ, [a]) not only preserve the sym-
plectic foliation (dictated by the Casimirs a) by merely reparametrizing the coordinate
description of points still not anyhow displacing the symplectic leaves, but also preserve
the tiling of the affine space Rd with respect to the zero locus of the inverse density ϱ
in P (ϱ, [a]). Both the foliation and tiling are thus rigid under the graph cocycle flows.
Appendix A. A class of (non)polynomial Poisson brackets on Rd
without global polynomial Casimir
First, let us recall a particular construction of homogeneous polynomial-coefficient Pois-
son brackets on Rd∑ with Cartesian coordina∑tes x
1, . . ., xd.
Denote by E⃗ the Euler vector field, E⃗ = xii · ∂/∂xi, and consider another nonzero
vector field V⃗ = j 1j V (x , . . ., xd)·∂/∂xj with homogeneous polynomial coefficients V j
of total degree k  1 (conveniently starting at k = 2). This homogeneity assumption
implies that E⃗(V⃗ ) = k · V⃗ and V⃗ (E⃗) = 1 · V⃗ , whence [E⃗, V⃗ ] = (k − 1) · V⃗ .
By definition, put P := V⃗ ∧ E⃗; this is a bi-vector with homogeneous-polynomial
coefficients (of degree k + 1).
Lemma 8. All such bi-vectors P = V⃗ ∧ E⃗ on Rd are Poisson.
Proof. Let us calculate the Schouten bracket [[P, P ]] = [[V⃗ ∧ E⃗, V⃗ ∧ E⃗]] by using its
inductive definition for decomposable multi-vectors and thus, reducing it to the calcu-
lation of commutators for 1-vector fields:
[[V⃗ ∧ E⃗, V⃗ ∧ E⃗]] = V⃗ ∧ [E⃗, V⃗ ]∧ E⃗ − V⃗ ∧ [E⃗, E⃗]∧ V⃗ − E⃗ ∧ [V⃗ , V⃗ ]∧ E⃗ + E⃗ ∧ [V⃗ , E⃗]∧ V⃗
= 2V⃗ ∧ [E⃗, V⃗ ] ∧ E⃗ = 2(k − 1) · V⃗ ∧ V⃗ ∧ E⃗ ≡ 0.
This proves that the Jacobi identity 1 [[P, P ]] = 0 holds, so P is Poisson. □
2
466 R. BURING, D. LIPPER, AND A. V. KISELEV
Remark 4. The above construction of Poisson bi-vectors P = V⃗ ∧ E⃗ naturally extends
to homogeneous vector fields V⃗ (possibly not on the entire Rd) with not necessarily
polynomial coefficients but still satisfying the condition E⃗(V⃗ ) = λ · V⃗ with λ 6= 0, 1.
We now claim that not all of these Poisson bi-vectors P = V⃗ ∧ E⃗ on Rd are Nambu-
determinant type. Specifically, let us produce a family of such Poisson bi-vectors
P = V⃗ ∧ E⃗ (with polynomial coefficients) which do not admit any global non-constant
polynomial Casimirs — and this is in contrast with the Nambu class P = ϱ(x) · da/dx
for polynomial parameters a = (a1, . . ., ad−2).
Indeed, suppose that there is a polynomial Casimir a for P = V⃗ ∧ E⃗ as above.5 By
the definition of Casimir, we have that
[[P, a]] = [[V⃗ ∧ E⃗, a]] = V⃗ · E⃗(a)− E⃗ · V⃗ (a) = 0,
whence we obtain the system of PDE: for each i running from 1 to d, the Casimir a
satisfies the equation ∑ ∑
V i · xj · ∂a/∂xj = xi · V j · ∂a/∂xj.
j j
An infinite family of counterexamples is now produced by taking the vector fields V⃗
with coefficients V i := (xi)k f∑or k ⩾ 2. Indeed, w∑e obtain that
(xi)k−1 · xℓ · ∂a/∂xℓ = (xj)k · ∂a/∂xj,
ℓ j
and the Casimir a is by assumption polynomial in all j′x for j′ 6= i in particular. With
respect to every ′xj at j′ 6= i for a fixed i, 1 ⩽ i ⩽ d, the degree of the left-hand side,
viewed as a polynomial in j′x , is strictly not equal to that degree of the right-hand side
(as k > 1) unless ′∂a/∂xj ≡ 0 for all j′ 6= i. Cycling over all the equations indexed by
i in the system, we conclude that every polynomial Casimir a for the Poisson bi-vector
P = V⃗ ∧E⃗ with V i = (xi)k is a constant over Rd. (For the Nambu-determinant brackets
da/ dvol(x) this means that the bi-vector vanishes identically on Rd.) □
Acknowledgements. The first and last authors thank M.Kontsevich for helpful advice
and discussions. The research of R.B. is supported by project 5020 at the Institute of
Mathematics, Johannes Gutenberg–Universität Mainz and by CRC-326 grant GAUS
“Geometry and Arithmetic of Uniformized Structures”. The travel of A.K. was par-
tially supported by project 135110 at the Bernoulli Institute, University of Groningen.
R. Buring and A.Kiselev are grateful to the IHÉS for hospitality and financial support.
References
[1] Banks P., Panzer E., Pym B. (2020) Multiple zeta values in deformation quanti-
zation, Invent. Math. 222:1, 79–159. (Preprint arXiv:1812.11649 [math.QA])
[2] Bouisaghouane A., Buring R., Kiselev A.V. (2017) The Kontsevich tetrahedral flow
revisited, J. Geom. Phys. 119, 272–285. (Preprint arXiv:1608.01710 [q-alg]).
5 From the homogeneity of objects it is readily seen that if a is polynomial, then it is homogeneous
of some degree D (for the homogeneous brackets from the Nambu class, D equals 1 + deg(P ij):
E⃗(a) = D · a).
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 467
[3] Buring R. (2022) The action of Kontsevich’s graph complex on Poisson structures
and star products: an implementation. PhD dissertation, Johannes Gutenberg–Uni-
versität Mainz, in preparation.
[4] Buring R., Kiselev A.V. (2019) The expansion ⋆ mod ō(ℏ4) and computer-
assisted proof schemes in the Kontsevich deformation quantization, Experi-
mental Math., 54 p. (doi:10.1080/10586458.2019.1680463, in press). (Preprint
arXiv:1702.00681 [math.CO])
[5] Buring R., Kiselev A.V. (2019) The orientation morphism: from graph cocycles
to deformations of Poisson structures, J. Phys.: Conf. Ser. 1194 Proc. 32nd Int.
colloquium on Group-theoretical methods in Physics: Group32 (9–13 July 2018,
CVUT Prague, Czech Republic), Paper 012017, 10 p. (Preprint arXiv:1811.07878
[math.CO])
[6] Buring R., Kiselev A.V. (2020) Universal cocycles and the graph complex action on
homogeneous Poisson brackets by diffeomorphisms, Physics of Particles and Nuclei
Letters 17:5 Supersymmetry and Quantum Symmetries 2019, 707–713. (Preprint
arXiv:1912.12664 [math.SG])
[7] Buring R., Kiselev A.V., Rutten N. J. (2017) The heptagon-wheel cocycle in the
Kontsevich graph complex, J. Nonlin. Math. Phys. 24:Suppl. 1 ‘Local & Nonlo-
cal Symmetries in Mathematical Physics’, 157–173. (Preprint arXiv:1710.00658
[math.CO])
[8] Buring R., Kiselev A.V., Rutten N. J. (2018) Infinitesimal deformations of Poisson
bi-vectors using the Kontsevich graph calculus, J. Phys.: Conf. Ser. 965 Proc.
XXV Int. conf. ‘Integrable Systems & Quantum Symmetries’ (6–10 June 2017,
CVUT Prague, Czech Republic), Paper 012010, 1–12. (Preprint arXiv:1710.02405
[math.CO])
[9] Buring R., Kiselev A.V., Rutten N. J. (2018) Poisson brackets symmetry from
the pentagon-wheel cocycle in the graph complex, Physics of Particles and Nu-
clei 49:5 Supersymmetry and Quantum Symmetries 2017, 924–928. (Preprint
arXiv:1712.05259 [math-ph])
[10] Donin J. (1998) On the quantization of quadratic Poisson brackets on a polynomial
algebra of four variables, in: Lie groups and Lie algebras, Math. Appl. 433, Kluwer
Acad. Publ., Dordrecht, 17–25.
[11] Grabowski J., Marmo G., Perelomov A.M. (1993) Poisson structures: towards a
classification, Modern Phys. Lett. A8:18, 1719–1733.
[12] Kiselev A.V. (2019) Open problems in the Kontsevich graph construction of Pois-
son bracket symmetries, J. Phys.: Conf. Ser. 1416 Proc. XXVI Int. conf. ‘Integrable
Systems & Quantum Symmetries’ (8–12 July 2019, CVUT Prague, Czech Repub-
lic), Paper 012018, 8 p. (Preprint arXiv:1910.05844 [math-ph])
[13] Kiselev A.V., Buring R. (2021) The Kontsevich graph orientation morphism re-
visited. Banach Center Publ. 123 Homotopy algebras, deformation theory & quan-
tization, 123–139. (Preprint arXiv:1904.13293 [math.CO])
[14] Kontsevich M. (1997) Formality conjecture. Deformation theory and symplectic
geometry (Ascona 1996, D. Sternheimer, J. Rawnsley and S.Gutt, eds), Math. Phys.
Stud. 20, Kluwer Acad. Publ., Dordrecht, 139–156.
468 R. BURING, D. LIPPER, AND A. V. KISELEV
[15] Kontsevich M. (1999) Operads and motives in deformation quantiza-
tion. Moshé Flato (1937–1998). Lett. Math. Phys. 48:1, 35–72. (Preprint
arXiv:q-alg/9904055)
[16] Kontsevich M. (2001) Deformation quantization of algebraic varieties. EuroCon-
férence Moshé Flato 2000, Part III (Dijon). Lett. Math. Phys. 56:3, 271–294.
(Preprint arXiv:math.AG/0106006)
[17] Kontsevich M. (2003) Deformation quantization of Poisson manifolds, Lett. Math.
Phys. 66:3, 157–216. (Preprint arXiv:q-alg/9709040)
[18] Kontsevich M. (2008) Hodge structures in non-commutative geometry (Notes by
E. Lupercio), in: XI Solomon Lefschetz Memorial Lecture series. Contemp. Math.
462 Non-commutative geometry in mathematics and physics, AMS, Providence RI,
1–21. (Preprint arXiv:math.AG/0801.4760)
[19] Kontsevich M. (2019) Derived Grothendieck–Teichmüller group and graph com-
plexes [after T.Willwacher]. Séminaire Bourbaki, Vol. 2016/2017. Exposés 1120–
1135. Astérisque 407, No. 1126, 183–211.
[20] Kontsevich M. (2019) Private communication, 13 December 2019, IHÉS.
[21] Marvan M. (2009) Sufficient set of integrability conditions of an orthonomic system,
Foundat. Comput. Math. 9, 651–674.
[22] Nambu Y. (1973) Generalized Hamiltonian dynamics, Phys. Rev. D7, 2405–2412.
[23] Willwacher T. (2015) M.Kontsevich’s graph complex and the Grothendieck–
Teichmüller Lie algebra, Invent. Math. 200:3, 671–760. (Preprint arXiv:1009.1654
[q-alg])
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 469
Appendix B. ȧ and ϱ̇ for the γ3-flow over R3
adot = -12*rho^2*a_x*rho_y*a_xy*a_zz*a_xyz+12*rho^2*a_x*rho_y*a_xy*a_xz*a_yzz+12*rho^2*a_x*rho_y*a_xy
*a_xzz*a_yz-12*rho^2*a_x*rho_y*a_xz*a_yy*a_xzz+12*rho^2*a_x*rho_y*a_xz*a_yz*a_xyz-12*rho^2*a_x
*rho_y*a_yzz*a_yz*a_xx+6*rho^2*a_x*rho_y*a_xx*a_zzz*a_yy+6*rho^2*a_x*rho_y*a_xx*a_zz*a_yyz
+6*rho^2*a_x*rho_y*a_zz*a_yy*a_xxz-12*rho^2*a_x*rho_z*a_xy*a_yz*a_xyz-12*rho^2*a_x*rho_z*a_xy
*a_xz*a_yyz+12*rho^2*a_x*rho_z*a_xy*a_zz*a_xyy-12*rho^2*a_x*rho_z*a_xz*a_xyy*a_yz+12*rho^2
*a_x*rho_z*a_xz*a_yy*a_xyz+12*rho^2*a_x*rho_z*a_yz*a_xx*a_yyz-6*rho^2*a_x*rho_z*a_xx*a_zz
*a_yyy-6*rho^2*a_x*rho_z*a_xx*a_yzz*a_yy-6*rho^2*a_x*rho_z*a_zz*a_yy*a_xxy-12*rho^2*a_y*rho_x
*a_xy*a_xz*a_yzz-12*rho^2*a_y*rho_x*a_xy*a_xzz*a_yz+12*rho^2*a_y*rho_x*a_xy*a_zz*a_xyz
-12*rho^2*a_y*rho_x*a_xz*a_yz*a_xyz+12*rho^2*a_y*rho_x*a_xz*a_yy*a_xzz+12*rho^2*a_y*rho_x
*a_yzz*a_yz*a_xx-6*rho^2*a_y*rho_x*a_xx*a_zz*a_yyz-6*rho^2*a_y*rho_x*a_xx*a_zzz*a_yy-6*rho^2
*a_y*rho_x*a_zz*a_yy*a_xxz+12*rho^2*a_y*rho_z*a_xy*a_xz*a_xyz+12*rho^2*a_y*rho_z*a_xy*a_yz
*a_xxz-12*rho^2*a_y*rho_z*a_xy*a_zz*a_xxy+12*rho*a_x*a_y*rho_z*rho_x*a_yy*a_xzz-24*rho*a_x
*a_y*rho_z*rho_x*a_yz*a_xyz+12*rho*a_x*a_y*rho_z*rho_x*a_zz*a_xyy+24*rho*a_x*a_y*rho_z*rho_y
*a_xz*a_xyz-12*rho*a_x*a_y*rho_z*rho_y*a_xx*a_yzz-12*rho*a_x*a_y*rho_z*rho_y*a_zz*a_xxy
+24*rho*a_x*a_z*rho_y*rho_x*a_yz*a_xyz-12*rho*a_x*a_z*rho_y*rho_x*a_zz*a_xyy-12*rho*a_x*a_z
*rho_y*rho_x*a_yy*a_xzz-24*rho*a_x*a_z*rho_z*rho_y*a_xyz*a_xy+12*rho*a_x*a_z*rho_z*rho_y*a_xx
*a_yyz+12*rho*a_x*a_z*rho_z*rho_y*a_xxz*a_yy+12*rho*a_z*a_y*rho_x*rho_y*a_xx*a_yzz-24*rho*a_z
*a_y*rho_x*rho_y*a_xz*a_xyz+12*rho*a_z*a_y*rho_x*rho_y*a_zz*a_xxy+24*rho*a_z*a_y*rho_x*rho_z
*a_xyz*a_xy-12*rho*a_z*a_y*rho_x*rho_z*a_xx*a_yyz-12*rho*a_z*a_y*rho_x*rho_z*a_xxz*a_yy
-6*rho^2*a_x*rho_y*a_zzz*a_xy^2-6*rho^2*a_x*rho_y*a_xz^2*a_yyz-6*rho^2*a_x*rho_y*a_yz^2*a_xxz
+6*rho^2*a_x*rho_z*a_yzz*a_xy^2+6*rho^2*a_x*rho_z*a_xz^2*a_yyy+6*rho^2*a_x*rho_z*a_yz^2*a_xxy
+6*rho^2*a_y*rho_x*a_zzz*a_xy^2+6*rho^2*a_y*rho_x*a_xz^2*a_yyz+6*rho^2*a_y*rho_x*a_yz^2*a_xxz
-6*rho^2*a_y*rho_z*a_xzz*a_xy^2-6*rho^2*a_y*rho_z*a_xz^2*a_xyy-6*rho^2*a_y*rho_z*a_yz^2*a_xxx
-6*rho^2*a_z*rho_x*a_yzz*a_xy^2-6*rho^2*a_z*rho_x*a_xz^2*a_yyy-6*rho^2*a_z*rho_x*a_yz^2*a_xxy
+6*rho^2*a_z*rho_y*a_xzz*a_xy^2+6*rho^2*a_z*rho_y*a_xz^2*a_xyy+6*rho^2*a_z*rho_y*a_yz^2*a_xxx
-6*rho*a_x^2*rho_y^2*a_zz*a_xyz-6*rho*a_x^2*rho_y^2*a_xy*a_zzz+6*rho*a_x^2*rho_y^2*a_xz*a_yzz
+6*rho*a_x^2*rho_y^2*a_xzz*a_yz-6*rho*a_x^2*rho_z^2*a_xyy*a_yz-6*rho*a_x^2*rho_z^2*a_xy*a_yyz
+6*rho*a_x^2*rho_z^2*a_xz*a_yyy+6*rho*a_x^2*rho_z^2*a_yy*a_xyz-6*rho*a_y^2*rho_x^2*a_xzz*a_yz
+6*rho*a_y^2*rho_x^2*a_zz*a_xyz+6*rho*a_y^2*rho_x^2*a_xy*a_zzz-6*rho*a_y^2*rho_x^2*a_xz*a_yzz
-6*rho*a_y^2*rho_z^2*a_xxx*a_yz+6*rho*a_y^2*rho_z^2*a_xxz*a_xy-6*rho*a_y^2*rho_z^2*a_xx*a_xyz
+6*rho*a_y^2*rho_z^2*a_xxy*a_xz+6*rho*a_z^2*rho_x^2*a_xy*a_yyz-6*rho*a_z^2*rho_x^2*a_xz*a_yyy
-6*rho*a_z^2*rho_x^2*a_yy*a_xyz+6*rho*a_z^2*rho_x^2*a_xyy*a_yz+6*rho*a_z^2*rho_y^2*a_xxx*a_yz
+6*rho*a_z^2*rho_y^2*a_xx*a_xyz-6*rho*a_z^2*rho_y^2*a_xxy*a_xz-6*rho*a_z^2*rho_y^2*a_xxz*a_xy
+6*a_x^2*a_y*rho_x*a_zzz*rho_y^2+6*a_x^2*a_y*rho_x*a_yyz*rho_z^2+12*a_x^2*a_y*a_xyz*rho_y
*rho_z^2-6*a_x^2*a_y*a_xzz*rho_z*rho_y^2-6*a_x^2*a_z*rho_x*rho_y^2*a_yzz-6*a_x^2*a_z*rho_x
*rho_z^2*a_yyy+6*a_x^2*a_z*a_xyy*rho_z^2*rho_y-12*a_x^2*a_z*a_xyz*rho_z*rho_y^2+6*a_x*a_y^2
*rho_x^2*rho_z*a_yzz-6*a_x*a_y^2*rho_x^2*a_zzz*rho_y-12*a_x*a_y^2*rho_x*a_xyz*rho_z^2-6*a_x
*a_y^2*a_xxz*rho_z^2*rho_y+6*a_x*a_z^2*rho_x^2*rho_z*a_yyy-6*a_x*a_z^2*rho_x^2*a_yyz*rho_y
+12*a_x*a_z^2*rho_x*a_xyz*rho_y^2+6*a_x*a_z^2*a_xxy*rho_z*rho_y^2+12*a_z*a_y^2*rho_x^2*rho_z
*a_xyz+6*a_z*a_y^2*rho_x^2*rho_y*a_xzz-6*a_z*a_y^2*rho_x*a_xxy*rho_z^2+6*a_z*a_y^2*a_xxx*rho_y
*rho_z^2-6*a_z^2*a_y*rho_x^2*a_xyy*rho_z-12*a_z^2*a_y*rho_x^2*a_xyz*rho_y+6*a_z^2*a_y*rho_x
*rho_y^2*a_xxz-6*a_z^2*a_y*a_xxx*rho_z*rho_y^2+6*a_x^3*a_yzz*rho_y^2*rho_z-6*a_x^3*a_yyz
*rho_z^2*rho_y-6*a_x^2*a_y*a_xyy*rho_z^3+6*a_x^2*a_z*a_xzz*rho_y^3+6*a_x*a_y^2*a_xxy*rho_z^3
-6*a_x*a_z^2*a_xxz*rho_y^3+6*a_y^3*a_xxz*rho_x*rho_z^2-6*a_y^3*a_xzz*rho_x^2*rho_z-6*a_z*a_y^2
*a_yzz*rho_x^3+6*a_z^2*a_y*a_yyz*rho_x^3+6*a_z^3*a_xyy*rho_x^2*rho_y-6*a_z^3*a_xxy*rho_x
*rho_y^2-12*rho^2*a_y*rho_z*a_xz*a_xxz*a_yy+12*rho^2*a_y*rho_z*a_xz*a_yz*a_xxy-12*rho^2*a_y
*rho_z*a_yz*a_xx*a_xyz+6*rho^2*a_y*rho_z*a_xx*a_zz*a_xyy+6*rho^2*a_y*rho_z*a_xx*a_yy*a_xzz
+6*rho^2*a_y*rho_z*a_zz*a_yy*a_xxx+12*rho^2*a_z*rho_x*a_xy*a_xz*a_yyz+12*rho^2*a_z*rho_x*a_xy
*a_yz*a_xyz-12*rho^2*a_z*rho_x*a_xy*a_zz*a_xyy-12*rho^2*a_z*rho_x*a_xz*a_yy*a_xyz+12*rho^2*a_z
*rho_x*a_xz*a_xyy*a_yz-12*rho^2*a_z*rho_x*a_yz*a_xx*a_yyz+6*rho^2*a_z*rho_x*a_xx*a_zz*a_yyy
+6*rho^2*a_z*rho_x*a_xx*a_yzz*a_yy+6*rho^2*a_z*rho_x*a_zz*a_yy*a_xxy-12*rho^2*a_z*rho_y*a_xy
*a_yz*a_xxz+12*rho^2*a_z*rho_y*a_xy*a_zz*a_xxy-12*rho^2*a_z*rho_y*a_xy*a_xz*a_xyz-12*rho^2*a_z
*rho_y*a_xz*a_yz*a_xxy+12*rho^2*a_z*rho_y*a_xz*a_xxz*a_yy+12*rho^2*a_z*rho_y*a_yz*a_xx*a_xyz
-6*rho^2*a_z*rho_y*a_xx*a_zz*a_xyy-6*rho^2*a_z*rho_y*a_xx*a_yy*a_xzz-6*rho^2*a_z*rho_y*a_zz
*a_yy*a_xxx+6*rho*a_x^2*rho_x*rho_y*a_zz*a_yyz+6*rho*a_x^2*rho_x*rho_y*a_zzz*a_yy-12*rho*a_x^2
*rho_x*rho_y*a_yzz*a_yz-6*rho*a_x^2*rho_x*rho_z*a_yzz*a_yy-6*rho*a_x^2*rho_x*rho_z*a_zz*a_yyy
+12*rho*a_x^2*rho_x*rho_z*a_yyz*a_yz+6*rho*a_x^2*rho_z*rho_y*a_zz*a_xyy+12*rho*a_x^2*rho_z
*rho_y*a_yzz*a_xy-6*rho*a_x^2*rho_z*rho_y*a_yy*a_xzz-12*rho*a_x^2*rho_z*rho_y*a_xz*a_yyz
-6*rho*a_x*a_y*rho_x^2*a_zz*a_yyz-6*rho*a_x*a_y*rho_x^2*a_zzz*a_yy+12*rho*a_x*a_y*rho_x^2
*a_yzz*a_yz+6*rho*a_x*a_y*rho_y^2*a_zz*a_xxz-12*rho*a_x*a_y*rho_y^2*a_xzz*a_xz+6*rho*a_x*a_y
*rho_y^2*a_xx*a_zzz-6*rho*a_x*a_y*rho_z^2*a_xxz*a_yy+6*rho*a_x*a_y*rho_z^2*a_xx*a_yyz+12*rho
*a_x*a_y*rho_z^2*a_yz*a_xxy-12*rho*a_x*a_y*rho_z^2*a_xz*a_xyy-12*rho*a_x*a_z*rho_x^2*a_yyz
*a_yz+6*rho*a_x*a_z*rho_x^2*a_zz*a_yyy+6*rho*a_x*a_z*rho_x^2*a_yzz*a_yy-6*rho*a_x*a_z*rho_y^2
470 R. BURING, D. LIPPER, AND A. V. KISELEV
*a_xx*a_yzz+6*rho*a_x*a_z*rho_y^2*a_zz*a_xxy+12*rho*a_x*a_z*rho_y^2*a_xzz*a_xy-12*rho*a_x*a_z
*rho_y^2*a_yz*a_xxz+12*rho*a_x*a_z*rho_z^2*a_xyy*a_xy-6*rho*a_x*a_z*rho_z^2*a_xxy*a_yy-6*rho
*a_x*a_z*rho_z^2*a_xx*a_yyy-6*rho*a_y^2*rho_x*rho_y*a_zz*a_xxz+12*rho*a_y^2*rho_x*rho_y*a_xzz
*a_xz-6*rho*a_y^2*rho_x*rho_y*a_xx*a_zzz+12*rho*a_y^2*rho_x*rho_z*a_yz*a_xxz+6*rho*a_y^2*rho_x
*rho_z*a_xx*a_yzz-6*rho*a_y^2*rho_x*rho_z*a_zz*a_xxy-12*rho*a_y^2*rho_x*rho_z*a_xzz*a_xy-12
*rho*a_y^2*rho_z*rho_y*a_xz*a_xxz+6*rho*a_y^2*rho_z*rho_y*a_xxx*a_zz+6*rho*a_y^2*rho_z*rho_y
*a_xzz*a_xx+12*rho*a_z*a_y*rho_x^2*a_xz*a_yyz-6*rho*a_z*a_y*rho_x^2*a_zz*a_xyy-12*rho*a_z*a_y
*rho_x^2*a_yzz*a_xy+6*rho*a_z*a_y*rho_x^2*a_yy*a_xzz+12*rho*a_z*a_y*rho_y^2*a_xz*a_xxz-6*rho
*a_z*a_y*rho_y^2*a_xxx*a_zz-6*rho*a_z*a_y*rho_y^2*a_xzz*a_xx+6*rho*a_z*a_y*rho_z^2*a_yy*a_xxx
+6*rho*a_z*a_y*rho_z^2*a_xyy*a_xx-12*rho*a_z*a_y*rho_z^2*a_xy*a_xxy-12*rho*a_z^2*rho_x*rho_y
*a_yz*a_xxy+6*rho*a_z^2*rho_x*rho_y*a_xxz*a_yy-6*rho*a_z^2*rho_x*rho_y*a_xx*a_yyz+12*rho*a_z^2
*rho_x*rho_y*a_xz*a_xyy+6*rho*a_z^2*rho_x*rho_z*a_xxy*a_yy+6*rho*a_z^2*rho_x*rho_z*a_xx*a_yyy
-12*rho*a_z^2*rho_x*rho_z*a_xyy*a_xy+12*rho*a_z^2*rho_z*rho_y*a_xy*a_xxy-6*rho*a_z^2*rho_z
*rho_y*a_xyy*a_xx-6*rho*a_z^2*rho_z*rho_y*a_yy*a_xxx-12*a_x^2*a_y*rho_x*rho_y*rho_z*a_yzz
+12*a_x^2*a_z*rho_x*rho_y*rho_z*a_yyz+12*a_x*a_y^2*rho_x*rho_y*rho_z*a_xzz+12*a_x*a_z*a_y
*rho_x^2*a_yzz*rho_y-12*a_x*a_z*a_y*rho_x^2*rho_z*a_yyz-12*a_x*a_z*a_y*rho_x*rho_y^2*a_xzz
+12*a_x*a_z*a_y*rho_x*rho_z^2*a_xyy-12*a_x*a_z*a_y*a_xxy*rho_z^2*rho_y+12*a_x*a_z*a_y*a_xxz
*rho_z*rho_y^2-12*a_x*a_z^2*rho_x*rho_y*rho_z*a_xyy-12*a_z*a_y^2*rho_x*rho_y*rho_z*a_xxz+12
*a_z^2*a_y*rho_x*rho_y*rho_z*a_xxy-2*a_x^3*a_zzz*rho_y^3+2*a_x^3*a_yyy*rho_z^3+2*a_y^3*a_zzz
*rho_x^3-2*a_y^3*a_xxx*rho_z^3+2*a_z^3*a_xxx*rho_y^3-2*a_z^3*a_yyy*rho_x^3
rhodot =
-12*rho*rho_x*rho_y*a_x*a_z*rho_xyy*a_zz-12*rho*rho_x*rho_y*a_x*a_z*rho_xzz*a_yy+24*rho*rho_x
*rho_y*a_x*a_z*rho_xyz*a_yz+12*rho*rho_x*rho_y*a_x*a_xy*rho_yz*a_zz-12*rho*rho_x*rho_y*a_x
*a_xy*a_yz*rho_zz-12*rho*rho_x*rho_y*a_x*a_xz*a_yz*rho_yz+6*rho^2*rho_x*a_y*rho_zzz*a_xy^2
+6*rho^2*rho_x*a_y*rho_yyz*a_xz^2+6*rho^2*rho_x*a_y*rho_xxz*a_yz^2-6*rho^2*rho_x*a_z*rho_yzz
*a_xy^2-6*rho^2*rho_x*a_z*rho_yyy*a_xz^2-6*rho^2*rho_x*a_z*rho_xxy*a_yz^2-6*rho^2*rho_y*a_x
*rho_zzz*a_xy^2-6*rho^2*rho_y*a_x*rho_yyz*a_xz^2-6*rho^2*rho_y*a_x*rho_xxz*a_yz^2+6*rho^2
*rho_y*a_z*rho_xzz*a_xy^2+6*rho^2*rho_y*a_z*rho_xyy*a_xz^2+6*rho^2*rho_y*a_z*rho_xxx*a_yz^2
+6*rho^2*rho_z*a_x*rho_yzz*a_xy^2+6*rho^2*rho_z*a_x*rho_yyy*a_xz^2+6*rho^2*rho_z*a_x*rho_xxy
*a_yz^2-6*rho^2*rho_z*a_y*rho_xzz*a_xy^2-6*rho^2*rho_z*a_y*rho_xyy*a_xz^2-6*rho^2*rho_z*a_y
*rho_xxx*a_yz^2+6*rho*rho_x^2*a_y^2*rho_zzz*a_xy-6*rho*rho_x^2*a_y^2*rho_xzz*a_yz+6*rho
*rho_x^2*a_y^2*rho_xyz*a_zz-6*rho*rho_x^2*a_y^2*rho_yzz*a_xz-12*rho*rho_x^2*a_y*rho_xz*a_yz^2
-6*rho*rho_x^2*a_z^2*a_xz*rho_yyy-6*rho*rho_x^2*a_z^2*rho_xyz*a_yy+6*rho*rho_x^2*a_z^2*rho_xyy
*a_yz+6*rho*rho_x^2*a_z^2*a_xy*rho_yyz+12*rho*rho_x^2*a_z*rho_xy*a_yz^2+6*rho*rho_y^2*a_x^2
*rho_xzz*a_yz+6*rho*rho_y^2*a_x^2*rho_yzz*a_xz-6*rho*rho_y^2*a_x^2*rho_zzz*a_xy-6*rho*rho_y^2
*a_x^2*rho_xyz*a_zz+12*rho*rho_y^2*a_x*rho_yz*a_xz^2-6*rho*rho_y^2*a_z^2*rho_xxy*a_xz-6*rho
*rho_y^2*a_z^2*a_xy*rho_xxz+6*rho*rho_y^2*a_z^2*rho_xyz*a_xx+6*rho*rho_y^2*a_z^2*a_yz*rho_xxx
-12*rho*rho_y^2*a_z*rho_xy*a_xz^2+6*rho*rho_z^2*a_x^2*a_xz*rho_yyy+6*rho*rho_z^2*a_x^2*rho_xyz
*a_yy-6*rho*rho_z^2*a_x^2*rho_xyy*a_yz-6*rho*rho_z^2*a_x^2*a_xy*rho_yyz-12*rho*rho_z^2*a_x
*a_xy^2*rho_yz-6*rho*rho_z^2*a_y^2*a_yz*rho_xxx+6*rho*rho_z^2*a_y^2*a_xy*rho_xxz-6*rho*rho_z^2
*a_y^2*rho_xyz*a_xx+6*rho*rho_z^2*a_y^2*rho_xxy*a_xz+12*rho*rho_z^2*a_y*rho_xz*a_xy^2
-6*rho_x^3*a_z*a_y*rho_zz*a_yy+6*rho_x^3*a_z*a_y*rho_yy*a_zz-6*rho_x^2*rho_y*a_x*rho_zzz*a_y^2
-6*rho_x^2*rho_y*a_x*rho_yyz*a_z^2-6*rho_x^2*rho_y*a_y^2*a_xz*rho_zz+6*rho_x^2*rho_y*a_y^2*a_z
*rho_xzz+6*rho_x^2*rho_y*a_y^2*rho_xz*a_zz-12*rho_x^2*rho_y*a_z^2*a_y*rho_xyz-12*rho_x^2*rho_y
*a_z^2*rho_yz*a_xy+12*rho_x^2*rho_y*a_z^2*a_yz*rho_xy-6*rho_x^2*rho_y*a_z^2*rho_xz*a_yy
+6*rho_x^2*rho_y*a_z^2*a_xz*rho_yy+6*rho_x^2*rho_z*a_x*rho_yzz*a_y^2+6*rho_x^2*rho_z*a_x*a_z^2
*rho_yyy+6*rho_x^2*rho_z*a_y^2*rho_xy*a_zz+12*rho_x^2*rho_z*a_y^2*rho_yz*a_xz+12*rho_x^2*rho_z
*a_y^2*a_z*rho_xyz-6*rho_x^2*rho_z*a_y^2*a_xy*rho_zz-12*rho_x^2*rho_z*a_y^2*rho_xz*a_yz
-6*rho_x^2*rho_z*a_z^2*a_y*rho_xyy-6*rho_x^2*rho_z*a_z^2*rho_xy*a_yy+6*rho_x^2*rho_z*a_z^2
*rho_yy*a_xy-6*rho_x*rho_y^2*a_x^2*rho_yzz*a_z+6*rho_x*rho_y^2*a_x^2*rho_zzz*a_y+6*rho_x
*rho_y^2*a_x^2*a_yz*rho_zz-6*rho_x*rho_y^2*a_x^2*rho_yz*a_zz+12*rho_x*rho_y^2*a_x*a_z^2
*rho_xyz+6*rho_x*rho_y^2*a_y*a_z^2*rho_xxz-12*rho_x*rho_y^2*a_z^2*rho_xy*a_xz+6*rho_x*rho_y^2
*a_z^2*rho_yz*a_xx-6*rho_x*rho_y^2*a_z^2*a_yz*rho_xx-6*rho_x^3*a_y^2*rho_yz*a_zz-6*rho_x^3
*a_y^2*rho_yzz*a_z+6*rho_x^3*a_y^2*a_yz*rho_zz+6*rho_x^3*a_z^2*a_y*rho_yyz-6*rho_x^3*a_z^2
*a_yz*rho_yy+6*rho_x^3*a_z^2*rho_yz*a_yy+6*rho_x^2*rho_y*a_z^3*rho_xyy-6*rho_x^2*rho_z*rho_xzz
*a_y^3-6*rho_x*rho_y^2*a_z^3*rho_xxy+6*rho_x*rho_z^2*rho_xxz*a_y^3-6*rho_y^3*a_x^2*a_xz*rho_zz
+6*rho_y^3*a_x^2*a_z*rho_xzz+6*rho_y^3*a_x^2*rho_xz*a_zz-6*rho_y^3*a_z^2*a_x*rho_xxz-6*rho_y^3
*a_z^2*a_xx*rho_xz+6*rho_y^3*a_z^2*a_xz*rho_xx+6*rho_z*rho_y^2*rho_yzz*a_x^3-6*rho_z^2*rho_y
*rho_yyz*a_x^3+6*rho_z^3*a_x^2*rho_yy*a_xy-6*rho_z^3*a_x^2*a_y*rho_xyy-6*rho_z^3*a_x^2*rho_xy
*a_yy+6*rho_z^3*a_y^2*a_x*rho_xxy-6*rho_z^3*a_y^2*a_xy*rho_xx-12*rho^2*rho_x*a_y*a_xy*rho_yzz
*a_xz-12*rho^2*rho_x*a_y*a_xy*rho_xzz*a_yz+12*rho^2*rho_x*a_y*a_xy*rho_xyz*a_zz+12*rho^2*rho_x
*a_y*a_xz*rho_xzz*a_yy-12*rho^2*rho_x*a_y*a_xz*rho_xyz*a_yz+12*rho^2*rho_x*a_y*rho_yzz*a_yz
THE HIDDEN SYMMETRY OF KONTSEVICH’S GRAPH FLOWS 471
*a_xx-6*rho^2*rho_x*a_y*a_xx*rho_yyz*a_zz-6*rho^2*rho_x*a_y*a_xx*rho_zzz*a_yy-6*rho^2*rho_x
*a_y*rho_xxz*a_zz*a_yy+12*rho^2*rho_x*a_z*a_xy*rho_yyz*a_xz-12*rho^2*rho_x*a_z*a_xy*rho_xyy
*a_zz+12*rho^2*rho_x*a_z*a_xy*rho_xyz*a_yz-12*rho^2*rho_x*a_z*a_xz*rho_xyz*a_yy+12*rho^2*rho_x
*a_z*a_xz*rho_xyy*a_yz-12*rho^2*rho_x*a_z*rho_yyz*a_yz*a_xx+6*rho^2*rho_x*a_z*a_xx*rho_yyy
*a_zz+6*rho^2*rho_x*a_z*a_xx*rho_yzz*a_yy+6*rho^2*rho_x*a_z*rho_xxy*a_zz*a_yy+12*rho^2*rho_y
*a_x*a_xy*rho_yzz*a_xz+12*rho^2*rho_y*a_x*a_xy*rho_xzz*a_yz-12*rho^2*rho_y*a_x*a_xy*rho_xyz
*a_zz-12*rho^2*rho_y*a_x*a_xz*rho_xzz*a_yy+12*rho^2*rho_y*a_x*a_xz*rho_xyz*a_yz-12*rho^2*rho_y
*a_x*rho_yzz*a_yz*a_xx+6*rho^2*rho_y*a_x*a_xx*rho_yyz*a_zz+6*rho^2*rho_y*a_x*a_xx*rho_zzz*a_yy
+6*rho^2*rho_y*a_x*rho_xxz*a_zz*a_yy+12*rho^2*rho_y*a_z*a_xy*rho_xxy*a_zz-12*rho^2*rho_y*a_z
*a_xy*rho_xxz*a_yz-12*rho^2*rho_y*a_z*a_xy*a_xz*rho_xyz+12*rho^2*rho_y*a_z*a_xz*rho_xxz*a_yy
-12*rho^2*rho_y*a_z*a_xz*rho_xxy*a_yz+12*rho^2*rho_y*a_z*rho_xyz*a_yz*a_xx-6*rho^2*rho_y*a_z
*a_xx*rho_xzz*a_yy-6*rho^2*rho_y*a_z*a_xx*rho_xyy*a_zz-6*rho^2*rho_y*a_z*rho_xxx*a_zz*a_yy
-12*rho^2*rho_z*a_x*a_xy*rho_yyz*a_xz-12*rho^2*rho_z*a_x*a_xy*rho_xyz*a_yz+12*rho^2*rho_z*a_x
*a_xy*rho_xyy*a_zz+12*rho^2*rho_z*a_x*a_xz*rho_xyz*a_yy-12*rho^2*rho_z*a_x*a_xz*rho_xyy*a_yz
+12*rho^2*rho_z*a_x*rho_yyz*a_yz*a_xx-6*rho^2*rho_z*a_x*a_xx*rho_yzz*a_yy-6*rho^2*rho_z*a_x
*a_xx*rho_yyy*a_zz-6*rho^2*rho_z*a_x*rho_xxy*a_zz*a_yy-12*rho^2*rho_z*a_y*a_xy*rho_xxy*a_zz
+12*rho^2*rho_z*a_y*a_xy*rho_xxz*a_yz+12*rho^2*rho_z*a_y*a_xy*a_xz*rho_xyz-12*rho^2*rho_z*a_y
*a_xz*rho_xxz*a_yy+12*rho^2*rho_z*a_y*a_xz*rho_xxy*a_yz-12*rho^2*rho_z*a_y*rho_xyz*a_yz*a_xx
+6*rho^2*rho_z*a_y*a_xx*rho_xzz*a_yy+6*rho^2*rho_z*a_y*a_xx*rho_xyy*a_zz+6*rho^2*rho_z*a_y
*rho_xxx*a_zz*a_yy-6*rho*rho_x^2*a_x*a_y*rho_yyz*a_zz-6*rho*rho_x^2*a_x*a_y*rho_zzz*a_yy
+12*rho*rho_x^2*a_x*a_y*rho_yzz*a_yz-12*rho*rho_x^2*a_x*a_z*rho_yyz*a_yz+6*rho*rho_x^2*a_x*a_z
*rho_yyy*a_zz+6*rho*rho_x^2*a_x*a_z*rho_yzz*a_yy+12*rho*rho_x^2*a_y*a_z*rho_yyz*a_xz+6*rho
*rho_x^2*a_y*a_z*rho_xzz*a_yy-6*rho*rho_x^2*a_y*a_z*rho_xyy*a_zz-12*rho*rho_x^2*a_y*a_z
*rho_yzz*a_xy-12*rho*rho_x^2*a_y*a_xy*rho_yz*a_zz+12*rho*rho_x^2*a_y*a_xy*a_yz*rho_zz
+12*rho*rho_x^2*a_y*a_xz*a_yz*rho_yz-12*rho*rho_x^2*a_y*a_xz*rho_zz*a_yy+12*rho*rho_x^2*a_y
*rho_xz*a_yy*a_zz-12*rho*rho_x^2*a_z*a_xy*a_yz*rho_yz+12*rho*rho_x^2*a_z*a_xy*rho_yy*a_zz
+12*rho*rho_x^2*a_z*a_xz*rho_yz*a_yy-12*rho*rho_x^2*a_z*a_xz*a_yz*rho_yy-12*rho*rho_x^2*a_z
*rho_xy*a_yy*a_zz+6*rho*rho_x*rho_y*a_x^2*rho_yyz*a_zz+6*rho*rho_x*rho_y*a_x^2*rho_zzz*a_yy
-12*rho*rho_x*rho_y*a_x^2*rho_yzz*a_yz+12*rho*rho_x*rho_y*a_x*rho_xz*a_yz^2-6*rho*rho_x*rho_y
*a_y^2*rho_xxz*a_zz+12*rho*rho_x*rho_y*a_y^2*a_xz*rho_xzz-6*rho*rho_x*rho_y*a_y^2*a_xx*rho_zzz
-12*rho*rho_x*rho_y*a_y*rho_yz*a_xz^2-6*rho*rho_x*rho_y*a_z^2*rho_yyz*a_xx+6*rho*rho_x*rho_y
*a_z^2*rho_xxz*a_yy-12*rho*rho_x*rho_y*a_z^2*rho_xxy*a_yz+12*rho*rho_x*rho_y*a_z^2*rho_xyy
*a_xz+12*rho*rho_x*rho_y*a_z*rho_yy*a_xz^2-12*rho*rho_x*rho_y*a_z*rho_xx*a_yz^2-6*rho*rho_x
*rho_z*a_x^2*rho_yyy*a_zz-6*rho*rho_x*rho_z*a_x^2*rho_yzz*a_yy+12*rho*rho_x*rho_z*a_x^2
*rho_yyz*a_yz-12*rho*rho_x*rho_z*a_x*rho_xy*a_yz^2-12*rho*rho_x*rho_z*a_y^2*rho_xzz*a_xy
+6*rho*rho_x*rho_z*a_y^2*rho_yzz*a_xx-6*rho*rho_x*rho_z*a_y^2*rho_xxy*a_zz+12*rho*rho_x*rho_z
*a_y^2*rho_xxz*a_yz-12*rho*rho_x*rho_z*a_y*rho_zz*a_xy^2+12*rho*rho_x*rho_z*a_y*rho_xx*a_yz^2
-12*rho*rho_x*rho_z*a_z^2*rho_xyy*a_xy+6*rho*rho_x*rho_z*a_z^2*a_xx*rho_yyy+6*rho*rho_x*rho_z
*a_z^2*rho_xxy*a_yy+12*rho*rho_x*rho_z*a_z*a_xy^2*rho_yz-12*rho*rho_y^2*a_x*a_y*a_xz*rho_xzz
+6*rho*rho_y^2*a_x*a_y*a_xx*rho_zzz+6*rho*rho_y^2*a_x*a_y*rho_xxz*a_zz+12*rho*rho_y^2*a_x*a_z
*rho_xzz*a_xy+6*rho*rho_y^2*a_x*a_z*rho_xxy*a_zz-6*rho*rho_y^2*a_x*a_z*rho_yzz*a_xx-12*rho
*rho_y^2*a_x*a_z*rho_xxz*a_yz-12*rho*rho_y^2*a_x*a_xy*a_xz*rho_zz+12*rho*rho_y^2*a_x*a_xy
*rho_xz*a_zz-12*rho*rho_y^2*a_x*a_xz*rho_xz*a_yz+12*rho*rho_y^2*a_x*a_xx*a_yz*rho_zz-12*rho
*rho_y^2*a_x*a_xx*rho_yz*a_zz-6*rho*rho_y^2*a_z*a_y*rho_xzz*a_xx-6*rho*rho_y^2*a_z*a_y*rho_xxx
*a_zz+12*rho*rho_y^2*a_z*a_y*rho_xxz*a_xz+12*rho*rho_y^2*a_z*a_xy*rho_xz*a_xz-12*rho*rho_y^2
*a_z*a_xy*a_zz*rho_xx+12*rho*rho_y^2*a_z*rho_xx*a_yz*a_xz+12*rho*rho_y^2*a_z*a_xx*rho_xy*a_zz
-12*rho*rho_y^2*a_z*a_xx*rho_xz*a_yz-12*rho*rho_z*rho_y*a_x^2*rho_yyz*a_xz+12*rho*rho_z*rho_y
*a_x^2*rho_yzz*a_xy+6*rho*rho_z*rho_y*a_x^2*rho_xyy*a_zz-6*rho*rho_z*rho_y*a_x^2*rho_xzz*a_yy
+12*rho*rho_z*rho_y*a_x*rho_zz*a_xy^2-12*rho*rho_z*rho_y*a_x*rho_yy*a_xz^2+6*rho*rho_z*rho_y
*a_y^2*rho_xzz*a_xx+6*rho*rho_z*rho_y*a_y^2*rho_xxx*a_zz-12*rho*rho_z*rho_y*a_y^2*rho_xxz*a_xz
+12*rho*rho_z*rho_y*a_y*rho_xy*a_xz^2+12*rho*rho_z*rho_y*a_z^2*rho_xxy*a_xy-6*rho*rho_z*rho_y
*a_z^2*a_yy*rho_xxx-6*rho*rho_z*rho_y*a_z^2*rho_xyy*a_xx-12*rho*rho_z*rho_y*a_z*rho_xz*a_xy^2
+12*rho*rho_z^2*a_x*a_y*rho_xxy*a_yz-12*rho*rho_z^2*a_x*a_y*rho_xyy*a_xz-6*rho*rho_z^2*a_x*a_y
*rho_xxz*a_yy+6*rho*rho_z^2*a_x*a_y*rho_yyz*a_xx+12*rho*rho_z^2*a_x*a_z*rho_xyy*a_xy-6*rho
*rho_z^2*a_x*a_z*a_xx*rho_yyy-6*rho*rho_z^2*a_x*a_z*rho_xxy*a_yy+12*rho*rho_z^2*a_x*a_xy*a_xz
*rho_yy+12*rho*rho_z^2*a_x*a_xy*a_yz*rho_xy-12*rho*rho_z^2*a_x*rho_xy*a_xz*a_yy+12*rho*rho_z^2
*a_x*a_xx*rho_yz*a_yy-12*rho*rho_z^2*a_x*a_xx*a_yz*rho_yy-12*rho*rho_z^2*a_y*a_z*rho_xxy*a_xy
+6*rho*rho_z^2*a_y*a_z*a_yy*rho_xxx+6*rho*rho_z^2*a_y*a_z*rho_xyy*a_xx-12*rho*rho_z^2*a_y*a_xy
*rho_xy*a_xz-12*rho*rho_z^2*a_y*a_xy*a_yz*rho_xx+12*rho*rho_z^2*a_y*rho_xx*a_yy*a_xz-12*rho
*rho_z^2*a_y*a_xx*rho_xz*a_yy+12*rho*rho_z^2*a_y*a_xx*a_yz*rho_xy+12*rho_x^2*rho_y*a_x*a_y
*rho_yz*a_zz-12*rho_x^2*rho_y*a_x*a_y*a_yz*rho_zz+12*rho_x^2*rho_y*a_x*a_y*rho_yzz*a_z
+6*rho_x^2*rho_y*a_x*a_z*rho_zz*a_yy-6*rho_x^2*rho_y*a_x*a_z*rho_yy*a_zz-12*rho_x^2*rho_y*a_z
*a_y*rho_xy*a_zz+12*rho_x^2*rho_y*a_z*a_y*a_xy*rho_zz+6*rho_x^2*rho_z*a_x*a_y*rho_zz*a_yy
472 R. BURING, D. LIPPER, AND A. V. KISELEV
-6*rho_x^2*rho_z*a_x*a_y*rho_yy*a_zz-12*rho_x^2*rho_z*a_x*a_y*rho_yyz*a_z-12*rho_x^2*rho_z*a_x
*a_z*rho_yz*a_yy+12*rho_x^2*rho_z*a_x*a_z*a_yz*rho_yy+12*rho_x^2*rho_z*a_z*a_y*rho_xz*a_yy
-12*rho_x^2*rho_z*a_z*a_y*a_xz*rho_yy+12*rho_x*rho_y^2*a_x*a_y*a_xz*rho_zz-12*rho_x*rho_y^2
*a_x*a_y*a_z*rho_xzz-12*rho_x*rho_y^2*a_x*a_y*rho_xz*a_zz+12*rho_x*rho_y^2*a_x*a_z*rho_xy*a_zz
-12*rho_x*rho_y^2*a_x*a_z*a_xy*rho_zz-6*rho_x*rho_y^2*a_z*a_y*a_xx*rho_zz+6*rho_x*rho_y^2*a_z
*a_y*a_zz*rho_xx-12*rho_x*rho_z*rho_y*a_x^2*a_y*rho_yzz+12*rho_x*rho_z*rho_y*a_x^2*rho_yyz*a_z
-6*rho_x*rho_z*rho_y*a_x^2*rho_zz*a_yy+6*rho_x*rho_z*rho_y*a_x^2*rho_yy*a_zz+12*rho_x*rho_z
*rho_y*a_x*rho_xzz*a_y^2-12*rho_x*rho_z*rho_y*a_x*a_z^2*rho_xyy-6*rho_x*rho_z*rho_y*a_y^2*a_zz
*rho_xx-12*rho_x*rho_z*rho_y*a_y^2*a_z*rho_xxz+6*rho_x*rho_z*rho_y*a_y^2*a_xx*rho_zz+12*rho_x
*rho_z*rho_y*a_z^2*a_y*rho_xxy+6*rho_x*rho_z*rho_y*a_z^2*a_yy*rho_xx-6*rho_x*rho_z*rho_y*a_z^2
*rho_yy*a_xx+12*rho_x*rho_z^2*a_x*a_y*a_z*rho_xyy-12*rho_x*rho_z^2*a_x*a_y*rho_xz*a_yy
+12*rho_x*rho_z^2*a_x*a_y*a_xz*rho_yy+12*rho_x*rho_z^2*a_x*a_z*rho_xy*a_yy-12*rho_x*rho_z^2
*a_x*a_z*rho_yy*a_xy-6*rho_x*rho_z^2*a_y*a_z*a_yy*rho_xx+6*rho_x*rho_z^2*a_y*a_z*rho_yy*a_xx
+6*rho_z*rho_y^2*a_x*a_y*a_zz*rho_xx+12*rho_z*rho_y^2*a_x*a_y*a_z*rho_xxz-6*rho_z*rho_y^2*a_x
*a_y*a_xx*rho_zz+12*rho_z*rho_y^2*a_x*a_z*a_yz*rho_xx-12*rho_z*rho_y^2*a_x*a_z*rho_yz*a_xx
+12*rho_z*rho_y^2*a_z*a_y*a_xx*rho_xz-12*rho_z*rho_y^2*a_z*a_y*a_xz*rho_xx+12*rho_z^2*rho_y
*a_x*a_y*rho_yz*a_xx-12*rho_z^2*rho_y*a_x*a_y*a_z*rho_xxy-12*rho_z^2*rho_y*a_x*a_y*a_yz*rho_xx
-6*rho_z^2*rho_y*a_x*a_z*a_yy*rho_xx+6*rho_z^2*rho_y*a_x*a_z*rho_yy*a_xx+12*rho_z^2*rho_y*a_y
*a_z*a_xy*rho_xx-12*rho_z^2*rho_y*a_y*a_z*a_xx*rho_xy+2*rho_x^3*rho_zzz*a_y^3-2*rho_x^3*a_z^3
*rho_yyy-2*rho_y^3*rho_zzz*a_x^3+2*rho_y^3*a_z^3*rho_xxx+2*rho_z^3*rho_yyy*a_x^3-2*rho_z^3
*a_y^3*rho_xxx+12*rho*rho_x*rho_y*a_x*a_xz*rho_zz*a_yy-12*rho*rho_x*rho_y*a_x*rho_xz*a_yy*a_zz
-24*rho*rho_x*rho_y*a_y*a_z*a_xz*rho_xyz+12*rho*rho_x*rho_y*a_y*a_z*rho_xxy*a_zz+12*rho*rho_x
*rho_y*a_y*a_z*rho_yzz*a_xx-12*rho*rho_x*rho_y*a_y*a_xy*rho_xz*a_zz+12*rho*rho_x*rho_y*a_y
*a_xy*a_xz*rho_zz+12*rho*rho_x*rho_y*a_y*a_xz*rho_xz*a_yz-12*rho*rho_x*rho_y*a_y*a_xx*a_yz
*rho_zz+12*rho*rho_x*rho_y*a_y*a_xx*rho_yz*a_zz+12*rho*rho_x*rho_y*a_z*a_xy*rho_xz*a_yz
-12*rho*rho_x*rho_y*a_z*a_xy*rho_yz*a_xz-12*rho*rho_x*rho_y*a_z*rho_xz*a_xz*a_yy+12*rho*rho_x
*rho_y*a_z*a_yz*a_xx*rho_yz+12*rho*rho_x*rho_y*a_z*a_zz*a_yy*rho_xx-12*rho*rho_x*rho_y*a_z
*a_zz*rho_yy*a_xx-24*rho*rho_x*rho_z*a_x*a_y*rho_xyz*a_yz+12*rho*rho_x*rho_z*a_x*a_y*rho_xzz
*a_yy+12*rho*rho_x*rho_z*a_x*a_y*rho_xyy*a_zz-12*rho*rho_x*rho_z*a_x*a_xy*rho_yy*a_z
z+12*rho*rho_x*rho_z*a_x*a_xy*a_yz*rho_yz-12*rho*rho_x*rho_z*a_x*a_xz*rho_yz*a_yy+12*rho*rho_x
*rho_z*a_x*a_xz*a_yz*rho_yy+12*rho*rho_x*rho_z*a_x*rho_xy*a_yy*a_zz-12*rho*rho_x*rho_z*a_y*a_z
*rho_xxz*a_yy-12*rho*rho_x*rho_z*a_y*a_z*rho_yyz*a_xx+24*rho*rho_x*rho_z*a_y*a_z*rho_xyz*a_xy
+12*rho*rho_x*rho_z*a_y*a_xy*rho_yz*a_xz+12*rho*rho_x*rho_z*a_y*a_xy*rho_xy*a_zz-12*rho*rho_x
*rho_z*a_y*a_yz*rho_xy*a_xz-12*rho*rho_x*rho_z*a_y*a_yz*a_xx*rho_yz-12*rho*rho_x*rho_z*a_y
*a_zz*a_yy*rho_xx+12*rho*rho_x*rho_z*a_y*a_yy*a_xx*rho_zz-12*rho*rho_x*rho_z*a_z*a_xy*a_xz
*rho_yy-12*rho*rho_x*rho_z*a_z*a_xy*a_yz*rho_xy+12*rho*rho_x*rho_z*a_z*rho_xy*a_xz*a_yy
-12*rho*rho_x*rho_z*a_z*a_xx*rho_yz*a_yy+12*rho*rho_x*rho_z*a_z*a_xx*a_yz*rho_yy-12*rho*rho_z
*rho_y*a_x*a_y*rho_yzz*a_xx-12*rho*rho_z*rho_y*a_x*a_y*rho_xxy*a_zz+24*rho*rho_z*rho_y*a_x*a_y
*a_xz*rho_xyz+12*rho*rho_z*rho_y*a_x*a_z*rho_yyz*a_xx-24*rho*rho_z*rho_y*a_x*a_z*rho_xyz*a_xy
+12*rho*rho_z*rho_y*a_x*a_z*rho_xxz*a_yy-12*rho*rho_z*rho_y*a_x*a_xy*rho_xz*a_yz-12*rho*rho_z
*rho_y*a_x*a_xy*rho_xy*a_zz+12*rho*rho_z*rho_y*a_x*rho_xz*a_xz*a_yy+12*rho*rho_z*rho_y*a_x
*a_yz*rho_xy*a_xz-12*rho*rho_z*rho_y*a_x*a_yy*a_xx*rho_zz+12*rho*rho_z*rho_y*a_x*a_zz*rho_yy
*a_xx-12*rho*rho_z*rho_y*a_y*a_xy*rho_xz*a_xz+12*rho*rho_z*rho_y*a_y*a_xy*a_zz*rho_xx
-12*rho*rho_z*rho_y*a_y*rho_xx*a_yz*a_xz-12*rho*rho_z*rho_y*a_y*a_xx*rho_xy*a_zz+12*rho*rho_z
*rho_y*a_y*a_xx*rho_xz*a_yz+12*rho*rho_z*rho_y*a_z*a_xy*rho_xy*a_xz+12*rho*rho_z*rho_y*a_z
*a_xy*a_yz*rho_xx-12*rho*rho_z*rho_y*a_z*rho_xx*a_yy*a_xz+12*rho*rho_z*rho_y*a_z*a_xx*rho_xz
*a_yy-12*rho*rho_z*rho_y*a_z*a_xx*a_yz*rho_xy+24*rho_x*rho_z*rho_y*a_x*a_y*rho_xz*a_yz
-24*rho_x*rho_z*rho_y*a_x*a_y*rho_yz*a_xz+24*rho_x*rho_z*rho_y*a_x*a_z*rho_yz*a_xy-24*rho_x*
rho_z*rho_y*a_x*a_z*a_yz*rho_xy+24*rho_x*rho_z*rho_y*a_z*a_y*rho_xy*a_xz-24*rho_x*rho_z*rho_y
*a_z*a_y*a_xy*rho_xz+12*rho_x*rho_y^2*a_z^2*a_xy*rho_xz+6*rho_x*rho_z^2*a_x^2*rho_yz*a_yy
+6*rho_x*rho_z^2*a_x^2*rho_yyz*a_y-6*rho_x*rho_z^2*a_x^2*a_yz*rho_yy-6*rho_x*rho_z^2*a_x^2
*a_z*rho_yyy-12*rho_x*rho_z^2*a_x*rho_xyz*a_y^2+6*rho_x*rho_z^2*a_y^2*a_yz*rho_xx+12*rho_x
*rho_z^2*a_y^2*a_xy*rho_xz-6*rho_x*rho_z^2*a_y^2*a_z*rho_xxy-12*rho_x*rho_z^2*a_y^2*rho_xy
*a_xz-6*rho_x*rho_z^2*a_y^2*rho_yz*a_xx+6*rho_y^3*a_z*a_x*a_xx*rho_zz-6*rho_y^3*a_z*a_x*a_zz
*rho_xx-12*rho_z*rho_y^2*a_x^2*rho_xz*a_yz-6*rho_z*rho_y^2*a_x^2*a_y*rho_xzz-6*rho_z*rho_y^2
*a_x^2*rho_xy*a_zz+6*rho_z*rho_y^2*a_x^2*a_xy*rho_zz+12*rho_z*rho_y^2*a_x^2*rho_yz*a_xz
-12*rho_z*rho_y^2*a_x^2*a_z*rho_xyz+6*rho_z*rho_y^2*a_x*a_z^2*rho_xxy-6*rho_z*rho_y^2*a_y
*a_z^2*rho_xxx-6*rho_z*rho_y^2*a_z^2*a_xy*rho_xx+6*rho_z*rho_y^2*a_z^2*a_xx*rho_xy-12*rho_z^2
*rho_y*a_x^2*rho_yz*a_xy-6*rho_z^2*rho_y*a_x^2*a_xz*rho_yy+6*rho_z^2*rho_y*a_x^2*rho_xz*a_yy
+12*rho_z^2*rho_y*a_x^2*a_yz*rho_xy+12*rho_z^2*rho_y*a_x^2*a_y*rho_xyz+6*rho_z^2*rho_y*a_x^2
*a_z*rho_xyy-6*rho_z^2*rho_y*a_x*rho_xxz*a_y^2+6*rho_z^2*rho_y*a_y^2*a_z*rho_xxx-6*rho_z^2
*rho_y*a_y^2*a_xx*rho_xz+6*rho_z^2*rho_y*a_y^2*a_xz*rho_xx+6*rho_z^3*a_y*a_x*a_yy*rho_xx
-6*rho_z^3*a_y*a_x*rho_yy*a_xx+6*rho_z^3*a_y^2*a_xx*rho_xy
Curriculum Vitae
Ricardo Buring was born in Groningen, the Netherlands on the 30th of May in 1992. He
received his B.Sc. diploma in Mathematics from the University of Groningen in 2013, and
his M.Sc. diploma in Mathematics from the same university in 2017. In 2017 he moved to
Johannes Gutenberg-University of Mainz in Germany for his Ph.D. studies in the group
led by Prof. dr. D. van Straten, supervised by dr. hab. A.V. Kiselev (Groningen) and
Prof. dr. D. van Straten (Mainz).
During his master’s and doctoral studies Ricardo Buring participated in eight inter-
national conferences, gave talks at five colloquia internationally, and spoke six times at
research seminars (in Mainz and abroad). He spent a week at the IHÉS in Bures-sur-
Yvette, France, in April 2017 and another week in December 2019, discussing his work
with M. Kontsevich. Ricardo Buring was selected to participate in the 7th Heidelberg
Laureate Forum (2019), and further selected as one of 10 participants to be interviewed.1
Ricardo Buring is a coauthor of ten publications in peer-reviewed journals and one
preprint, jointly with A.V. Kiselev and other collaborators (A. Bouisaghouane, N.J. Rut-
ten, and D. Lipper). He is the author of the MIT-licensed kontsevich_graph_series-cpp
and gcaops software packages, which have been used to obtain results contained in the
aforementioned publications.
By using this software, in the Autumn semester 2020/21 he contributed 15 tutorials
and two PC demo sessions to a master’s course “Deformation Quantization, Graph Com-
plex, and Number Theory”, read by A.V. Kiselev, in the Dutch national Mastermath
programme. During the four year Ph.D. term in Mainz, Ricardo Buring taught tutorials
in undergraduate courses Computeralgebra, Discrete Mathematics for Computer Scien-
tists, and “Geometry, Algebra, and Number Theory”. Ricardo Buring co-supervised two
bachelor projects (N.J. Rutten in 2017/18, D. Lipper in 2020/21), and helped—e.g. by
using software—with master theses of A. Bouisaghouane (2016/17) and Willem de Kok
(2020/21), and B.Sc. work of S. Kerkhove (2020).
Currently, Ricardo Buring is a research associate at the Institute of Mathematics,
Johannes Gutenberg-University Mainz.
1https://scilogs.spektrum.de/hlf/10-out-of-200-from-diagrams-to-formulas-via-
computers-ricardo-buring-loves-teaching-math/
473

Acknowledgements
First I express my thanks to Duco van Straten, who made this Ph.D. thesis possible.
Equally, I thank my supervisor Arthemy Kiselev, for his continued guidance and his
seemingly inexhaustible stream of good ideas, practical advice, and relevant stories.
I am grateful to all the members of the examination committee, namely
, Duco van Straten and Arthemy Kiselev, for
their attention, time, criticism, and feedback. Thanks to for writing
the minutes at my doctoral examination. Special thanks to for asking
particularly good questions during my defense.
Thanks to for creating this entire theory, for helpful discussions,
for suggesting areas of research, and for foretelling answers before we knew how to
tackle the problem.
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475
476 Acknowledgements
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Many thanks to my colleagues and friends in Mainz for the great times we spent
together; thank you in particular Tom, Matthias, Raymond, Laura and Markus, Philipp,
Nutsa, and Lina. Not least, I am deeply grateful to my friends and my family for their
patience and kindness.
Mainz, July 2022
Zusammenfassung
Poisson-Klammern treten auf, wenn das punktweise Produkt von Skalarfunktionen auf
einer affinen Mannigfaltigkeit so deformiert wird, dass es assoziativ bleibt. Kontsevich
bewies das Gegenteil: eine universelle Formel ordnet jeder Poisson-Klammer eine solche
assoziative Deformation zu. Ebenso können Poisson-Klammern durch universelle Formeln
deformiert werden. In beiden Konstruktionen werden die universellen Formeln mit Hilfe
von Graphen gebildet.
Um die Tausende von Graphen zu handhaben, entwickeln und präsentieren wir das
Softwarepaket gcaops (Graph Complex Action on Poisson Structures) für SageMath. Mit
diesem Paket, • entwickeln wir Kontsevich’s ⋆-Produkt bis auf ō(h̄4); • wir setzen ⋆
mod ō(h̄6) aus externen Daten von Banks–Panzer–Pym zusammen, und wir erhalten das
Sternprodukt ⋆aff mod ō(h̄7) für affine Poisson-Klammern; • wir verifizieren dass die von
Banks–Panzer–Pym gefundene Graphgewichte viele bekannte Gleichungen erfüllen; • wir
illustrieren den expliziten Beweis der Assoziativität für das vollständige Sternprodukt
modulo ō(h̄6) und für das affine Sternprodukt modulo ō(h̄7); • wir finden neue explizite
Formeln für Graphencozyklen und universelle Poisson-Cozyklen, und • wir beweisen die
Faktorisierung der Poisson-Cozyklus-Bedingung über die Jacobi-Identität in jedem Fall.
477

Samenvatting
Poisson-haakjes duiken op wanneer het puntsgewijze product van scalaire functies op
een affiene variëteit zo gedeformeerd wordt dat het associatief blijft. Kontsevich be-
wees het omgekeerde: een universele formule wijst aan elk Poisson-haakje zo’n asso-
ciatieve deformatie toe. Op een soortgelijke manier kunnen Poisson-haakjes zelf door
universele formules gedeformeerd worden. In beide constructies worden de universele
formules opgesteld met behulp van grafen.
Om de duizenden grafen te handhaven, ontwikkelen en presenteren we het soft-
warepakket gcaops (Graph Complex Action on Poisson Structures) voor SageMath. Met
behulp van dit pakket, • ontwikkelen we Kontsevich’s ⋆-product tot op ō(h̄4); • we bouwen
⋆ mod ō(h̄6) met externe data van Banks–Panzer–Pym, en we verkrijgen het sterproduct
⋆aff mod ō(h̄7) voor affiene Poisson-haakjes; • we bevestigen dat de gewichten van grafen
gevonden door Banks–Panzer–Pym aan vele bekende vergelijkingen voldoen; • we illus-
treren het expliciete bewijs van de associativiteit voor het volledige sterproduct modulo
ō(h̄6) en voor het affiene sterproduct modulo ō(h̄7); • we vinden nieuwe expliciete for-
mules voor graaf-cocykels en universele Poisson-cocykels, en • we bewijzen de factorisatie
van de Poisson-cocykel conditie via de Jacobi-identiteit in elk afzonderlijk geval.
479

Summary for Laymen
This is a dissertation in fundamental mathematics research. The area of mathematics
under study is Kontsevich’s deformation quantization of Poisson structures and the re-
lated universal flows on spaces of Poisson structures. These topics are distinguished by
their involvement of graphs (i.e. objects consisting of vertices and edges, visualized as
dots and lines respectively) in a place of mathematics where they were previously not
expected: graphs are used to construct formulas. This domain of mathematics has re-
ceived a lot of attention in the form of abstract theory building, but examples have so
far remained scarce. Part of the reason for the lack of examples is that producing them
requires computational power and programming effort. We believe that finally having
a way to easily generate those examples would benefit not only those who first come
in contact with the material, but also the experts. Experimentally observed properties
can—and actually do—lead to the recognition of patterns and forming new conjectures;
old conjectures can be checked in particular cases. In this dissertation we develop the
gcaops software (Graph Complex Action on Poisson Structures), package for SageMath,
that constructs the needed examples and verifies some conjectures.
Let us discuss the mathematical component of this result, now produced by using the
new software. The(Kontsevich ⋆)-produ(ct is a formula that looks as follows:
f ⋆ g = f · g + h̄ P ij · ∂if · ∂jg + h̄2 1P ij · P k` ·)∂k∂if( · ∂ ∂ g + 1` j ∂`P ij · P k` · ∂k∂2 3 i · f∂jg
− 1∂ P ij ·P k` ·∂ f∂ ·∂ g− 1 ij` i k j ∂`P ·∂ P k`j ·∂if ·∂kg + h̄3 1P ij ·P k` ·Pmn ·∂3 6 6 m∂k∂if ·∂n∂`∂jg
− 1∂ ij k` mn 1 ij k`m∂`P · ∂n∂jP · P · ∂if · ∂kg − P · ∂nP · ∂`Pmn · ∂k∂if · ∂m∂6 6 jg
− 1∂m∂`P ij · ∂ P k` · Pmn · ∂ ∂ 1 ij k` mn6 n k if · ∂jg − ∂m∂`P · ∂nP · P · ∂if · ∂6 k∂jg
+ 1∂ ∂ P ij · P k` · Pmn · ∂ ∂ ∂ f · ∂ g + 1∂ ∂ P ij · P k`n ` m k i j n ` · Pmn · ∂if · ∂m∂k∂6 6 jg
+ 1∂ ijnP · P k` · Pmn · ∂m∂k∂if · ∂`∂jg − 1∂ ijnP · P k` · Pmn · ∂k∂if · ∂m∂`∂jg3 3
− 1∂ P ij · ∂ ∂ P k` · Pmn · ∂ ∂ f · ∂ g + 1∂ ∂ P ij · ∂ P k` · Pmn · ∂ f · ∂
6 ` n j m i k 6 n ` j i )m∂kg
− 1∂ ij k`nP · P ∂ · Pmn · ∂ 1 ij k` mn` k∂if · ∂m∂jg − ∂`P · ∂nP · P · ∂k∂if · ∂m∂jg + ō(h̄3).6 6
Here the inputs f, g are scalar functions on Rn, the P ij are the function coefficients of
a Poisson structure and the sum over powers of the variable h̄ extends infinitely (i.e. it
is a power series). The ⋆-product deforms the ordinary (pointwise) product of functions
(f ·g)(x) = f(x)·g(x) in such a way that the associativity (f⋆g)⋆h = f⋆(g⋆h) is preserved.
If the ⋆-product is known up to a certain order, then its associativity is guaranteed up
to the same order. We find Kontsevich’s star-product up to the order 4, by using the
method of undetermined coefficients (for the coefficients of graphs) and finding many
(new) relations between these weights. Here we also use the Shoikhet–Felder–Willwacher
cyclic weight relations. We verify the associativity of Kontsevich’s ⋆-product up to the
order 4, by expanding the associator (f ⋆ g) ⋆ h− f ⋆ (g ⋆ h) mod ō(h̄4) in terms of graphs
481
482 Summary for Laymen
and collecting those sums of graphs which are known to be zero by the Jacobi identity
for the Poisson bracket P . In this context we also verify that the graph weights found by
Banks–Panzer–Pym (2018) satisfy many relations.
Secondly we investigated the Kontsevich universal graph flows on the spaces of Poisson
structures. A Poisson structure P on Rn can be considered as an n× n skew-symmetric
matrix of functions satisfying a system of partial differential equations [[P, P ]] = 0. This
is the master equation, given by the Schouten bracket of bi-vectors. To deform a Poisson
structure means to add a power series in ε with a leading deformation term Q, that is
P 7→ P + εQ + ō(ε); the linearity and graded symmetry of the Schouten bracket yields
[[P + εQ + ō(ε), P + εQ + ō(ε)]] = [[P, P ]] + 2ε[[P,Q]] + ō(ε), hence the condition on the
leading deformation term Q is [[P,Q]] = 0, which means for Q to be a 2-cocycle in the
Poisson complex of P . Kontsevich (1996) wrote universal formulas mapping Poisson
structures to their deformations: P 7→ Q(P ) where [[P,Q(P )]] = 0. To illustrate them,
we produce explicit formulas and weighted graph encodings of new examples of flows.
We examine the Poisson-(non)triviality of the tetrahedral flow for particular Poisson
structures P , i.e. we inspect whether Q(P ) can be expressed as [[P,X]] for a vector field
X (in coordinates, a column vector with functions as coefficients). For this we use different
classes of Poisson structures of different origin (e.g. quadratic and cubic brackets from
Li–Parmentier, as well as brackets with differential polynomial coefficients from Nambu).
In all cases considered, the flow was Poisson-trivial. On the other hand, there is no
known mechanism for it to be universally trivial. We investigate in detail what happens
in dimension two (n = 2). There all the flows considered are trivial, with their trivializing
vector field realized in terms of Kontsevich’s directed graphs. To deform ⋆ mod ō(h4)
exactly at h̄4 by using the tetrahedral flow, we found no mechanism for it to be universally
trivial with respect to gauge transformations of star products. The produced formulas
for Poisson 2-cocycles may be of interest to the larger Poisson geometry community.
Thirdly we calculated graph cohomology in several vertex-edge bi-gradings, and we
found new explicit examples of graph cocycles. In particular, we expressed the heptagon-
wheel cocycle and we calculated the bracket [γ3, γ5] of two previously known cocycles.
This result can be used to produce graph flows for Poisson structures. Independently, this
result is useful for the study of the graded Lie algebra of graphs. Indeed, it is easier to
study with examples than without. Also, we recall that Willwacher established a bridge
between the Grothendieck–Teichmüller Lie algebra grt and the Kontsevich graph complex
GC, but only one example of this transition (for the tetrahedron) is available from all the
previous work. This dissertation provides handy examples of what one has on the graph
side of the bridge.
We hope that in the future the experimental facts and the revealed properties will be
explained theoretically. Likewise, this research allows us to pose new problems.
The results in this dissertation were obtained using diverse methods. First it was
necessary to learn the theory and to implement it accurately. When an algorithm was
designed and the computer program ready, it was run. The output and results are
reported in this dissertation: e.g. tables with the count of graphs, allowing one to evaluate
the size of the problem. We used the method of undetermined coefficients whenever
appropriate. That is, in the beginning we preview the use of all (in hindsight, too many)
potentially needed structures, form equations, and solve them, so that new solutions
collapse to low dimensional subspaces. (Examples of solutions which are small are graph
cocycles and factorizations via Leibniz graphs.) Next, we systematically try the theory on
examples and detect patterns; this is specific in particular to the graph flows. Naturally,
Summary for Laymen 483
we practice strict rigorous mathematical proof of identities and (graph) equalities, e.g.
for the Nambu–Poisson brackets.
To understand the mechanism at work in the algebraic theory of star-products de-
scribed using graphs and in the geometric and algebraic theory of flows, again described
by using graphs, I had to learn many adjacent domains of mathematics, such as (i) su-
peralgebra, superanalysis, and Lie superalgebras, (ii) differential graded Lie algebras and
their L∞-morphisms, (iii) operads and endomorphisms of the space of multivector fields,
and (iv) elements of Poisson geometry and Poisson cohomology. It is wonderful that all
these different domains are brought together in the study of Kontsevich’s deformation
quantization.
This dissertation allowed me to better learn and practice scientific communication: I
went to conferences, listened to talks, chatted informally with experts, and so on. Par-
ticularly memorable to me are the conference GADEIS VIII in Larnaca, the Groenewold
symposium in Groningen, the Oxford seminar, as well as the workshop in Banff, where
people in the audience also had something to say. There could be expected and there
indeed was much feedback. Besides, I enjoyed teaching, for I like answering questions.
Let us keep in mind that that the Mastermath course tutorials contributed to Part I
(Computer Demonstrations), amounting to 40% of this dissertation.
All the relevant feedback was incorporated in this dissertation, for which I am grateful
to everyone in reference. This dissertation is now based on 10 peer-reviewed publications
and one preprint, and on many conference talks. The new gcaops software is available from
https://github.com/rburing/gcaops and external data files (which can be appreciated
separately) are stored at https://rburing.nl/gcaops.

Appendix A
Introduction to SageMath
This supplementary chapter is based on a document originally written and developed by
the dissertant for the first two exercise classes in Computeralgebra at JGU Mainz (SoSe
2018, 2019, 2020, 2021).
1. What to do with this document
This document consists of two parts:
• An introduction to SageMath,
• Miscellaneous topics (that you can consider if you are interested).
This document was written as a SageMath Jupyter notebook, containing both text (also
LATEX) and executable code cells. (See below.)
If you have opened this document as a notebook (the .ipynb file, available separately
from the .pdf file), then you can run the cells by giving them focus (e.g. clicking on
them) and pressing Shift+Enter.
(Note: Double-clicking on a text cell like this one opens its editor. Press Shift+Enter to
save it and exit that editor.)
If you are viewing this document as a PDF then you should type the commands into
a SageMath session (beware that copying symbols and whitespace from a PDF can be
problematic) and run them there (or open the notebook instead).
To open this notebook
• using a local installation:
– on Linux, run sage -n jupyter; on Windows, press Start and then Sage
Jupyter notebook,
– open your web browser and navigate to http://localhost:8888 (if it was
not done automatically),
– press ‘Upload’, select the .ipynb file, and press Upload again,
– click the notebook to open it.
485
486 APPENDIX A. INTRODUCTION TO SAGEMATH
• on CoCalc:
– create a project,
– press ‘New’
– click to upload the .ipynb file,
– open it.
Have a look through this document, run the cells which have been prepared, and write
your own code in the empty cells for the exercises.
2. SageMath
SageMath is a free open-source mathematics software system. It builds on top of existing
open-source packages, such as Singular, PARI/GP, GAP, and many more. In SageMath
we can access their combined power through a common language, based on Python 3
(since SageMath 9.0).
The SageMath language is just Python 3 with a preprocessing step to allow some “more
mathematical” syntax:
[1]: 1/3
[1]: 1/3
[2]: 2^3
[2]: 8
The SageMath inputs and outputs above will come as no surprise to a pure mathemati-
cian.
• In SageMath, when you enter 1/3, two objects (1 and 3) of type Integer are
created, and the operation / of division of an Integer by an Integer is defined to
be the appropriate rational (of type Rational), hence the result is 1/3.
This is in contrast to Python 3:
• In Python 3, when you enter 1/3, two objects (1 and 3) of type int are created,
and the operation / of division of an int by an int is defined to be the floating
point approximation (of type float), hence the result is 0.3333333333333333.
The way SageMath works is that the preparser replaces literal numbers such as 1 in
your code (which would normally become an int object) by Integer(1), so that an
object of type Integer is created instead, on which operations are defined in the way
mathematicians expect.
In the SageMath language, many pre-defined functions, symbols, and constants are al-
ready imported for your convenience, so you can access them immediately. We will see
examples of this below. While knowledge about Python would be advantageous, it is not
required for this guide.
487
3. Introduction to SageMath
By working through the examples and small exercises in this section, you will gradually
learn the basics of SageMath. Ultimately you will be able to write your own (simple)
functions. That is a very practical skill, as you will see. Good luck!
3.1. Calculations with integers
SageMath can be used as a calculator. Try to understand the results of the following
calculations, which involve only integers (whole numbers). If something is not clear, see
the next section.
[ ]: 1 + 1
[ ]: 101 // 25
[ ]: 101 % 25
[ ]: lcm(101,25)
[ ]: gcd(101,25)
[ ]: next_prime(1000)
[ ]: nth_prime(26)
3.1.1. Exercise
Calculate 3385 mod 2048.
[ ]:
3.1.2. Exercise
Find the smallest prime with 20 digits.
[ ]:
3.2. Help with functions
By running the cells above you were calling functions such as gcd, by writing the name
of the function, followed by opening and closing parentheses, with arguments (inputs to
the function, separated by commas) in between the parentheses: e.g. gcd(101,25) calls
gcd with arguments 101 and 25.
SageMath contains a built-in help system for functions. To find out what a function does,
enter its name followed by a question mark:
[ ]: gcd?
You will get a short description, a list of INPUT that the function accepts, what OUTPUT
the function returns, and usually also some EXAMPLES. When you are finished reading,
488 APPENDIX A. INTRODUCTION TO SAGEMATH
you can close the documentation (sub)window in the notebook interface by pressing the
x button on the right, and in the command line interface by pressing the q button.
How do you find out which functions exist in the first place? The SageMath reference
manual contains documentation for (almost) all of Sage’s features, and you can search
through it. Also, the next section of this intro is a mainly a collection of functions that
will be useful later in the course.
You can also start typing the name of a function, and press the TAB key for autocom-
pletion:
[ ]: next_p
3.3. Variables
Defining and (re)assigning variables is done with the = sign.
[ ]: a = 3
[ ]: b = 4
Variables can be defined in terms of (the values of) other variables:
[ ]: c = sqrt(a^2 + b^2)
To retrieve the value of a variable, enter its name:
[ ]: c
Variables can be used in expressions:
[ ]: a^2 + b^2
Note that the following is valid code:
[ ]: x = 1
[ ]: x = x + 1
[ ]: x
In an assignment such as x = x + 1, the right-hand side is evaluated first, and then
assigned to the variable in the left-hand side. Effectively, this statement increments the
value of x by one. A shorthand notation that achieves the same effect is x += 1.
3.4. Methods
Each type of object (such as Integer) also has a bunch of methods associated to it, which
can be called on all objects of that type.
[ ]: 5.factorial()
[ ]: 8.nth_root(3)
489
In general: enter (the name of) an object, followed by a dot (.), followed by the name
of the method, followed by opening and closing parentheses (with the arguments of the
method, if any, in between the parentheses).
Warning: To call a method, the parentheses (as above) are always required. It’s possible
to refer to a method by name (omitting the parentheses):
[ ]: 5.factorial
The output of the cell above just tells you that the method exists; it does not call the
method, i.e. it does not tell you what the value of 5 factorial is.
3.5. Help with methods
For technical reasons, there is a (minor) limitation on the help system for getting help
with methods.
It requires you to first define a variable of the desired type, and then you can get help
with the methods which are available for that variable.
[ ]: z = 101
[ ]: z.factorial?
In the documentation of a method, the object on which it is called (z here) is referred to
as self.
Given a variable, it is possible to list all its publicly available methods by typing its name,
followed by a dot (.), and pressing the TAB key.
[ ]: z.
You can scroll through the list by using the arrow keys.
3.5.1. Exercise
Find a method to obtain the list of decimal digits of an integer. (We will learn more
about lists later.)
[ ]:
3.5.2. Exercise
Find a method to obtain the list of binary digits of an integer. Hint: Check the docu-
mentation of the method you found before.
[ ]:
3.6. Functions versus methods
For convenience there exist some functions that (internally) call the respective methods:
[ ]: factorial(5)
490 APPENDIX A. INTRODUCTION TO SAGEMATH
This is just an alias for 5.factorial(). You might prefer to write factorial(5), e.g. be-
cause it looks better or because it is one keystroke shorter.
Similarly:
[ ]: is_prime(25)
Warning: It is not always the case that for every method there is a corresponding
function that calls the method. Use the help system and TAB completion (as explained
above) to discover functions and methods.
3.7. Logical expressions
Truth values in SageMath (as in Python) are called True and False; aliases are true and
false respectively.
For equality testing in SageMath you use the double equals sign.
[ ]: 1 + 1 == 3
[ ]: 3^2 + 4^2 == 5^2
Inequalities are written in the usual way:
[ ]: 1 < 3
[ ]: 2 >= 3
For non-equality testing you use the != operator:
[ ]: 1 != 2
More logical expressions:
[ ]: 5 % 2 == 0
[ ]: 5 % 2 == 1
[ ]: gcd(3, 12) == 1
[ ]: 25.is_prime()
[ ]: 9.is_square()
Several truth values can be combined using logical AND and logical OR.
[ ]: 8 % 2 == 0 and 8.is_square()
[ ]: 8 % 2 == 0 or 8.is_square()
3.7.1. Exercise
Translate the (true) expression “3 is an odd prime number” into a line of code that
evaluates to True.
491
[ ]:
3.8. Order of execution and multi-line programs
Clearly the order of execution is important:
[ ]: x = 3
[ ]: x = 4
[ ]: x.is_prime()
Here, the result of x.is_prime() depends on how x is defined, i.e. which definition of x
was executed last.
So far we have only executed (cells containing) a single line of code at a time. We can
use more than one line in a cell:
[ ]: x = 3
x.is_prime()
The code is executed one line at a time, from top to bottom.
In a program with more than one line, the value of the last line (if it is not None) is
displayed in the output (in the notebook interface).
3.9. Strings and printing
Another type of variable is str, for a string of characters, written between single or
double quotation marks:
[ ]: 'Hello, World'
The print function is used to output things to the screen:
[ ]: print('Hello, World')
Note the difference in the output of the above two code cells. In the first, the string value
is displayed as the output value. In the second, the string is displayed but there is no
output value (which would be marked by Out[n]: in the notebook interface). This is
because print does the work of displaying the value, and then returns the value None,
which is not displayed.
Here is another example, where there is output from print, and the value True is dis-
played because it is the value of the last line:
[ ]: x = 3
print('Is 3 a prime?')
x.is_prime()
Strings can also be stored in variables, and then used:
[ ]: message = 'Hi'
print(message)
Passing several values to print outputs them in succession, with a space in between:
492 APPENDIX A. INTRODUCTION TO SAGEMATH
[ ]: print('Hi', 'there')
Variables of other types can also be printed:
[ ]: print(a,b,c)
[ ]: print('a =', a, 'and b =', b, 'and c =', c)
[ ]: print('Does a^2 + b^2 = c^2 hold?', a^2 + b^2 == c^2)
[ ]: print('Is c prime?', c.is_prime())
Big Tip: When using print to display the values of multiple variables, always make it
clear which value belongs to which variable (e.g. as above).
It’s also possible to define strings spanning multiple lines, by using triple quotes as de-
limiters:
[ ]: print(""" We're no strangers to love
You know the rules and so do I""")
[ ]: lyrics = """ Never gonna give you up
Never gonna let you down
Never gonna run around and desert you"""
print(lyrics)
You can build strings from variables using the format method of str:
[ ]: name = 'Ricardo'
age = 2022 - 1992
sentence = 'My name is {} and I am {} years old.'.format(name, age)
print(sentence)
Addition of strings is concatenation, and in this way we can also build strings dynamically:
[ ]: sentence = 'My name is ' + name + ' and I am ' + str(age) + ' years old.'
print(sentence)
Note that you cannot add str and Integer objects directly: you must convert the
Integer to a str explicitly, using str(...).
3.9.1. Exercise
Define an integer variable k, print it, and print some questions with answers about it: is
it even, is it prime, is it a square?
[ ]:
3.10. Escape sequences
The backslash character \ has a special meaning inside a string. It is used for escape
sequences such as \n which is replaced by a newline, \t which is replaced by a tab,
\' which is replaced by a literal quotation mark, and \\ which is replaced by a literal
backslash.
493
[ ]: print('Hello,\nWorld')
[ ]: print('I = R \\ Q')
To write a raw string without escape sequences, write r before the opening quote sign.
[ ]: print(r'I = R \ Q')
3.11. Comments
Text written after a pound/hash/octothorpe sign # is ignored by the Python/SageMath
interpreter (except when # is used inside a string).
[ ]: 1 + 2 + 3 + 4 + 5 # the 5th triangular number
Such text is called a comment. Comments are often used to explain why code is written
the way it is. Comments that explain what code does are mostly useless:
[ ]: n = 3 # define n to be 3
n += 1 # add 1 to n
print(n) # print n to the screen
Those comments are unlikely to be helpful. Another use for comments is to “disable” a
piece of code, without removing it entirely:
[ ]: n = 3
#n+= 1
print(n)
This is called “commenting out” a piece of code.
To comment out multiple lines without writing a lot of # signs, we can surround the lines
by triple quotes:
[ ]: x = 1
y = 2
z = 3
"""x += 1
y += 1
z += 1"""
print(x,y,z)
This makes use of the fact that triple quoted strings on their own are valid statements
that don’t do anything.
3.12. Conditionals
For n ∈ N, the integer Cn is defined by n/2 if n is even and by 3n + 1 if n is odd. In
code, such a conditional can be expressed by an if-statement:
[ ]: n = 12
if n % 2 == 0:
C = n // 2
else:
C = 3*n + 1
494 APPENDIX A. INTRODUCTION TO SAGEMATH
print('C_{} = {}'.format(n, C))
The line starts with if, followed by the condition (a truth value, possibly depending on
variables), followed by a colon (:). The next line(s), the body of the if-statement, are
indented by four spaces. That means each such line starts with four spaces. The body
can contain multiple lines (all indented). The body of the if-statement is evaluated only
if the condition is true.
The else: (at the same level of indentation as the corresponding if) specifies what
should be done if the condition is not true. Again, the body of the else-clause must be
indented, and can contain multiple lines.
An else-clause is optional:
[ ]: n = 6
print('A perfect number is', n)
if n % 3 == 0:
print("By the way, it's a threeven number")
It is also possible to switch between more than two alternatives, by adding
elif <other condition>:
<other code>
after an if-statement.
3.12.1. Exercise
Define an integer variable k and print a string that says k is negative, k is zero, or k is
positive (depending on the value of k).
[ ]:
3.13. Range-based for loops
We can print the integers from 1 up to and including 10:
[ ]: for n in range(1,11):
print(n)
Each line in the body of a for-loop is indented by four spaces. The body can contain
multiple lines.
The range(a,b) consists of [a, a+1, ..., b-1] (excluding the endpoint). The notation
range(n) is shorthand for range(0,n).
We will see later why this convention (starting at 0 and excluding the endpoint) is con-
venient, e.g. in the context of lists.
The range function is from Python 3. In the loop above, n is a Python int object. To
get a SageMath Integer we can use srange:
[ ]: for n in srange(1,11):
print(n)
495
[ ]: for n in srange(1,11):
print('{}! = {}'.format(n, n.factorial()))
Note: The code n.factorial() does not work when n is a Python int (try it), but it
does when n is a SageMath Integer.
If n is a Python int, then you can also write Integer(n).factorial(), i.e. first con-
verting to a SageMath Integer.
3.13.1. Exercise
Print the squares of the first 10 non-negative integers.
[ ]:
3.14. Range-based for loops and conditionals
Which numbers y from 0 to 12 satisfy gcd(y, 12) = 1?
[ ]: for y in range(0,12):
if gcd(y,12) == 1:
print(y)
We will see later how we can store these in a list. (Now we can only look at them.)
3.14.1. Exercise
Print the numbers from 1 to 10 which are squarefree.
Hint: Use m.is_squarefree() for an Integer object m (in the body of a loop over m).
[ ]:
3.15. Range-based for loops and variables
Let’s sum the first 100 positive integers:
[ ]: n = 100
total = 0
for k in range(1,n+1):
total += k
print('The sum of the first', n, 'positive integers is', total)
Here we used the += operator to add k to total; this is equivalent to (but shorter than)
total = total + k.
(Other shorthand notations are -=, *=, %=.)
Compare:
[ ]: n = 100
n*(n+1)/2
496 APPENDIX A. INTRODUCTION TO SAGEMATH
3.15.1. Exercise
Define variables n, oddsum, evensum, and make a single for-loop (containing a con-
ditional) so that, at the end oddsum is the sum of all odd integers from 1 to n (and
similarly evensum). Add some nice print statements at the end.
[ ]:
3.16. While loops
Let’s find the first square number greater than 200 (in a brute-force way).
[ ]: n = 1
while n^2 < 200:
n += 1
print(n, 'squared is', n^2)
print('Just to be sure:', n-1, 'squared is', (n-1)^2)
In a while-loop, the body (n += 1 here) is repeatedly executed, as long as the condition
(n^2 < 200 here) is true.
Warning: Take care that your while-loops are written in such a way that they actually
terminate in finite time.
To interrupt the execution of Sage code, press the “Stop” (black square) button in the
notebook interface or press Ctrl+C on the command-line.
3.16.1. Exercise
Define a variable n and create a while loop to find the nth prime (e.g. in a naive way,
trying every integer).
Hint: define a variable k which you increment (k += 1) with every iteration, use
is_prime() to determine if you have a prime, and use another variable prime_count to
keep track of how many primes have been seen so far. What should the while-condition
be?
[ ]:
3.17. Break and continue
It’s also possible to use the break statement in a loop (for or while) to exit the loop
immediately (and proceed with the rest of the program).
[ ]: n = 1
while True:
if (n+1)^2 > 200:
break
n += 1
print(n, 'squared is', n^2)
You can also use the continue statement to skip the current iteration of a loop and
continue with the next one.
497
[ ]: for n in range(10):
if n % 3 == 0:
continue
print(n)
This is often useful to avoid “deep” indentation of the main body of your loop.
3.18. Lists
Lists can be defined by specifying their elements explicitly, between square brackets and
separated by commas:
[ ]: Lst = [4,8,15,16,23,42]
Lst
Or by a list comprehension:
[ ]: [n^2 for n in range(10)]
These may include a condition at the end:
[ ]: [a for a in range(12) if gcd(a,12) == 1]
Access individual elements:
[ ]: Lst[0]
[ ]: Lst[1]
[ ]: Lst[-1]
Test membership:
[ ]: 23 in Lst
Slicing:
[ ]: Lst[1:4]
[ ]: Lst[3:0:-1]
[ ]: Lst[-4:-1]
[ ]: Lst[::-1]
Lists may contain elements of different types, e.g. other lists.
[ ]: [ 1, 2, 3, [4,5], 6, True, None]
3.18.1. Exercise
Create a list of even numbers up to some bound. Do the same for odd numbers. Then
create a list containing both of these lists.
[ ]: 498 APPENDIX A. INTRODUCTION TO SAGEMATH
3.19. Operations on lists
[ ]: len(Lst)
[ ]: sum(Lst)
[ ]: prod(Lst)
[ ]: sorted(Lst)
[ ]: [1,2,5] + [4,8,10,12]
[ ]: list(zip([0,1,2,3,4],[10,100,1000,10000,100000]))
3.20. Modifying lists
Lists can be modified.
[ ]: L = list(primes(20))
L
[ ]: L[0] = 1
L
[ ]: L[0] = 2
L
[ ]: L.append(23)
L
[ ]: L.pop(0)
L
[ ]: L.remove(13)
L
[ ]: L.insert(0, 2)
L
3.20.1. Exercise
Find out what else you can do with lists, by typing L. and pressing the TAB key.
[ ]: L.
To find out what a method does, enter its name followed by a question mark, as before.
(The documentation of these Python methods is rather brief.)
3.21. Lists and loops
Let’s build a list iteratively, using append in a range-based for loop.
499
[ ]: unitsmod12 = []
for a in range(0,12):
if gcd(a, 12) == 1:
unitsmod12.append(a)
print(unitsmod12)
You can loop over the elements of a list:
[ ]: L = [1,3,5,7,9]
for x in L:
print(x)
This is much more convenient than (but equivalent to) the index-based alternative:
[ ]: L = [1,3,5,7,9]
for i in range(len(L)):
print(L[i])
Tip: If you have written a range-based for loop where the index is only used to access
elements of a list (as above), rewrite it as a loop over list elements to make it more
readable.
There is also the following shorthand notation:
[ ]: for x,y in [ [1,2], [3,4] ]:
print(x)
This is equivalent to:
[ ]: for P in [ [1,2], [3,4] ]:
x,y = P
print(x)
3.21.1. Exercise
Create a list of square numbers which are less than some (variable) bound.
[ ]:
3.22. Functions
A function can be defined as follows:
[ ]: def squared_plus_one(x):
return x^2 + 1
The definition starts with def, followed by the name of the function, followed by the list
of (input) arguments between parentheses, followed by a colon (:). The following lines
(each indented by four spaces) form the body of the function. The body of a function
can contain several statements. It is optional (but highly recommended) that a function
return some result.
Calling a function executes the body, and the returned value is the result:
[ ]: squared_plus_one(3)
500 APPENDIX A. INTRODUCTION TO SAGEMATH
[ ]: squared_plus_one(5)
A function needs to be defined only once, and can then be called any number of times.
(Re-defining a function overwrites the previous definition, just like it is with variables.)
Using functions is a great time saver. Instead of writing the code in the body of the
function over and over again, we just write it once, and call it by its name.
You can turn any piece of code into a function by indenting it (adding four spaces in front
of each line), giving it a name, specifying the arguments and adding a return statement.
Recall this code from the section about lists:
[ ]: unitsmod12 = []
for a in range(0,12):
if gcd(a, 12) == 1:
unitsmod12.append(a)
print(unitsmod12)
Here it is as a function, now depending on the modulus n (which used to be the constant
12):
[ ]: def units_mod(n):
units = []
for a in range(0,n):
if gcd(a, n) == 1:
units.append(a)
return units
[ ]: units_mod(12)
Big Tip: If your function calculates something, return the result and don’t just print it.
If you do not return it, then you cannot use it in further calculations.
[ ]: list(reversed(units_mod(12)))
The above would not be possible if units_mod only printed the units instead of returning
them.
For more information about printing inside functions, see Section A below.
3.22.1. Exercise
Define a function that returns the list of square numbers up to some bound (the bound
should be the input). Name your function and its arguments appropriately.
Hint: Use the code you wrote in the exercise about lists and loops.
[ ]:
3.22.2. Exercise
Define another function, and call it.
[ ]:
[ ]:
501
4. Miscellaneous
This section contains other things which are nice to know.
4.1. Documenting your code
We have already seen the documentation of Sage’s own functions, accessed using the
question mark. We can also add documentation to our own functions. This is done by
using a docstring, between triple quotes, starting at the first line in the definition of your
function:
[ ]: def multiplicative_order_mod(a,n):
"""Return the multiplicative order of a mod n."""
order = 1
g = a % n
while g != 1:
g *= a # shorthand for g = g * a
g %= n # shorthand for g = g % n
order += 1
return order
[ ]: multiplicative_order_mod(5,12)
The docstring can be accessed using the question mark:
[ ]: multiplicative_order_mod?
The docstrings in SageMath are written according to a format like this:
[ ]: def multiplicative_order_mod(a,n):
r"""
Return the multiplicative order of $a$ mod $n$.
INPUT:
- ``a`` - integer
- ``n`` - integer
OUTPUT:
- the multiplicative order of $a$ mod $n$, i.e. the smallest
positive integer $k$ such that $a^k = 1$ mod $n$.
ASSUMPTIONS:
- a mod n is a unit, i.e. $\gcd(a,n) = 1$.
EXAMPLES::
sage: multiplicative_order_mod(5,12)
2
sage: multiplicative_order_mod(2,13)
12
"""
order = 1
502 APPENDIX A. INTRODUCTION TO SAGEMATH
g = a % n
while g != 1:
g *= a # shorthand for g = g * a
g %= n # shorthand for g = g % n
order += 1
return order
[ ]: multiplicative_order_mod?
If you write a function which is interesting enough, then it is a good idea to add docu-
mentation like that.
[ ]:
4.2. Local variables
Typically, the behavior of a function should depend only on its arguments. Variables
defined inside a function are called local to that function. These variables do not affect
the values of variables with the same name which are defined outside the function.
[ ]: order = 'One pepperoni pizza, please.'
print('the multiplicative order of 3 mod 11 is', multiplicative_order_mod(3,11))
print(order)
The above works fine, even though order is also the name of a local variable inside the
function multiplicative_order_mod (defined above).
That is the nice thing about functions which use local variables (and do not modify
external variables): you can just use them without worrying about what happens inside
them.
4.3. Benchmarking your code
Check how long your code takes by adding %time in front of a line:
[ ]: %time factor(randint(10^50, 10^51))
Or, by using %timeit, run the code several times and take the best time:
[ ]: %timeit factor(randint(10^50, 10^51))
4.4. Types and parents
We already mentioned in the very beginning that each object in SageMath has a type
such as Integer, Rational, etc.
Most objects which are “elements” of some kind have a parent in SageMath.
[ ]: 5.parent()
The parent object ZZ represents the ring of integers Z.
[ ]: ZZ
503
[ ]: 5 in ZZ
[ ]: 1/2 in ZZ
The parent of a polynomial is a polynomial ring.
[ ]: R.<x> = PolynomialRing(QQ)
(x^2 + 1).parent()
[ ]: (x^2 + 1).parent() is R
The parent of a matrix is a matrix space.
[ ]: A = Matrix(QQ, [[1,1],[1,1]])
A.parent()
[ ]: A.parent() is MatrixSpace(QQ, 2)
This system of “parents” makes e.g. the following change_ring functionality possible:
[ ]: B = A.change_ring(ZZ)
B.parent()
[ ]: B.parent() is MatrixSpace(ZZ, 2)
See Parents, Conversion and Coercion in the Sage Tutorial if you are interested in the
technical details.
4.5. LATEX output
In the notebook interface, the show function displays an object using LATEX in math
mode, in a pretty way if possible. The LatexExpr function can be used to define custom
LaTeX expressions.
[ ]: show(ZZ)
show(QQ)
show(LatexExpr(r'\LaTeX\text{ is }\LaTeX\text{, la la la la la.}'))
[ ]: R.<x> = QQ[]
show(R)
show(x^2 + 2)
show(latex(x^2 + 2) + LatexExpr(r'\in') + latex(R))
[ ]: S.<x,y> = QQ[]
show(S)
show(x^2 + y^2 - 1)
show(latex(x^2 + y^2 - 1) + LatexExpr(r'\in') + latex(S))
[ ]: A = Matrix(QQ, [[0,-1],[1,0]])
show(A.parent())
show(A)
show(latex(A) + LatexExpr(r'\in') + latex(A.parent()))
504 APPENDIX A. INTRODUCTION TO SAGEMATH
4.6. Verbosity
In long computations it is sometimes useful to display intermediate results. This can of
course be done using print. But sometimes you just want to use a function without
seeing the intermediate results. How to get the best of both worlds? Use verbose and
pass a level.
[ ]: def multiplicative_order_mod(a,n):
order = 1
g = a % n
while g != 1:
verbose("{}^{} = {}".format(a,order,g), level=2)
g *= a # shorthand for g = g * a
g %= n # shorthand for g = g % n
order += 1
return order
The function verbose prints its first argument only if the current verbosity level is at
least level.
You can get the current verbosity level (default: 0) with get_verbose() and set it with
e.g. set_verbose(100).
[ ]: get_verbose()
[ ]: set_verbose(100)
multiplicative_order_mod(3,17)
[ ]: set_verbose(0)
multiplicative_order_mod(3,17)
4.7. Copying lists
There is some subtlety involved in copying a list. Try to predict the output of the following
code:
[ ]: L1 = [1,2,3]
L2 = L1
L1[0] = 5
L2
What happens here is that = for lists is assignment by reference. Here’s how to make a
copy instead:
[ ]: L1 = [1,2,3]
L2 = list(L1)
L1[0] = 5
L2
In case of e.g. nested lists even this is not enough, and you have to ensure that also the
sub-objects are copied:
[ ]: from copy import deepcopy
L1 = [ [0, 1], [-1, 0] ]
L2 = deepcopy(L1)
505
L1[0][0] = 1
L2
Don’t worry about this for now, but remember it if you run into trouble with lists.
4.8. Tuples
Tuples are like lists, but they cannot be modified and cannot contain modifiable elements.
[ ]: T = (1,2,3,4)
T
[ ]: T[0]
[ ]: T[-1]
4.9. Accessing the source code
SageMath is free software. Its source code is freely available at e.g. https://github.com/
sagemath/sage. In the notebook interface and on the command line, the source code of
functions and methods can be accessed in a similar way to accessing the documentation,
using ?? instead of ?:
[ ]: n = 10
n.factor??
At the end of the displayed source code, you find the name of the file where the
method or function is defined. In the example above, it looks something like
~/src/SageMath-9.0/local/lib/python3.7/site-packages/sage/rings/integer.pyx.
You can use this path to find the file on your own machine or on GitHub under src/sage.
Continuing the example, that file would be src/sage/rings/integer.pyx. Seeing the
whole file is useful e.g. if the source code uses some imported names (which you couldn’t
see by only looking at the function).
In the source code you will often find calls to external libraries, which are also open
source, so you can continue your investigation by reading the source code of that library.
4.10. Error messages are your friends
Each cell in this section contains a mistake, and executing it will result in an error message
in the output.
Don’t be afraid, just try it! The error messages are intended to be informative and helpful
(have a look at their last line first).
[ ]: (1+2
That was a syntax error, meaning the input was not well-formed. In particular, “unex-
pected EOF” (end-of-file) means that SageMath was expecting something more at the
end of the input (i.e. the input was not complete). If you get this kind of error, check the
balance of your parentheses and such things.
506 APPENDIX A. INTRODUCTION TO SAGEMATH
[ ]: 1/(1-2^0)
That was division by zero.
[ ]: 2 + ZZ
That error means that addition (the operator +) is not defined for the two objects (the
operands of +) which we tried to add.
Usually for addition to make sense, the two operands should have a common parent
object.
[ ]: numerical_approx(ZZ)
The code above tries to get a numerical approximation of Z, which is nonsense because
Z is not a number.
[ ]: ZZ.n()
The code above tries to get a numerical approximation of Z in another way. This time it
doesn’t work because ZZ does not have the method n().
Appendix B
Kontsevich’s star product
⋆ mod ō(h̄6)
B.1 Kontsevich’s star product ⋆ mod ō(h̄4)
507
508 APPENDIX B. KONTSEVICH’S STAR PRODUCT ⋆ mod ō(h̄6)
Encoding 1. In the format described in 2 4 1 0 1 0 1 1 2 2 3 -1/9
Chapter 11, Implementation 1: 2 4 1 0 1 0 1 1 2 3 4 -1/18
2 4 1 0 1 0 2 1 2 1 2 1/30
h^0: 2 4 1 0 1 0 4 1 2 1 2 13/90
# 0 0 2 4 1 0 1 0 2 1 2 1 4 -1/45
2 0 1 1 2 4 1 0 1 0 4 1 2 1 4 -1/30
h^1: 2 4 1 0 1 0 4 1 5 1 2 1/90
# 1 1 2 4 1 0 1 0 2 1 2 1 3 1/30
2 1 1 0 1 1 2 4 1 0 1 0 4 1 2 1 3 1/15
h^2: 2 4 1 0 1 0 2 1 3 1 3 1/15
# 1 1 2 4 1 0 1 0 2 1 3 1 4 1/90
2 2 1 0 3 1 2 -1/6 # 3 2
# 1 2 2 4 1 0 1 0 1 0 2 2 4 -1/6
2 2 1 0 1 1 2 -1/3 2 4 1 0 1 0 4 1 3 0 4 -1/6
# 2 1 2 4 1 0 1 0 2 0 5 1 4 -1/18
2 2 1 0 1 0 2 1/3 2 4 1 0 1 0 1 0 2 2 3 -1/9
# 2 2 2 4 1 0 1 0 1 0 2 3 4 -1/18
2 2 1 0 1 0 1 1/2 2 4 1 0 1 0 2 0 2 1 2 -1/30
h^3: 2 4 1 0 1 0 2 0 2 1 3 -13/90
# 1 1 2 4 1 0 1 0 2 0 3 1 2 1/45
2 3 1 0 3 1 2 2 3 -1/6 2 4 1 0 1 0 2 0 3 1 3 1/30
# 1 3 2 4 1 0 1 0 2 0 3 1 4 -1/90
2 3 1 0 1 1 2 1 2 1/6 2 4 1 0 1 0 2 0 5 1 2 -1/30
# 2 3 2 4 1 0 1 0 2 0 5 1 3 -1/15
2 3 1 0 1 0 1 1 2 -1/3 2 4 1 0 1 0 4 1 2 0 4 -1/15
# 3 1 2 4 1 0 1 0 4 0 5 1 2 -1/90
2 3 1 0 1 0 2 0 2 1/6 # 1 3
# 3 2 2 4 1 0 1 1 2 1 2 2 3 7/90
2 3 1 0 1 0 1 0 2 1/3 2 4 1 0 1 1 2 1 3 2 3 -1/90
# 3 3 2 4 1 0 1 1 2 1 3 2 4 1/30
2 3 1 0 1 0 1 0 1 1/6 2 4 1 0 1 1 2 1 3 3 4 2/45
# 1 2 2 4 1 0 1 1 2 1 5 2 3 1/30
2 3 1 0 1 1 2 2 3 -1/6 2 4 1 0 1 1 2 1 5 2 4 1/45
2 3 1 0 3 1 2 1 2 1/6 2 4 1 0 1 1 2 1 5 3 4 -1/90
# 2 1 2 4 1 0 1 1 4 1 5 2 3 -1/90
2 3 1 0 1 0 2 2 3 -1/6 2 4 1 0 1 1 4 1 5 2 4 -1/90
2 3 1 0 3 1 2 0 3 -1/6 2 4 1 0 1 1 4 2 3 1 4 1/30
# 2 2 2 4 1 0 3 1 4 1 5 1 2 1/90
2 3 1 0 1 0 4 1 3 -1/6 2 4 1 0 3 1 4 1 2 1 3 1/90
2 3 1 0 1 0 2 1 3 -1/6 2 4 1 0 3 1 2 1 3 1 4 1/90
2 3 1 0 1 0 4 1 2 -1/6 2 4 1 0 3 1 4 1 2 1 4 -1/30
h^4: 2 4 1 0 3 1 4 1 2 1 2 -1/45
# 3 4 2 4 1 0 3 1 2 1 3 1 3 -1/60
2 4 1 0 1 0 1 0 1 1 2 -1/6 2 4 1 0 3 1 2 1 2 1 3 -1/45
# 4 3 2 4 1 0 3 1 2 1 2 1 4 -1/45
2 4 1 0 1 0 1 0 1 0 2 1/6 2 4 1 0 3 1 2 1 2 1 2 -1/20
# 4 4 # 3 1
2 4 1 0 1 0 1 0 1 0 1 1/24 2 4 1 0 1 0 2 0 2 2 3 -7/90
# 2 4 2 4 1 0 1 0 2 0 3 2 3 1/90
2 4 1 0 1 0 1 1 2 1 2 1/6 2 4 1 0 1 0 2 0 3 2 4 -1/30
2 4 1 0 1 0 1 1 2 1 3 1/18 2 4 1 0 1 0 2 0 3 3 4 -2/45
# 4 2 2 4 1 0 1 0 2 0 5 2 3 -1/30
2 4 1 0 1 0 1 0 2 0 2 1/6 2 4 1 0 1 0 2 0 5 2 4 -1/45
2 4 1 0 1 0 1 0 2 0 3 1/18 2 4 1 0 1 0 2 0 5 3 4 1/90
# 1 4 2 4 1 0 1 0 4 0 5 2 3 1/90
2 4 1 0 1 1 2 1 2 1 2 -1/30 2 4 1 0 1 0 4 0 5 2 4 1/90
2 4 1 0 1 1 2 1 2 1 3 -2/45 2 4 1 0 1 0 4 2 3 0 4 -1/30
2 4 1 0 1 1 2 1 3 1 3 -1/30 2 4 1 0 3 0 4 0 5 1 2 1/90
2 4 1 0 1 1 2 1 3 1 4 1/45 2 4 1 0 3 0 4 0 5 1 3 1/90
# 3 3 2 4 1 0 3 0 4 0 5 1 4 1/90
2 4 1 0 1 0 1 0 5 1 4 -1/12 2 4 1 0 3 0 4 1 2 0 3 -1/30
2 4 1 0 1 0 1 0 2 1 4 -1/6 2 4 1 0 3 0 4 1 2 0 4 -1/45
2 4 1 0 1 0 1 0 5 1 2 -1/6 2 4 1 0 3 0 4 1 3 0 3 -1/60
2 4 1 0 1 0 1 0 2 1 3 -1/9 2 4 1 0 3 0 4 1 3 0 4 -1/45
# 4 1 2 4 1 0 3 0 4 1 5 0 4 -1/45
2 4 1 0 1 0 2 0 2 0 2 1/30 2 4 1 0 3 1 2 0 3 0 3 -1/20
2 4 1 0 1 0 2 0 2 0 3 2/45 # 1 1
2 4 1 0 1 0 2 0 3 0 3 1/30 2 4 1 0 3 1 2 2 3 2 3 -1/40
2 4 1 0 1 0 2 0 3 0 4 -1/45 2 4 1 0 3 1 2 2 3 2 4 -1/72
# 2 3 2 4 1 0 3 1 2 2 3 3 4 1/72
2 4 1 0 1 0 1 1 2 2 4 -1/6 2 4 1 0 3 1 2 2 5 3 4 1/360
2 4 1 0 1 0 4 1 3 1 3 1/6 2 4 1 0 3 1 4 2 3 2 3 -1/60
2 4 1 0 1 0 4 1 3 1 2 1/18 2 4 1 0 3 2 4 1 2 2 4 -1/60
B.1. KONTSEVICH’S STAR PRODUCT ⋆ mod ō(h̄4) 509
2 4 1 0 3 1 4 2 3 3 4 17/720 2 4 1 0 1 0 2 2 3 3 4 -1/30
2 4 1 0 3 2 4 1 2 2 3 -17/720 2 4 1 0 1 0 2 2 5 3 4 1/720
2 4 1 0 3 1 4 2 5 2 3 -1/180 2 4 1 0 1 0 4 2 3 2 3 19/720
2 4 1 0 3 4 5 1 2 2 4 1/180 2 4 1 0 1 0 4 2 3 2 4 -1/180
2 4 1 0 3 1 4 2 5 2 4 1/360 2 4 1 0 1 0 4 2 3 3 4 -13/360
2 4 1 0 3 4 5 1 2 3 4 -1/360 2 4 1 0 1 0 4 2 5 2 3 1/720
2 4 1 0 3 1 4 2 5 3 4 1/160 2 4 1 0 1 0 4 2 5 2 4 -1/360
2 4 1 0 3 4 5 1 2 2 3 1/160 2 4 1 0 1 0 4 2 5 3 4 -1/720
2 4 1 0 3 1 4 3 5 2 3 -17/1440 2 4 1 0 1 0 4 3 5 2 3 1/80
2 4 1 0 3 2 4 2 5 1 2 17/1440 2 4 1 0 1 0 4 3 5 2 4 1/180
2 4 1 0 3 1 4 3 5 2 4 -1/360 2 4 1 0 3 0 4 1 2 2 3 -1/36
2 4 1 0 3 2 4 3 5 1 2 1/360 2 4 1 0 3 0 4 1 2 2 4 -1/60
2 4 1 0 3 2 4 1 3 2 3 -13/720 2 4 1 0 3 0 4 1 2 3 4 1/720
2 4 1 0 3 2 4 1 3 3 4 -13/720 2 4 1 0 3 0 4 1 3 2 3 -1/45
2 4 1 0 3 2 4 1 3 2 4 -1/60 2 4 1 0 3 0 4 1 3 2 4 -17/720
2 4 1 0 3 2 4 1 5 2 3 -7/720 2 4 1 0 3 0 4 1 3 3 4 -13/360
2 4 1 0 3 4 5 1 5 2 4 7/720 2 4 1 0 3 0 4 1 5 2 3 -1/120
2 4 1 0 3 2 4 1 5 2 4 -1/180 2 4 1 0 3 0 4 1 5 2 4 -1/240
2 4 1 0 3 2 4 1 5 3 4 1/160 2 4 1 0 3 0 4 1 5 3 4 1/80
2 4 1 0 3 2 4 2 5 1 4 1/160 2 4 1 0 3 0 4 2 5 1 2 1/72
2 4 1 0 3 2 4 2 5 1 3 13/1440 2 4 1 0 3 0 4 2 5 1 3 1/90
2 4 1 0 3 4 5 1 3 2 4 -13/1440 2 4 1 0 3 0 4 2 5 1 4 1/180
2 4 1 0 3 2 4 3 5 1 3 -1/1440 2 4 1 0 3 0 4 3 5 1 2 1/360
2 4 1 0 3 4 5 1 3 2 3 1/1440 2 4 1 0 3 0 4 3 5 1 3 -1/720
2 4 1 0 3 2 4 3 5 1 4 1/360 2 4 1 0 3 0 4 3 5 1 4 1/180
2 4 1 0 3 4 5 1 5 2 3 1/240 2 4 1 0 3 1 2 0 3 2 3 -17/180
# 1 2 2 4 1 0 3 1 2 0 3 2 4 -1/72
2 4 1 0 1 1 2 2 3 2 3 -13/360 2 4 1 0 3 1 2 0 3 3 4 7/180
2 4 1 0 1 1 2 2 3 2 4 -1/720 2 4 1 0 3 1 2 0 5 2 3 -7/180
2 4 1 0 1 1 2 2 3 3 4 1/30 2 4 1 0 3 1 2 0 5 2 4 -1/180
2 4 1 0 1 1 2 2 5 3 4 -1/720 2 4 1 0 3 1 2 0 5 3 4 1/90
2 4 1 0 1 1 4 2 3 2 3 -19/720 2 4 1 0 3 1 4 0 5 2 3 -1/80
2 4 1 0 1 1 4 2 3 2 4 1/180 2 4 1 0 3 1 4 0 5 2 4 -1/120
2 4 1 0 1 1 4 2 3 3 4 13/360 2 4 1 0 3 1 4 0 5 3 4 1/40
2 4 1 0 1 1 4 2 5 2 3 -1/720 2 4 1 0 3 1 4 2 3 0 3 1/360
2 4 1 0 1 1 4 2 5 2 4 1/360 2 4 1 0 3 1 4 2 3 0 4 11/720
2 4 1 0 1 1 4 2 5 3 4 1/720 2 4 1 0 3 2 4 1 2 0 3 -7/240
2 4 1 0 1 1 4 3 5 2 3 -1/80 2 4 1 0 3 1 4 2 5 0 4 -1/180
2 4 1 0 1 1 4 3 5 2 4 -1/180 2 4 1 0 3 2 4 1 5 0 3 -1/60
2 4 1 0 3 1 4 1 2 3 4 -1/36 2 4 1 0 3 1 4 3 5 0 4 1/90
2 4 1 0 3 1 4 1 2 2 3 1/60 2 4 1 0 3 2 4 1 3 0 3 -1/60
2 4 1 0 3 1 4 1 2 2 4 -1/720 # 2 2
2 4 1 0 3 1 2 1 3 3 4 1/45 2 4 1 0 1 0 4 1 3 3 4 -1/6
2 4 1 0 3 1 2 1 3 2 4 17/720 2 4 1 0 3 1 2 0 5 1 4 1/72
2 4 1 0 3 1 2 1 3 2 3 13/360 2 4 1 0 1 0 1 2 3 2 3 17/360
2 4 1 0 3 4 5 1 2 1 4 1/120 2 4 1 0 1 0 1 2 3 2 4 1/24
2 4 1 0 3 2 4 1 5 1 2 1/240 2 4 1 0 1 0 1 2 5 3 4 1/180
2 4 1 0 3 2 4 1 2 1 4 -1/80 2 4 1 0 1 0 2 1 2 2 3 2/45
2 4 1 0 3 1 4 1 5 2 3 -1/72 2 4 1 0 1 0 2 1 2 2 4 -2/45
2 4 1 0 3 1 4 2 5 1 3 -1/90 2 4 1 0 1 0 2 1 2 3 4 -1/30
2 4 1 0 3 2 4 1 5 1 3 -1/180 2 4 1 0 1 0 2 1 3 2 3 1/8
2 4 1 0 3 1 4 1 5 2 4 -1/360 2 4 1 0 1 0 4 1 2 2 4 -1/8
2 4 1 0 3 1 4 2 3 1 3 1/720 2 4 1 0 1 0 2 1 3 2 4 1/720
2 4 1 0 3 2 4 1 3 1 4 -1/180 2 4 1 0 1 0 4 1 2 2 3 -1/720
2 4 1 0 3 1 2 1 2 2 3 17/180 2 4 1 0 1 0 2 1 3 3 4 -11/180
2 4 1 0 3 1 2 1 2 3 4 -1/72 2 4 1 0 1 0 4 1 2 3 4 -11/180
2 4 1 0 3 1 2 1 2 2 4 7/180 2 4 1 0 1 0 2 1 5 2 3 1/36
2 4 1 0 3 1 2 1 5 2 3 7/180 2 4 1 0 1 0 4 2 5 1 2 -1/36
2 4 1 0 3 1 2 1 5 3 4 -1/180 2 4 1 0 1 0 2 1 5 2 4 1/90
2 4 1 0 3 1 2 1 5 2 4 1/90 2 4 1 0 1 0 4 2 3 1 2 -1/90
2 4 1 0 3 1 4 2 5 1 2 1/80 2 4 1 0 1 0 2 1 5 3 4 -1/180
2 4 1 0 3 1 4 3 5 1 2 1/120 2 4 1 0 1 0 4 3 5 1 2 -1/180
2 4 1 0 3 1 4 2 3 1 2 1/40 2 4 1 0 1 0 4 1 3 2 3 1/18
2 4 1 0 3 2 4 1 2 1 2 -1/360 2 4 1 0 1 0 4 1 3 2 4 -1/18
2 4 1 0 3 2 4 1 2 1 3 -11/720 2 4 1 0 1 0 4 1 5 2 3 1/144
2 4 1 0 3 1 4 2 3 1 4 7/240 2 4 1 0 1 0 4 2 5 1 3 -1/144
2 4 1 0 3 4 5 1 2 1 3 -1/180 2 4 1 0 1 0 4 1 5 2 4 -1/90
2 4 1 0 3 1 4 2 5 1 4 1/60 2 4 1 0 1 0 4 2 3 1 3 1/90
2 4 1 0 3 2 4 1 3 1 2 1/90 2 4 1 0 1 0 4 2 3 1 4 -1/60
2 4 1 0 3 2 4 1 3 1 3 1/60 2 4 1 0 1 0 4 2 5 1 4 1/60
# 2 1 2 4 1 0 3 0 4 1 2 1 2 -1/240
2 4 1 0 1 0 2 2 3 2 3 13/360 2 4 1 0 3 1 4 1 2 0 3 -1/240
2 4 1 0 1 0 2 2 3 2 4 1/720 2 4 1 0 3 0 4 1 2 1 3 -13/720
510 APPENDIX B. KONTSEVICH’S STAR PRODUCT ⋆ mod ō(h̄6)
2 4 1 0 3 1 4 0 5 1 3 -13/720
2 4 1 0 3 0 4 1 2 1 4 -1/90
2 4 1 0 3 0 4 1 5 1 3 -1/90
2 4 1 0 3 0 4 1 3 1 2 1/60
2 4 1 0 3 1 2 0 5 1 3 1/60
2 4 1 0 3 0 4 1 3 1 3 1/30
2 4 1 0 3 1 2 0 3 1 3 1/30
2 4 1 0 3 0 4 1 3 1 4 -1/90
2 4 1 0 3 0 4 1 5 1 2 1/360
2 4 1 0 3 1 2 0 3 1 2 13/90
2 4 1 0 3 1 2 0 3 1 4 13/180
2 4 1 0 3 1 2 0 5 1 2 13/180
2 4 1 0 3 1 4 0 5 1 2 1/72
B.2. ASSOCIATIVITY OF KONTSEVICH’S ⋆ mod ō(h̄6) 511
B.2 Associativity of Kontsevich’s ⋆ mod ō(h̄6)
Here we show the associativity of Kontsevich’s ⋆ mod ō(h̄6), up to ō(h̄6).
First we split the associator into two parts, and show their vanishing separately.
The rational part vanishes because it can be expressed as a sum of Leibniz graphs (from
the 0th layer):
[1]: from gcaops.graph.formality_graph_complex import FormalityGraphComplex
FGC = FormalityGraphComplex(QQ, lazy=True); FGC
n = 6
assoc = FGC.element_from_kgs_encoding(open('data/assoc{}_ratpart.txt'.format(n)).
↪→read().rstrip())
assoc_n = assoc.homogeneous_part(3,n,2*n)
print('Number of Kontsevich graphs:', len(assoc_n), flush=True)
diff_orders = list(assoc_n.differential_orders())
print('Number of differential orders:', len(diff_orders), flush=True)
from gcaops.graph.leibniz_graph_expansion import␣
↪→kontsevich_graph_sum_to_leibniz_graph_sum
from gcaops.graph.leibniz_graph_expansion import␣
↪→leibniz_graph_sum_to_kontsevich_graph_sum
for diff_order in diff_orders:
print(diff_order, end=': ', flush=True)
part = assoc_n.part_of_differential_order(diff_order)
part_Leibniz = kontsevich_graph_sum_to_leibniz_graph_sum(part, verbose=True)
print(leibniz_graph_sum_to_kontsevich_graph_sum(part_Leibniz) == part, flush=True)
Number of Kontsevich graphs: 290243
Number of differential orders: 105
(3, 1, 4): 449K -> +220L -> +26K
True
(2, 2, 4): 829K -> +424L -> +71K
True
(2, 1, 4): 1524K -> +780L -> +115K
True
(1, 3, 4): 443K -> +220L -> +32K
True
(1, 2, 4): 1515K -> +780L -> +124K
True
(1, 1, 4): 2315K -> +1135L -> +281K
True
(3, 2, 4): 208K -> +98L -> +17K
True
(2, 3, 4): 203K -> +98L -> +22K
True
(1, 4, 4): 75K -> +36L -> +11K
True
(4, 1, 4): 82K -> +36L -> +4K
True
(4, 2, 4): 32K -> +14L -> +7K
True
(3, 3, 4): 38K -> +16L -> +7K
True
(2, 4, 4): 32K -> +14L -> +7K
True
(4, 2, 3): 208K -> +98L -> +17K
512 APPENDIX B. KONTSEVICH’S STAR PRODUCT ⋆ mod ō(h̄6)
True
(4, 1, 3): 449K -> +220L -> +26K
True
(3, 3, 3): 362K -> +175L -> +30K
True
(3, 2, 3): 1424K -> +810L -> +161K
True
(3, 1, 3): 2612K -> +1475L -> +199K
True
(2, 4, 3): 199K -> +97L -> +26K
True
(2, 3, 3): 1423K -> +810L -> +162K
True
(2, 2, 3): 4984K -> +2947L -> +451K
True
(2, 1, 3): 7702K -> +4353L -> +618K
True
(1, 4, 3): 417K -> +215L -> +58K
True
(1, 3, 3): 2583K -> +1469L -> +216K
True
(1, 2, 3): 7659K -> +4350L -> +661K
True
(1, 1, 3): 10263K -> +5295L -> +1217K
True
(4, 3, 3): 38K -> +16L -> +7K
True
(3, 4, 3): 38K -> +16L -> +7K
True
(2, 5, 3): 10K -> +5L -> +5K
True
(1, 5, 3): 36K -> +14L -> +6K
True
(5, 2, 3): 15K -> +5L -> +0K
True
(5, 1, 3): 41K -> +14L -> +1K
True
(5, 3, 3): 3K -> +1L -> +0K
True
(4, 4, 3): 3K -> +1L -> +0K
True
(3, 5, 3): 3K -> +1L -> +0K
True
(1, 5, 4): 7K -> +4L -> +5K
True
(5, 1, 4): 12K -> +4L -> +0K
True
(5, 2, 4): 3K -> +1L -> +0K
True
(4, 3, 4): 3K -> +1L -> +0K
True
(3, 4, 4): 3K -> +1L -> +0K
True
(2, 5, 4): 3K -> +1L -> +0K
True
(4, 2, 2): 829K -> +424L -> +71K
True
(4, 1, 2): 1524K -> +780L -> +115K
B.2. ASSOCIATIVITY OF KONTSEVICH’S ⋆ mod ō(h̄6) 513
True
(3, 3, 2): 1423K -> +810L -> +162K
True
(3, 2, 2): 4984K -> +2947L -> +451K
True
(3, 1, 2): 7702K -> +4353L -> +618K
True
(2, 3, 2): 4779K -> +2908L -> +631K
True
(2, 2, 2): 14046K -> +8416L -> +1618K
True
(2, 1, 2): 18894K -> +10298L -> +1904K
True
(1, 4, 2): 1338K -> +752L -> +266K
True
(1, 3, 2): 7297K -> +4282L -> +956K
True
(1, 2, 2): 19000K -> +10368L -> +1796K
True
(1, 1, 2): 22789K -> +10742L -> +2227K
True
(4, 3, 2): 203K -> +98L -> +22K
True
(3, 4, 2): 199K -> +97L -> +26K
True
(2, 4, 2): 758K -> +421L -> +141K
True
(1, 5, 2): 96K -> +43L -> +33K
True
(5, 2, 2): 53K -> +18L -> +1K
True
(5, 1, 2): 121K -> +43L -> +8K
True
(5, 3, 2): 15K -> +5L -> +0K
True
(4, 4, 2): 32K -> +14L -> +7K
True
(2, 5, 2): 43K -> +18L -> +11K
True
(3, 5, 2): 10K -> +5L -> +5K
True
(4, 2, 1): 1515K -> +780L -> +124K
True
(4, 1, 1): 2315K -> +1135L -> +281K
True
(3, 3, 1): 2583K -> +1469L -> +216K
True
(3, 2, 1): 7659K -> +4350L -> +661K
True
(3, 1, 1): 10263K -> +5295L -> +1217K
True
(2, 4, 1): 1338K -> +752L -> +266K
True
(2, 3, 1): 7297K -> +4282L -> +956K
True
(2, 2, 1): 19000K -> +10368L -> +1796K
True
(2, 1, 1): 22789K -> +10742L -> +2227K
514 APPENDIX B. KONTSEVICH’S STAR PRODUCT ⋆ mod ō(h̄6)
True
(1, 4, 1): 2223K -> +1135L -> +373K
True
(1, 3, 1): 10068K -> +5290L -> +1412K
True
(1, 2, 1): 22591K -> +10736L -> +2424K
True
(1, 1, 1): 23814K -> +9358L -> +2709K
True
(4, 3, 1): 443K -> +220L -> +32K
True
(3, 4, 1): 417K -> +215L -> +58K
True
(2, 5, 1): 96K -> +43L -> +33K
True
(1, 5, 1): 234K -> +81L -> +8K
True
(5, 2, 1): 121K -> +43L -> +8K
True
(5, 1, 1): 234K -> +81L -> +8K
True
(5, 3, 1): 41K -> +14L -> +1K
True
(4, 4, 1): 75K -> +36L -> +11K
True
(3, 5, 1): 36K -> +14L -> +6K
True
(5, 4, 1): 12K -> +4L -> +0K
True
(5, 4, 2): 3K -> +1L -> +0K
True
(4, 5, 2): 3K -> +1L -> +0K
True
(1, 1, 5): 234K -> +81L -> +8K
True
(5, 1, 5): 3K -> +1L -> +0K
True
(4, 2, 5): 3K -> +1L -> +0K
True
(3, 3, 5): 3K -> +1L -> +0K
True
(2, 4, 5): 3K -> +1L -> +0K
True
(1, 5, 5): 3K -> +1L -> +0K
True
(2, 1, 5): 121K -> +43L -> +8K
True
(1, 2, 5): 121K -> +43L -> +8K
True
(4, 1, 5): 12K -> +4L -> +0K
True
(3, 2, 5): 15K -> +5L -> +0K
True
(2, 3, 5): 15K -> +5L -> +0K
True
(1, 4, 5): 12K -> +4L -> +0K
True
(3, 1, 5): 41K -> +14L -> +1K
B.2. ASSOCIATIVITY OF KONTSEVICH’S ⋆ mod ō(h̄6) 515
True
(2, 2, 5): 53K -> +18L -> +1K
True
(1, 3, 5): 41K -> +14L -> +1K
True
(4, 5, 1): 7K -> +4L -> +5K
True
(5, 5, 1): 3K -> +1L -> +0K
True
The part proportional to ζ(3)2/π6 vanishes because it is a sum of Leibniz graphs (from
the 0th layer, and from the 1st layer in 6 exceptional cases):
[2]: from gcaops.graph.formality_graph_complex import FormalityGraphComplex
FGC = FormalityGraphComplex(QQ, lazy=True); FGC
n = 6
assoc = FGC.element_from_kgs_encoding(open('data/assoc{}_zetapart.txt'.format(n)).
↪→read().rstrip())
assoc_n = assoc.homogeneous_part(3,n,2*n)
print('Number of Kontsevich graphs:', len(assoc_n), flush=True)
diff_orders = list(assoc_n.differential_orders())
print('Number of differential orders:', len(diff_orders), flush=True)
from gcaops.graph.leibniz_graph_expansion import␣
↪→kontsevich_graph_sum_to_leibniz_graph_sum
from gcaops.graph.leibniz_graph_expansion import␣
↪→leibniz_graph_sum_to_kontsevich_graph_sum
for diff_order in diff_orders:
print(diff_order, end=': ', flush=True)
part = assoc_n.part_of_differential_order(diff_order)
part_Leibniz = kontsevich_graph_sum_to_leibniz_graph_sum(part, verbose=True)
print(leibniz_graph_sum_to_kontsevich_graph_sum(part_Leibniz) == part, flush=True)
Number of Kontsevich graphs: 194060
Number of differential orders: 28
(2, 1, 3): 4987K -> +3481L -> +1447K
True
(1, 1, 3): 8899K -> +5099L -> +2489K -> +648L -> +201K
True
(3, 1, 3): 732K -> +592L -> +488K
True
(2, 2, 2): 6240K -> +5047L -> +4038K
True
(3, 1, 2): 4987K -> +3481L -> +1447K
True
(2, 1, 2): 16100K -> +9665L -> +2912K -> +575L -> +84K
True
(1, 2, 2): 14200K -> +9001L -> +4579K -> +1224L -> +311K
True
(1, 1, 2): 21813K -> +10699L -> +3178K
True
(3, 2, 2): 988K -> +904L -> +844K
True
(4, 1, 2): 520K -> +392L -> +299K
True
(4, 1, 1): 1363K -> +876L -> +1061K
True
516 APPENDIX B. KONTSEVICH’S STAR PRODUCT ⋆ mod ō(h̄6)
(3, 2, 1): 4173K -> +3076L -> +2051K
True
(3, 1, 1): 8899K -> +5099L -> +2489K -> +648L -> +201K
True
(2, 3, 1): 2620K -> +2084L -> +2797K
True
(2, 2, 1): 14200K -> +9001L -> +4579K -> +1224L -> +311K
True
(2, 1, 1): 21813K -> +10699L -> +3178K
True
(1, 3, 1): 5913K -> +3834L -> +4472K -> +1749L -> +1122K
True
(1, 2, 1): 20238K -> +10386L -> +4612K
True
(1, 1, 1): 23331K -> +9345L -> +3180K
True
(4, 2, 1): 520K -> +392L -> +299K
True
(3, 3, 1): 670K -> +566L -> +487K
True
(1, 1, 4): 1363K -> +876L -> +1061K
True
(1, 3, 2): 2620K -> +2084L -> +2797K
True
(1, 2, 3): 4173K -> +3076L -> +2051K
True
(2, 1, 4): 520K -> +392L -> +299K
True
(1, 2, 4): 520K -> +392L -> +299K
True
(2, 2, 3): 988K -> +904L -> +844K
True
(1, 3, 3): 670K -> +566L -> +487K
True
In fact the need of the 1st layer in the 6 exceptional cases is an artefact of our splitting
of the associator. At each of those 6 tri-differential orders, taking the two parts together
yields a factorization using Leibniz graphs from the 0th layer:
[ ]: from gcaops.graph.formality_graph_complex import FormalityGraphComplex
FGC = FormalityGraphComplex(SR, lazy=True); FGC
n = 6
#star = FGC.element_from_kgs_encoding(open('data/star{}.txt'.format(n)).read().
↪→rstrip())
#%time
#assoc = star.insertion(0, star, max_num_aerial=n) - star.insertion(1, star,␣
↪→max_num_aerial=n)
#with open('data/assoc{}.txt'.format(n), 'w') as f:
# f.write(assoc.kgs_encoding())
#%time
assoc = FGC.element_from_kgs_encoding(open('data/assoc{}.txt'.format(n)).read().
↪→rstrip())
assoc_n = assoc.homogeneous_part(3,n,2*n)
print('Number of Kontsevich graphs:', len(assoc_n), flush=True)
#diff_orders = list(assoc_n.differential_orders())
#print('Number of differential orders:', len(diff_orders), flush=True)
B.2. ASSOCIATIVITY OF KONTSEVICH’S ⋆ mod ō(h̄6) 517
from gcaops.graph.leibniz_graph_expansion import␣
↪→kontsevich_graph_sum_to_leibniz_graph_sum
from gcaops.graph.leibniz_graph_expansion import␣
↪→leibniz_graph_sum_to_kontsevich_graph_sum
def coefficient_to_vector(c):
f = QQ['zzz'](str(SR(c).expand()).replace('zeta(3)^2/pi^6', 'zzz'))
return vector(QQ, [f.constant_coefficient(), f.monomial_coefficient(QQ['zzz'].
↪→gen())])
def vector_to_coefficient(v):
return v[0] + v[1]*zeta(3)^2/pi^6
for diff_order in reversed([(2,1,2), (1,2,2), (2,2,1), (1,1,3), (3,1,1), (1,3,1)]):
print(diff_order, end=': ', flush=True)
part = assoc_n.part_of_differential_order(diff_order)
part_Leibniz = kontsevich_graph_sum_to_leibniz_graph_sum(part,␣
↪→coefficient_to_vector=coefficient_to_vector,␣
↪→vector_to_coefficient=vector_to_coefficient, verbose=True)
print(leibniz_graph_sum_to_kontsevich_graph_sum(part_Leibniz) == part, flush=True)
Number of Kontsevich graphs: 290305
(1, 3, 1): 10068K -> +5290L -> +1412K
True
(3, 1, 1): 10264K -> +5295L -> +1216K
True
(1, 1, 3): 10264K -> +5295L -> +1216K
True
(2, 2, 1): 19006K -> +10368L -> +1790K
True
(1, 2, 2): 19006K -> +10368L -> +1790K
True
(2, 1, 2): 18901K -> +10298L -> +1897K
True
Hence Kontsevich’s ⋆ mod ō(h̄6) is associative; its associator up to ō(h̄6) is a sum of
Leibniz graphs from the 0th layer.

Appendix C
Kontsevich affine star product
⋆ mod ō(h̄7)
C.1 Original expansion ⋆aff mod ō(h̄7)
Encoding 1. In the format described in Chapter 11, Implementation 1:
h^0:
2 0 1 1
h^1:
2 1 1 0 1 1
h^2:
2 2 1 1 3 0 2 -1/6
2 2 1 1 3 0 1 -1/3
2 2 1 0 3 0 1 1/3
2 2 1 0 1 0 1 1/2
h^3:
2 3 1 1 4 0 1 0 1 -1/3
2 3 1 0 4 0 1 0 1 1/3
2 3 1 0 1 0 1 0 1 1/6
2 3 1 1 3 0 2 0 1 -1/6
2 3 1 1 3 0 4 0 1 -1/6
2 3 1 1 4 0 2 0 1 -1/6
h^4:
2 4 1 1 5 0 1 0 1 0 1 -1/6
2 4 1 0 5 0 1 0 1 0 1 1/6
2 4 1 0 1 0 1 0 1 0 1 1/24
2 4 1 1 5 1 4 0 1 0 1 1/18
2 4 1 0 5 0 4 0 1 0 1 1/18
2 4 1 1 4 1 5 1 3 0 1 1/45
2 4 1 1 3 0 2 0 1 0 1 -1/12
2 4 1 1 3 0 5 0 1 0 1 -1/6
2 4 1 1 5 0 2 0 1 0 1 -1/6
2 4 1 1 5 0 4 0 1 0 1 -1/9
2 4 1 0 4 0 5 0 3 0 1 -1/45
2 4 1 1 5 1 4 0 3 0 1 1/18
2 4 1 3 5 1 4 0 1 0 1 1/18
2 4 1 1 3 1 5 0 2 0 1 1/90
2 4 1 1 4 1 5 0 3 0 1 1/15
2 4 1 1 3 1 4 0 5 0 1 1/90
2 4 1 1 4 0 5 0 2 0 1 -1/18
2 4 1 3 5 0 4 0 1 0 1 1/18
2 4 1 1 3 0 4 0 5 0 1 -1/90
2 4 1 1 4 0 2 0 5 0 1 -1/15
2 4 1 1 5 0 4 0 2 0 1 -1/90
2 4 1 3 4 1 2 1 5 0 1 1/90
2 4 1 3 5 1 4 1 2 0 1 1/90
2 4 1 1 4 1 5 1 3 0 2 1/90
2 4 1 3 4 0 5 0 2 0 1 1/90
2 4 1 3 5 0 4 0 2 0 1 -1/90
2 4 1 1 3 0 5 0 2 0 4 1/90
2 4 1 1 5 1 4 0 3 0 2 1/72
519
520 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 4 1 3 5 2 4 0 1 0 1 1/180
2 4 1 3 4 1 2 0 5 0 1 1/180
2 4 1 3 4 1 5 0 2 0 1 1/180
2 4 1 4 5 1 2 0 3 0 1 -1/144
2 4 1 3 5 1 4 0 2 0 1 1/144
2 4 1 1 3 1 4 0 5 0 2 1/360
2 4 1 1 5 1 4 0 2 0 3 1/72
h^5:
2 5 1 0 1 0 1 0 1 0 1 0 1 1/120
2 5 1 1 6 0 1 0 1 0 1 0 1 -1/18
2 5 1 0 6 0 1 0 1 0 1 0 1 1/18
2 5 1 1 3 0 6 0 1 0 1 0 1 -1/12
2 5 1 1 6 0 2 0 1 0 1 0 1 -1/12
2 5 1 1 6 1 5 0 1 0 1 0 1 1/18
2 5 1 0 6 0 5 0 1 0 1 0 1 1/18
2 5 1 1 6 0 5 0 1 0 1 0 1 -1/9
2 5 1 1 4 1 6 1 3 0 1 0 1 1/45
2 5 1 0 4 0 6 0 3 0 1 0 1 -1/45
2 5 1 1 4 1 6 0 3 0 1 0 1 1/15
2 5 1 1 4 0 2 0 6 0 1 0 1 -1/15
2 5 1 3 6 1 5 0 1 0 1 0 1 1/18
2 5 1 3 6 0 5 0 1 0 1 0 1 1/18
2 5 1 4 5 1 2 1 6 0 1 0 1 -1/90
2 5 1 4 5 0 2 0 6 0 1 0 1 1/90
2 5 1 1 3 1 6 0 2 0 1 0 1 1/90
2 5 1 1 3 0 4 0 6 0 1 0 1 -1/90
2 5 1 1 5 1 6 0 3 0 1 0 1 1/18
2 5 1 1 4 0 5 0 6 0 1 0 1 -1/18
2 5 1 1 4 1 5 1 3 0 2 0 1 1/90
2 5 1 1 3 0 5 0 2 0 4 0 1 1/90
2 5 1 3 6 1 4 1 2 0 1 0 1 1/90
2 5 1 3 6 0 4 0 2 0 1 0 1 -1/90
2 5 1 1 4 1 5 1 3 0 6 0 1 1/90
2 5 1 1 6 0 5 0 2 0 4 0 1 1/90
2 5 1 1 3 1 4 0 6 0 1 0 1 1/90
2 5 1 1 6 0 4 0 2 0 1 0 1 -1/90
2 5 1 1 4 1 5 0 6 0 1 0 1 1/18
2 5 1 1 5 0 6 0 2 0 1 0 1 -1/18
2 5 1 3 6 1 5 1 2 0 4 0 1 -1/240
2 5 1 5 6 1 4 0 2 0 3 0 1 -1/240
2 5 1 1 5 1 4 0 2 0 3 0 1 1/72
2 5 1 3 6 1 4 1 5 0 1 0 1 -1/90
2 5 1 3 6 0 4 0 5 0 1 0 1 1/90
2 5 1 1 4 1 6 1 3 0 2 0 1 1/90
2 5 1 1 3 0 5 0 6 0 4 0 1 1/90
2 5 1 4 6 1 2 0 5 0 3 0 1 -1/720
2 5 1 3 6 1 4 1 5 0 2 0 1 -1/720
2 5 1 3 6 1 4 0 5 0 2 0 1 -1/1440
2 5 1 5 6 1 4 1 2 0 3 0 1 -1/1440
2 5 1 1 4 1 5 0 6 0 2 0 3 1/480
2 5 1 1 4 1 6 1 5 0 3 0 2 -1/480
2 5 1 1 6 1 5 0 3 0 2 0 1 43/1440
2 5 1 1 5 1 4 0 6 0 3 0 1 43/1440
2 5 1 3 6 1 4 0 5 0 1 0 1 7/360
2 5 1 4 6 1 5 0 3 0 1 0 1 -7/360
2 5 1 4 6 1 2 0 5 0 1 0 1 -11/1440
2 5 1 3 6 1 5 0 2 0 1 0 1 11/1440
2 5 1 3 6 4 5 1 2 0 1 0 1 -1/288
2 5 1 3 6 4 5 0 2 0 1 0 1 1/288
2 5 1 4 6 1 2 0 3 0 1 0 1 -1/144
2 5 1 3 6 1 4 0 2 0 1 0 1 1/144
2 5 1 3 4 1 6 1 5 0 2 0 1 -1/1440
2 5 1 4 5 1 2 0 6 0 3 0 1 -1/1440
2 5 1 1 4 1 6 0 5 0 3 0 1 19/1440
2 5 1 1 5 1 2 0 3 0 6 0 1 19/1440
2 5 1 1 4 1 2 0 5 0 3 0 1 1/360
2 5 1 3 5 1 4 0 2 0 6 0 1 1/240
2 5 1 3 5 1 6 1 2 0 4 0 1 -1/240
2 5 1 3 4 1 6 0 5 0 1 0 1 23/720
2 5 1 1 4 1 2 0 6 0 3 0 5 -1/480
2 5 1 1 4 1 6 1 3 0 2 0 5 1/480
2 5 1 3 5 1 4 1 2 0 6 0 1 1/180
2 5 1 3 4 1 6 0 5 0 2 0 1 -1/180
C.1. ORIGINAL EXPANSION ⋆ 7aff mod ō(h̄ ) 521
2 5 1 1 6 1 2 0 5 0 3 0 1 1/720
2 5 1 1 3 1 4 0 5 0 6 0 1 1/720
2 5 1 1 3 0 2 0 1 0 1 0 1 -1/36
2 5 1 1 4 1 6 0 2 0 1 0 1 1/18
2 5 1 1 4 0 6 0 2 0 1 0 1 -1/18
2 5 1 3 4 1 6 1 2 0 1 0 1 -1/90
2 5 1 3 4 0 6 0 2 0 1 0 1 1/90
2 5 1 3 6 2 5 0 1 0 1 0 1 1/180
2 5 1 4 5 1 6 0 3 0 2 0 1 -1/180
2 5 1 3 4 1 2 1 5 0 6 0 1 1/180
2 5 1 3 4 1 2 0 5 0 6 0 1 1/1440
2 5 1 3 5 1 4 1 6 0 2 0 1 -1/1440
2 5 1 1 6 1 4 0 3 0 2 0 1 1/36
2 5 1 1 5 1 4 0 3 0 6 0 1 1/36
2 5 1 3 6 2 4 1 5 0 1 0 1 -1/240
2 5 1 3 4 2 5 0 6 0 1 0 1 1/240
2 5 1 3 4 1 2 0 6 0 1 0 1 1/180
2 5 1 3 4 1 6 0 2 0 1 0 1 1/180
2 5 1 1 5 1 4 0 3 0 2 0 1 1/72
h^6:
2 6 1 0 1 0 1 0 1 0 1 0 1 0 1 1/720
2 6 1 1 7 0 1 0 1 0 1 0 1 0 1 -1/72
2 6 1 0 7 0 1 0 1 0 1 0 1 0 1 1/72
2 6 1 1 3 0 7 0 1 0 1 0 1 0 1 -1/36
2 6 1 1 7 0 2 0 1 0 1 0 1 0 1 -1/36
2 6 1 1 7 1 6 0 1 0 1 0 1 0 1 1/36
2 6 1 0 7 0 6 0 1 0 1 0 1 0 1 1/36
2 6 1 1 7 0 6 0 1 0 1 0 1 0 1 -1/18
2 6 1 1 4 1 7 1 3 0 1 0 1 0 1 1/90
2 6 1 0 4 0 7 0 3 0 1 0 1 0 1 -1/90
2 6 1 1 4 1 7 0 3 0 1 0 1 0 1 1/30
2 6 1 1 4 0 2 0 7 0 1 0 1 0 1 -1/30
2 6 1 1 3 1 4 0 7 0 1 0 1 0 1 1/180
2 6 1 1 7 0 4 0 2 0 1 0 1 0 1 -1/180
2 6 1 3 7 1 6 0 1 0 1 0 1 0 1 1/36
2 6 1 3 7 0 6 0 1 0 1 0 1 0 1 1/36
2 6 1 1 3 1 7 0 2 0 1 0 1 0 1 1/180
2 6 1 1 3 0 4 0 7 0 1 0 1 0 1 -1/180
2 6 1 1 6 1 7 0 3 0 1 0 1 0 1 1/18
2 6 1 1 4 0 7 0 6 0 1 0 1 0 1 -1/18
2 6 1 1 4 1 6 0 7 0 1 0 1 0 1 1/18
2 6 1 1 7 0 2 0 6 0 1 0 1 0 1 -1/18
2 6 1 4 6 1 2 1 7 0 1 0 1 0 1 -1/90
2 6 1 4 7 0 2 0 6 0 1 0 1 0 1 1/90
2 6 1 4 7 1 2 0 6 0 1 0 1 0 1 -11/1440
2 6 1 3 7 1 6 0 2 0 1 0 1 0 1 11/1440
2 6 1 3 7 1 4 1 2 0 1 0 1 0 1 1/180
2 6 1 4 7 0 2 0 3 0 1 0 1 0 1 -1/180
2 6 1 4 7 1 2 0 3 0 1 0 1 0 1 -1/288
2 6 1 3 7 1 4 0 2 0 1 0 1 0 1 1/288
2 6 1 1 5 1 4 0 2 0 3 0 1 0 1 1/144
2 6 1 1 4 1 2 0 5 0 3 0 1 0 1 1/720
2 6 1 1 6 1 7 1 5 1 3 1 4 0 1 -2/945
2 6 1 0 6 0 7 0 5 0 3 0 4 0 1 2/945
2 6 1 1 7 1 4 1 5 1 6 0 1 0 1 -1/135
2 6 1 0 7 0 4 0 5 0 6 0 1 0 1 -1/135
2 6 1 3 7 1 4 1 6 0 1 0 1 0 1 -1/90
2 6 1 3 7 0 4 0 6 0 1 0 1 0 1 1/90
2 6 1 1 6 1 7 1 5 1 3 1 4 0 2 -1/945
2 6 1 1 3 0 7 0 2 0 6 0 4 0 5 -1/945
2 6 1 1 4 1 7 1 3 0 2 0 1 0 1 1/90
2 6 1 1 3 0 5 0 7 0 4 0 1 0 1 1/90
2 6 1 1 6 1 4 1 5 1 7 0 3 0 1 -1/210
2 6 1 1 4 0 2 0 5 0 6 0 7 0 1 1/210
2 6 1 4 7 1 6 0 3 0 1 0 1 0 1 -7/360
2 6 1 3 7 1 4 0 6 0 1 0 1 0 1 7/360
2 6 1 1 3 1 4 0 5 0 7 0 1 0 1 1/720
2 6 1 1 7 1 2 0 5 0 3 0 1 0 1 1/720
2 6 1 1 4 1 7 0 5 0 3 0 1 0 1 19/1440
2 6 1 1 5 1 2 0 3 0 7 0 1 0 1 19/1440
2 6 1 3 4 1 7 0 6 0 1 0 1 0 1 23/720
2 6 1 1 5 1 4 0 7 0 3 0 1 0 1 43/1440
2 6 1 1 7 1 5 0 3 0 2 0 1 0 1 43/1440
522 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 6 1 3 4 1 5 1 7 1 6 1 2 0 1 -1/945
2 6 1 3 4 0 5 0 7 0 6 0 2 0 1 1/945
2 6 1 1 7 1 4 1 5 1 6 0 3 0 1 -1/270
2 6 1 1 4 0 7 0 5 0 6 0 2 0 1 1/270
2 6 1 1 4 1 7 1 5 1 6 0 3 0 1 -1/210
2 6 1 1 4 0 5 0 7 0 6 0 2 0 1 1/210
2 6 1 1 4 1 5 1 3 0 7 0 1 0 1 1/90
2 6 1 1 7 0 5 0 2 0 4 0 1 0 1 1/90
2 6 1 1 4 1 5 1 3 0 2 0 1 0 1 1/180
2 6 1 1 3 0 5 0 2 0 4 0 1 0 1 1/180
2 6 1 1 7 1 6 0 3 0 2 0 1 0 1 1/72
2 6 1 1 5 1 4 0 6 0 7 0 1 0 1 1/72
2 6 1 1 5 1 6 0 3 0 7 0 1 0 1 1/36
2 6 1 5 7 1 4 1 2 0 3 0 1 0 1 -1/1440
2 6 1 3 7 1 4 0 5 0 2 0 1 0 1 -1/1440
2 6 1 5 6 1 7 1 2 0 3 0 1 0 1 15/4*zeta(3)^2/pi^6-19/3240
2 6 1 3 6 1 4 0 7 0 2 0 1 0 1 15/4*zeta(3)^2/pi^6-19/3240
2 6 1 3 7 1 4 1 5 0 2 0 1 0 1 -1/720
2 6 1 4 7 1 2 0 5 0 3 0 1 0 1 -1/720
2 6 1 1 6 1 7 1 5 0 3 0 4 0 1 27/8*zeta(3)^2/pi^6-53/3024
2 6 1 1 5 1 6 0 3 0 7 0 2 0 1 -27/8*zeta(3)^2/pi^6+53/3024
2 6 1 1 7 1 6 1 5 0 3 0 4 0 1 -1/216
2 6 1 1 6 1 5 0 7 0 2 0 3 0 1 1/216
2 6 1 5 7 4 6 1 2 1 3 0 1 0 1 17/2*zeta(3)^2/pi^6-29/2268
2 6 1 5 7 4 6 0 2 0 3 0 1 0 1 17/2*zeta(3)^2/pi^6-29/2268
2 6 1 1 4 1 7 1 5 1 6 0 3 0 2 -33/2*zeta(3)^2/pi^6+103/4536
2 6 1 1 5 1 4 0 6 0 3 0 7 0 2 -33/2*zeta(3)^2/pi^6+103/4536
2 6 1 3 7 1 4 1 6 0 2 0 1 0 1 15/4*zeta(3)^2/pi^6-103/12960
2 6 1 4 7 1 2 0 5 0 6 0 1 0 1 15/4*zeta(3)^2/pi^6-103/12960
2 6 1 3 4 1 7 1 6 1 2 0 5 0 1 4*zeta(3)^2/pi^6-43/7560
2 6 1 5 6 1 4 0 2 0 7 0 3 0 1 -4*zeta(3)^2/pi^6+43/7560
2 6 1 4 7 1 2 1 5 0 6 0 1 0 1 15/4*zeta(3)^2/pi^6-157/12960
2 6 1 5 7 1 6 0 2 0 3 0 1 0 1 15/4*zeta(3)^2/pi^6-157/12960
2 6 1 3 7 4 6 1 2 0 1 0 1 0 1 -1/288
2 6 1 3 7 4 6 0 2 0 1 0 1 0 1 1/288
2 6 1 4 7 1 2 1 5 0 3 0 1 0 1 -1/240
2 6 1 5 7 1 4 0 2 0 3 0 1 0 1 -1/240
2 6 1 4 6 5 7 1 3 0 2 0 1 0 1 49/4*zeta(3)^2/pi^6-47/2520
2 6 1 1 7 1 6 1 5 0 3 0 2 0 4 -105/16*zeta(3)^2/pi^6+731/90720
2 6 1 1 7 1 6 1 5 0 1 0 1 0 1 -1/162
2 6 1 0 7 0 6 0 5 0 1 0 1 0 1 1/162
2 6 1 1 7 1 6 0 5 0 1 0 1 0 1 1/54
2 6 1 1 7 0 6 0 5 0 1 0 1 0 1 -1/54
2 6 1 1 5 1 6 1 7 0 4 0 1 0 1 -1/45
2 6 1 1 5 0 6 0 2 0 7 0 1 0 1 -1/45
2 6 1 1 7 1 4 1 5 0 6 0 1 0 1 -1/270
2 6 1 1 6 0 7 0 5 0 2 0 1 0 1 -1/270
2 6 1 4 5 1 6 1 7 0 1 0 1 0 1 -1/54
2 6 1 4 7 0 6 0 5 0 1 0 1 0 1 1/54
2 6 1 4 6 1 5 1 7 1 2 0 1 0 1 -1/270
2 6 1 4 6 0 5 0 7 0 2 0 1 0 1 -1/270
2 6 1 3 7 1 6 1 2 1 4 1 5 0 1 -1/945
2 6 1 3 7 0 6 0 2 0 4 0 5 0 1 1/945
2 6 1 1 5 1 4 1 7 1 6 0 3 0 1 -1/945
2 6 1 1 4 0 6 0 5 0 7 0 2 0 1 1/945
2 6 1 1 7 1 4 1 6 0 3 0 1 0 1 -1/270
2 6 1 1 4 0 7 0 5 0 6 0 1 0 1 -1/270
2 6 1 4 6 1 7 1 5 1 2 0 1 0 1 -1/270
2 6 1 4 6 0 7 0 5 0 2 0 1 0 1 -1/270
2 6 1 1 5 1 6 1 3 1 4 0 7 0 1 -1/945
2 6 1 1 7 0 6 0 2 0 4 0 5 0 1 1/945
2 6 1 1 4 1 7 1 3 0 6 0 1 0 1 1/135
2 6 1 1 6 0 5 0 7 0 4 0 1 0 1 1/135
2 6 1 3 4 1 6 1 5 1 2 1 7 0 1 1/945
2 6 1 3 4 0 6 0 5 0 7 0 2 0 1 1/945
2 6 1 1 5 1 6 0 7 0 3 0 1 0 1 1/45
2 6 1 1 6 1 5 0 3 0 7 0 1 0 1 1/45
2 6 1 1 3 1 5 0 6 0 7 0 1 0 1 1/270
2 6 1 1 7 1 6 0 5 0 3 0 1 0 1 1/270
2 6 1 3 7 1 6 0 5 0 1 0 1 0 1 1/54
2 6 1 4 7 1 6 0 5 0 1 0 1 0 1 -1/54
2 6 1 1 6 1 2 0 7 0 3 0 1 0 1 1/270
2 6 1 1 7 1 4 0 5 0 6 0 1 0 1 1/270
C.1. ORIGINAL EXPANSION ⋆ mod ō(h̄7aff ) 523
2 6 1 4 6 1 5 1 7 0 2 0 1 0 1 -3*zeta(3)^2/pi^6-13/6480
2 6 1 5 6 1 2 0 3 0 7 0 1 0 1 -3*zeta(3)^2/pi^6-13/6480
2 6 1 1 4 1 6 1 5 0 3 0 2 0 1 -1/480
2 6 1 1 4 1 5 0 6 0 2 0 3 0 1 1/480
2 6 1 4 5 1 2 1 7 0 3 0 1 0 1 -1/240
2 6 1 3 5 1 4 0 2 0 7 0 1 0 1 1/240
2 6 1 4 5 1 2 1 7 0 6 0 1 0 1 -59/12960
2 6 1 3 5 1 6 0 2 0 7 0 1 0 1 59/12960
2 6 1 3 4 1 7 1 5 0 2 0 1 0 1 -1/1440
2 6 1 4 5 1 2 0 7 0 3 0 1 0 1 -1/1440
2 6 1 1 6 1 4 1 5 0 7 0 3 0 1 15/8*zeta(3)^2/pi^6-173/30240
2 6 1 1 7 1 5 0 3 0 6 0 2 0 1 -15/8*zeta(3)^2/pi^6+173/30240
2 6 1 3 7 1 5 1 2 1 6 0 4 0 1 -273/16*zeta(3)^2/pi^6+4703/181440
2 6 1 5 7 1 4 0 2 0 6 0 3 0 1 273/16*zeta(3)^2/pi^6-4703/181440
2 6 1 4 6 1 2 1 7 0 3 0 1 0 1 -3*zeta(3)^2/pi^6-11/4320
2 6 1 5 6 1 4 0 2 0 7 0 1 0 1 -3*zeta(3)^2/pi^6-11/4320
2 6 1 3 7 1 6 1 5 1 2 0 4 0 1 287/16*zeta(3)^2/pi^6-1013/36288
2 6 1 5 7 1 4 0 6 0 3 0 2 0 1 -287/16*zeta(3)^2/pi^6+1013/36288
2 6 1 3 7 4 5 1 6 0 1 0 1 0 1 19/12960
2 6 1 3 7 4 5 0 6 0 1 0 1 0 1 -19/12960
2 6 1 1 5 1 4 1 7 1 6 0 3 0 2 23/2*zeta(3)^2/pi^6-1583/90720
2 6 1 1 5 1 4 0 7 0 6 0 3 0 2 23/2*zeta(3)^2/pi^6-1583/90720
2 6 1 5 7 1 4 1 2 0 6 0 1 0 1 -3*zeta(3)^2/pi^6+11/12960
2 6 1 3 7 1 6 0 5 0 2 0 1 0 1 -3*zeta(3)^2/pi^6+11/12960
2 6 1 5 6 1 7 1 2 0 4 0 1 0 1 1/432
2 6 1 3 6 1 5 0 7 0 2 0 1 0 1 1/432
2 6 1 4 6 1 7 1 5 0 2 0 1 0 1 -1/432
2 6 1 5 6 1 2 0 7 0 3 0 1 0 1 -1/432
2 6 1 1 4 1 6 1 5 0 3 0 7 0 1 -3*zeta(3)^2/pi^6+377/90720
2 6 1 1 7 1 5 0 6 0 2 0 3 0 1 3*zeta(3)^2/pi^6-377/90720
2 6 1 3 5 1 4 1 2 0 7 0 1 0 1 1/180
2 6 1 3 4 1 7 0 5 0 2 0 1 0 1 -1/180
2 6 1 3 4 1 5 1 6 1 2 0 7 0 1 -1/1890
2 6 1 4 5 1 7 0 6 0 3 0 2 0 1 1/1890
2 6 1 3 7 1 4 1 2 0 6 0 1 0 1 1/270
2 6 1 4 7 1 6 0 5 0 2 0 1 0 1 1/270
2 6 1 3 6 1 5 1 7 1 4 0 1 0 1 -1/270
2 6 1 3 6 0 5 0 7 0 4 0 1 0 1 -1/270
2 6 1 3 4 1 7 1 5 1 6 0 1 0 1 -1/270
2 6 1 3 4 0 7 0 5 0 6 0 1 0 1 -1/270
2 6 1 1 5 1 7 1 3 1 4 0 2 0 1 -1/945
2 6 1 1 3 0 6 0 7 0 4 0 5 0 1 1/945
2 6 1 3 7 4 5 1 6 1 2 0 1 0 1 37/16*zeta(3)^2/pi^6-701/181440
2 6 1 4 5 2 6 0 7 0 3 0 1 0 1 37/16*zeta(3)^2/pi^6-701/181440
2 6 1 3 7 1 4 1 5 0 6 0 1 0 1 -9/8*zeta(3)^2/pi^6-89/12960
2 6 1 4 7 1 6 0 5 0 3 0 1 0 1 -9/8*zeta(3)^2/pi^6-89/12960
2 6 1 3 5 4 7 1 2 1 6 0 1 0 1 -1/16*zeta(3)^2/pi^6+43/181440
2 6 1 5 6 2 4 0 7 0 3 0 1 0 1 -1/16*zeta(3)^2/pi^6+43/181440
2 6 1 3 4 1 7 1 5 0 6 0 1 0 1 3/4*zeta(3)^2/pi^6-1/360
2 6 1 4 5 1 6 0 7 0 3 0 1 0 1 3/4*zeta(3)^2/pi^6-1/360
2 6 1 3 6 1 5 1 2 1 7 0 4 0 1 11/8*zeta(3)^2/pi^6-53/30240
2 6 1 3 5 1 4 0 2 0 6 0 7 0 1 11/8*zeta(3)^2/pi^6-53/30240
2 6 1 3 5 1 4 1 6 0 7 0 1 0 1 -3/4*zeta(3)^2/pi^6-13/6480
2 6 1 3 4 1 7 0 5 0 6 0 1 0 1 3/4*zeta(3)^2/pi^6+13/6480
2 6 1 3 4 1 5 1 6 1 7 0 2 0 1 -11/8*zeta(3)^2/pi^6+137/45360
2 6 1 4 5 1 2 0 6 0 3 0 7 0 1 11/8*zeta(3)^2/pi^6-137/45360
2 6 1 1 4 1 6 1 3 0 2 0 7 0 1 -3/8*zeta(3)^2/pi^6+11/2268
2 6 1 1 4 1 7 0 6 0 3 0 5 0 1 3/8*zeta(3)^2/pi^6-11/2268
2 6 1 1 5 1 7 1 3 1 4 0 2 0 6 5*zeta(3)^2/pi^6-239/30240
2 6 1 1 4 1 2 0 7 0 3 0 5 0 6 5*zeta(3)^2/pi^6-239/30240
2 6 1 1 4 1 7 1 3 0 6 0 2 0 1 3/2*zeta(3)^2/pi^6-53/45360
2 6 1 1 3 1 4 0 6 0 7 0 5 0 1 -3/2*zeta(3)^2/pi^6+53/45360
2 6 1 6 7 1 5 1 2 1 4 0 3 0 1 -7*zeta(3)^2/pi^6+1963/181440
2 6 1 3 7 1 4 0 6 0 2 0 5 0 1 7*zeta(3)^2/pi^6-1963/181440
2 6 1 5 7 1 4 1 6 0 3 0 1 0 1 -9/8*zeta(3)^2/pi^6+7/6480
2 6 1 3 7 1 4 0 5 0 6 0 1 0 1 -9/8*zeta(3)^2/pi^6+7/6480
2 6 1 3 7 1 5 1 6 1 4 0 2 0 1 13/2*zeta(3)^2/pi^6-31/2880
2 6 1 4 7 1 2 0 6 0 3 0 5 0 1 -13/2*zeta(3)^2/pi^6+31/2880
2 6 1 1 4 1 6 1 7 0 3 0 2 0 1 -27/8*zeta(3)^2/pi^6+377/90720
2 6 1 1 5 1 4 0 6 0 3 0 7 0 1 27/8*zeta(3)^2/pi^6-377/90720
2 6 1 4 5 1 6 1 7 0 3 0 1 0 1 3/4*zeta(3)^2/pi^6-91/6480
2 6 1 3 5 1 4 0 6 0 7 0 1 0 1 -3/4*zeta(3)^2/pi^6+91/6480
2 6 1 1 4 1 7 1 6 0 2 0 3 0 1 9/8*zeta(3)^2/pi^6-173/30240
524 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 6 1 1 5 1 4 0 7 0 6 0 3 0 1 -9/8*zeta(3)^2/pi^6+173/30240
2 6 1 3 6 1 7 1 5 1 2 0 4 0 1 -17/16*zeta(3)^2/pi^6+131/181440
2 6 1 3 5 1 4 0 6 0 7 0 2 0 1 -17/16*zeta(3)^2/pi^6+131/181440
2 6 1 3 4 1 5 1 7 1 6 0 2 0 1 35/16*zeta(3)^2/pi^6-37/12096
2 6 1 4 5 1 2 0 6 0 7 0 3 0 1 -35/16*zeta(3)^2/pi^6+37/12096
2 6 1 4 6 1 7 1 5 0 3 0 1 0 1 -9/8*zeta(3)^2/pi^6-13/1440
2 6 1 5 6 1 4 0 7 0 3 0 1 0 1 -9/8*zeta(3)^2/pi^6-13/1440
2 6 1 3 7 4 6 1 5 1 2 0 1 0 1 -41/4*zeta(3)^2/pi^6+163/11340
2 6 1 4 7 2 6 0 5 0 3 0 1 0 1 -41/4*zeta(3)^2/pi^6+163/11340
2 6 1 1 6 1 4 1 7 0 3 0 5 0 1 -9/8*zeta(3)^2/pi^6+11/18144
2 6 1 1 5 1 2 0 3 0 6 0 7 0 1 9/8*zeta(3)^2/pi^6-11/18144
2 6 1 1 4 1 6 1 3 0 2 0 5 0 1 1/480
2 6 1 1 4 1 2 0 6 0 3 0 5 0 1 -1/480
2 6 1 1 4 1 7 1 5 0 6 0 2 0 1 -1/1080
2 6 1 1 5 1 2 0 7 0 6 0 3 0 1 1/1080
2 6 1 1 4 1 7 1 5 0 6 0 3 0 1 -3/2*zeta(3)^2/pi^6+11/18144
2 6 1 1 5 1 2 0 6 0 7 0 3 0 1 3/2*zeta(3)^2/pi^6-11/18144
2 6 1 3 7 5 6 1 2 0 4 0 1 0 1 -115/16*zeta(3)^2/pi^6+487/45360
2 6 1 3 7 4 6 1 5 0 2 0 1 0 1 -115/16*zeta(3)^2/pi^6+487/45360
2 6 1 5 7 1 2 1 6 0 4 0 3 0 1 -379/32*zeta(3)^2/pi^6+3463/181440
2 6 1 3 7 1 6 1 5 0 2 0 4 0 1 379/32*zeta(3)^2/pi^6-3463/181440
2 6 1 3 6 1 5 1 7 1 2 0 4 0 1 -1/420
2 6 1 3 5 1 4 0 7 0 6 0 2 0 1 -1/420
2 6 1 3 6 1 5 1 2 1 4 0 7 0 1 -1/1890
2 6 1 3 4 1 7 0 6 0 2 0 5 0 1 -1/1890
2 6 1 1 4 1 6 1 3 0 7 0 2 0 1 1/270
2 6 1 1 7 1 4 0 6 0 3 0 5 0 1 -1/270
2 6 1 5 6 1 4 1 2 0 7 0 3 0 1 -59/32*zeta(3)^2/pi^6+779/362880
2 6 1 3 4 1 6 1 7 0 2 0 5 0 1 -59/32*zeta(3)^2/pi^6+779/362880
2 6 1 5 6 1 7 1 2 0 3 0 4 0 1 2*zeta(3)^2/pi^6-299/90720
2 6 1 3 4 1 5 1 6 0 7 0 2 0 1 -2*zeta(3)^2/pi^6+299/90720
2 6 1 3 7 4 5 1 2 0 6 0 1 0 1 11/32*zeta(3)^2/pi^6-811/362880
2 6 1 3 7 4 5 1 6 0 2 0 1 0 1 -11/32*zeta(3)^2/pi^6+811/362880
2 6 1 1 7 1 4 1 6 0 3 0 2 0 5 17/4*zeta(3)^2/pi^6-643/90720
2 6 1 1 7 1 4 1 5 0 6 0 2 0 3 17/4*zeta(3)^2/pi^6-643/90720
2 6 1 3 5 1 6 1 2 0 4 0 7 0 1 11/4*zeta(3)^2/pi^6-31/5040
2 6 1 3 5 1 6 1 7 0 4 0 2 0 1 11/4*zeta(3)^2/pi^6-31/5040
2 6 1 3 6 1 4 1 5 0 2 0 7 0 1 -25/32*zeta(3)^2/pi^6+13/13440
2 6 1 4 5 1 2 1 7 0 6 0 3 0 1 -25/32*zeta(3)^2/pi^6+13/13440
2 6 1 3 7 1 4 1 5 0 6 0 2 0 1 -151/32*zeta(3)^2/pi^6+191/25920
2 6 1 5 7 1 4 1 2 0 6 0 3 0 1 151/32*zeta(3)^2/pi^6-191/25920
2 6 1 3 5 4 6 1 2 0 7 0 1 0 1 -19/32*zeta(3)^2/pi^6-307/362880
2 6 1 3 4 5 6 1 7 0 2 0 1 0 1 -19/32*zeta(3)^2/pi^6-307/362880
2 6 1 1 4 1 7 1 3 0 6 0 2 0 5 -31/16*zeta(3)^2/pi^6+47/18144
2 6 1 3 5 1 4 1 2 0 6 0 7 0 1 3/4*zeta(3)^2/pi^6-53/90720
2 6 1 3 5 1 4 1 7 0 6 0 2 0 1 3/4*zeta(3)^2/pi^6-53/90720
2 6 1 4 5 1 2 1 6 0 7 0 3 0 1 -17/16*zeta(3)^2/pi^6-37/90720
2 6 1 3 5 1 7 1 6 0 4 0 2 0 1 -17/16*zeta(3)^2/pi^6-37/90720
2 6 1 6 7 1 4 1 5 0 2 0 3 0 1 -1/4*zeta(3)^2/pi^6+41/36288
2 6 1 3 7 1 6 1 2 0 4 0 5 0 1 1/4*zeta(3)^2/pi^6-41/36288
2 6 1 1 4 1 6 1 3 0 7 0 5 0 1 3/2*zeta(3)^2/pi^6-53/45360
2 6 1 1 7 1 2 0 6 0 3 0 5 0 1 -3/2*zeta(3)^2/pi^6+53/45360
2 6 1 1 3 0 2 0 1 0 1 0 1 0 1 -1/144
2 6 1 1 4 1 7 0 2 0 1 0 1 0 1 1/36
2 6 1 1 3 0 2 0 7 0 1 0 1 0 1 -1/36
2 6 1 3 4 1 7 1 2 0 1 0 1 0 1 -1/180
2 6 1 3 4 0 2 0 7 0 1 0 1 0 1 -1/180
2 6 1 3 4 1 2 0 7 0 1 0 1 0 1 1/360
2 6 1 3 4 1 7 0 2 0 1 0 1 0 1 1/360
2 6 1 3 7 2 6 0 1 0 1 0 1 0 1 1/360
2 6 1 1 5 1 4 0 3 0 7 0 1 0 1 1/36
2 6 1 1 7 1 4 0 3 0 2 0 1 0 1 1/36
2 6 1 3 4 1 5 1 2 1 6 1 7 0 1 1/945
2 6 1 3 4 0 5 0 2 0 6 0 7 0 1 -1/945
2 6 1 1 4 1 6 1 5 1 7 0 3 0 1 -1/270
2 6 1 1 4 0 5 0 2 0 6 0 7 0 1 1/270
2 6 1 3 5 1 2 1 6 1 7 0 1 0 1 -1/270
2 6 1 4 5 0 7 0 6 0 2 0 1 0 1 1/270
2 6 1 3 5 1 4 1 7 0 2 0 1 0 1 -1/1440
2 6 1 3 4 1 2 0 5 0 7 0 1 0 1 1/1440
2 6 1 3 7 2 4 1 6 0 1 0 1 0 1 -1/240
2 6 1 3 4 2 6 0 7 0 1 0 1 0 1 1/240
2 6 1 1 7 1 4 1 5 1 6 0 3 0 2 -1/540
C.1. ORIGINAL EXPANSION ⋆ mod ō(h̄7aff ) 525
2 6 1 1 4 1 5 0 6 0 3 0 7 0 2 -1/540
2 6 1 3 4 1 2 1 6 1 7 0 5 0 1 -1/420
2 6 1 5 6 1 4 0 7 0 3 0 2 0 1 -1/420
2 6 1 3 4 1 5 1 2 0 7 0 1 0 1 -1/180
2 6 1 4 5 1 7 0 3 0 2 0 1 0 1 -1/180
2 6 1 1 7 1 5 1 6 0 3 0 2 0 4 -1/432
2 6 1 1 5 1 7 1 6 0 2 0 1 0 1 -1/108
2 6 1 1 5 0 7 0 6 0 2 0 1 0 1 -1/108
2 6 1 1 6 1 5 0 7 0 3 0 1 0 1 1/54
2 6 1 1 6 1 7 1 5 0 4 0 3 0 1 -1/90
2 6 1 1 6 1 5 0 3 0 7 0 2 0 1 1/90
2 6 1 4 5 1 7 1 2 0 6 0 1 0 1 -1/540
2 6 1 3 5 1 6 0 7 0 2 0 1 0 1 1/540
2 6 1 1 6 1 4 1 5 0 7 0 2 0 1 -1/540
2 6 1 1 7 1 4 0 3 0 6 0 2 0 1 1/540
2 6 1 4 5 1 6 1 7 0 2 0 1 0 1 -1/540
2 6 1 3 5 1 2 0 6 0 7 0 1 0 1 1/540
2 6 1 4 6 1 5 1 7 0 3 0 1 0 1 -1/108
2 6 1 5 6 1 4 0 3 0 7 0 1 0 1 -1/108
2 6 1 3 5 4 6 1 7 1 2 0 1 0 1 1/540
2 6 1 5 6 2 4 0 3 0 7 0 1 0 1 1/540
2 6 1 3 7 2 6 1 5 0 1 0 1 0 1 -1/540
2 6 1 3 7 2 6 0 5 0 1 0 1 0 1 1/540
2 6 1 3 6 1 2 1 5 1 7 0 4 0 1 -1/1890
2 6 1 3 5 1 4 0 6 0 2 0 7 0 1 -1/1890
2 6 1 1 6 1 4 1 7 0 3 0 2 0 1 -1/540
2 6 1 1 5 1 4 0 3 0 6 0 7 0 1 1/540
2 6 1 3 7 1 5 1 6 1 2 0 4 0 1 -1/540
2 6 1 5 7 1 4 0 3 0 6 0 2 0 1 1/540
2 6 1 3 4 1 5 1 2 1 6 0 7 0 1 1/1890
2 6 1 4 5 1 7 0 6 0 2 0 3 0 1 -1/1890
2 6 1 3 4 1 7 1 2 0 6 0 1 0 1 -1/270
2 6 1 4 5 1 6 0 7 0 2 0 1 0 1 -1/270
2 6 1 3 7 2 5 1 6 1 4 0 1 0 1 -3/8*zeta(3)^2/pi^6+17/90720
2 6 1 3 7 2 4 0 5 0 6 0 1 0 1 -3/8*zeta(3)^2/pi^6+17/90720
2 6 1 3 6 1 5 1 7 1 4 0 2 0 1 -3/4*zeta(3)^2/pi^6+53/90720
2 6 1 3 4 1 2 0 6 0 7 0 5 0 1 -3/4*zeta(3)^2/pi^6+53/90720
2 6 1 3 5 2 4 1 6 1 7 0 1 0 1 -9/8*zeta(3)^2/pi^6+1/360
2 6 1 3 5 2 4 0 6 0 7 0 1 0 1 -9/8*zeta(3)^2/pi^6+1/360
2 6 1 3 5 2 4 1 7 0 6 0 1 0 1 -15/8*zeta(3)^2/pi^6+13/6480
2 6 1 3 6 1 4 1 5 0 7 0 2 0 1 -3/4*zeta(3)^2/pi^6+11/36288
2 6 1 4 5 1 7 1 2 0 6 0 3 0 1 -3/4*zeta(3)^2/pi^6+11/36288
2 6 1 1 7 1 4 1 5 0 6 0 3 0 2 -1/2160
2 6 1 3 4 5 6 1 2 0 7 0 1 0 1 3/2*zeta(3)^2/pi^6-11/25920
2 6 1 3 5 4 6 1 7 0 2 0 1 0 1 3/2*zeta(3)^2/pi^6-11/25920
2 6 1 5 7 1 6 1 2 0 4 0 3 0 1 1/864
2 6 1 4 7 1 6 1 5 0 2 0 3 0 1 -1/864
2 6 1 3 7 2 4 1 5 0 6 0 1 0 1 -3/8*zeta(3)^2/pi^6-11/5040
2 6 1 3 7 2 5 1 6 0 4 0 1 0 1 -3/8*zeta(3)^2/pi^6-11/5040
2 6 1 4 5 1 7 1 6 0 2 0 3 0 1 -3/2*zeta(3)^2/pi^6+377/181440
2 6 1 3 5 1 2 1 6 0 4 0 7 0 1 -3/2*zeta(3)^2/pi^6+377/181440
2 6 1 5 6 1 4 1 7 0 2 0 3 0 1 -3/4*zeta(3)^2/pi^6+53/90720
2 6 1 3 4 1 6 1 2 0 7 0 5 0 1 -3/4*zeta(3)^2/pi^6+53/90720
2 6 1 1 5 1 4 0 3 0 2 0 1 0 1 1/144
2 6 1 3 4 1 7 1 2 1 6 0 5 0 1 1/540
2 6 1 5 6 1 4 0 3 0 7 0 2 0 1 -1/540
2 6 1 1 7 1 6 1 5 0 4 0 3 0 1 -1/216
2 6 1 1 6 1 5 0 7 0 3 0 2 0 1 1/216
2 6 1 3 7 2 6 1 5 0 4 0 1 0 1 -1/1080
2 6 1 4 5 1 6 1 7 0 2 0 3 0 1 -1/1080
2 6 1 4 5 1 6 1 2 0 7 0 3 0 1 -1/1080
2 6 1 1 7 1 6 1 5 0 4 0 3 0 2 -1/1296
h^7:
2 7 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1/5040
2 7 1 1 8 0 1 0 1 0 1 0 1 0 1 0 1 -1/360
2 7 1 0 8 0 1 0 1 0 1 0 1 0 1 0 1 1/360
2 7 1 1 3 0 8 0 1 0 1 0 1 0 1 0 1 -1/144
2 7 1 1 8 0 2 0 1 0 1 0 1 0 1 0 1 -1/144
2 7 1 1 8 1 7 0 1 0 1 0 1 0 1 0 1 1/108
2 7 1 0 8 0 7 0 1 0 1 0 1 0 1 0 1 1/108
2 7 1 1 8 0 7 0 1 0 1 0 1 0 1 0 1 -1/54
2 7 1 1 3 1 4 1 8 0 1 0 1 0 1 0 1 1/270
2 7 1 0 3 0 4 0 8 0 1 0 1 0 1 0 1 -1/270
526 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 7 1 1 4 1 8 0 3 0 1 0 1 0 1 0 1 1/90
2 7 1 1 4 0 2 0 8 0 1 0 1 0 1 0 1 -1/90
2 7 1 1 3 1 4 0 8 0 1 0 1 0 1 0 1 1/540
2 7 1 1 8 0 4 0 2 0 1 0 1 0 1 0 1 -1/540
2 7 1 3 8 1 7 0 1 0 1 0 1 0 1 0 1 1/108
2 7 1 3 8 0 7 0 1 0 1 0 1 0 1 0 1 1/108
2 7 1 1 3 1 8 0 2 0 1 0 1 0 1 0 1 1/540
2 7 1 1 3 0 4 0 8 0 1 0 1 0 1 0 1 -1/540
2 7 1 1 7 1 8 0 3 0 1 0 1 0 1 0 1 1/36
2 7 1 1 4 0 8 0 7 0 1 0 1 0 1 0 1 -1/36
2 7 1 1 4 1 7 0 8 0 1 0 1 0 1 0 1 1/36
2 7 1 1 8 0 2 0 7 0 1 0 1 0 1 0 1 -1/36
2 7 1 4 8 1 2 1 7 0 1 0 1 0 1 0 1 -1/180
2 7 1 4 8 0 2 0 7 0 1 0 1 0 1 0 1 1/180
2 7 1 4 8 1 2 0 7 0 1 0 1 0 1 0 1 -11/2880
2 7 1 3 8 1 7 0 2 0 1 0 1 0 1 0 1 11/2880
2 7 1 3 8 1 4 1 2 0 1 0 1 0 1 0 1 1/540
2 7 1 4 8 0 2 0 3 0 1 0 1 0 1 0 1 -1/540
2 7 1 4 8 1 2 0 3 0 1 0 1 0 1 0 1 -1/864
2 7 1 3 8 1 4 0 2 0 1 0 1 0 1 0 1 1/864
2 7 1 1 4 1 5 1 3 0 2 0 1 0 1 0 1 1/540
2 7 1 1 3 0 5 0 2 0 4 0 1 0 1 0 1 1/540
2 7 1 1 5 1 4 0 2 0 3 0 1 0 1 0 1 1/432
2 7 1 1 3 1 4 0 5 0 2 0 1 0 1 0 1 1/2160
2 7 1 3 8 1 4 1 7 0 1 0 1 0 1 0 1 -1/180
2 7 1 3 8 0 4 0 7 0 1 0 1 0 1 0 1 1/180
2 7 1 1 3 1 4 1 8 0 2 0 1 0 1 0 1 1/180
2 7 1 1 3 0 5 0 8 0 4 0 1 0 1 0 1 1/180
2 7 1 1 4 1 5 1 3 0 8 0 1 0 1 0 1 1/180
2 7 1 1 8 0 5 0 2 0 4 0 1 0 1 0 1 1/180
2 7 1 4 8 1 7 0 3 0 1 0 1 0 1 0 1 -7/720
2 7 1 3 8 1 4 0 7 0 1 0 1 0 1 0 1 7/720
2 7 1 1 3 1 4 0 5 0 8 0 1 0 1 0 1 1/1440
2 7 1 1 3 1 8 0 5 0 2 0 1 0 1 0 1 1/1440
2 7 1 1 5 1 8 0 3 0 4 0 1 0 1 0 1 19/2880
2 7 1 1 3 1 5 0 2 0 8 0 1 0 1 0 1 19/2880
2 7 1 3 4 1 8 0 7 0 1 0 1 0 1 0 1 23/1440
2 7 1 1 4 1 5 0 3 0 8 0 1 0 1 0 1 43/2880
2 7 1 1 5 1 8 0 2 0 3 0 1 0 1 0 1 43/2880
2 7 1 1 5 1 8 0 3 0 7 0 1 0 1 0 1 1/36
2 7 1 1 5 1 4 0 7 0 8 0 1 0 1 0 1 1/72
2 7 1 1 8 1 7 0 3 0 2 0 1 0 1 0 1 1/72
2 7 1 5 8 1 4 1 2 0 3 0 1 0 1 0 1 -1/2880
2 7 1 3 8 1 4 0 5 0 2 0 1 0 1 0 1 -1/2880
2 7 1 3 8 1 4 1 5 0 2 0 1 0 1 0 1 -1/1440
2 7 1 5 8 1 2 0 3 0 4 0 1 0 1 0 1 -1/1440
2 7 1 4 8 1 2 1 5 0 3 0 1 0 1 0 1 -1/480
2 7 1 5 8 1 4 0 2 0 3 0 1 0 1 0 1 -1/480
2 7 1 5 8 1 2 1 7 0 4 0 1 0 1 0 1 15/4*zeta(3)^2/pi^6-19/3240
2 7 1 3 8 1 5 0 2 0 7 0 1 0 1 0 1 15/4*zeta(3)^2/pi^6-19/3240
2 7 1 3 8 1 4 1 7 0 2 0 1 0 1 0 1 15/4*zeta(3)^2/pi^6-103/12960
2 7 1 4 8 1 2 0 5 0 7 0 1 0 1 0 1 15/4*zeta(3)^2/pi^6-103/12960
2 7 1 4 8 1 2 1 5 0 7 0 1 0 1 0 1 15/4*zeta(3)^2/pi^6-157/12960
2 7 1 5 8 1 7 0 2 0 3 0 1 0 1 0 1 15/4*zeta(3)^2/pi^6-157/12960
2 7 1 3 8 4 7 1 2 0 1 0 1 0 1 0 1 -1/576
2 7 1 3 8 4 7 0 2 0 1 0 1 0 1 0 1 1/576
2 7 1 4 7 5 8 1 3 0 2 0 1 0 1 0 1 49/4*zeta(3)^2/pi^6-47/2520
2 7 1 5 8 4 7 1 2 1 3 0 1 0 1 0 1 17/2*zeta(3)^2/pi^6-29/2268
2 7 1 5 8 4 7 0 2 0 3 0 1 0 1 0 1 17/2*zeta(3)^2/pi^6-29/2268
2 7 1 1 8 1 7 1 6 0 1 0 1 0 1 0 1 -1/162
2 7 1 0 8 0 7 0 6 0 1 0 1 0 1 0 1 1/162
2 7 1 1 8 1 7 0 6 0 1 0 1 0 1 0 1 1/54
2 7 1 1 8 0 7 0 6 0 1 0 1 0 1 0 1 -1/54
2 7 1 1 8 1 4 1 5 1 7 0 1 0 1 0 1 -1/135
2 7 1 0 8 0 4 0 5 0 7 0 1 0 1 0 1 -1/135
2 7 1 1 5 1 7 1 8 0 4 0 1 0 1 0 1 -1/45
2 7 1 1 5 0 7 0 2 0 8 0 1 0 1 0 1 -1/45
2 7 1 1 3 1 5 1 7 0 8 0 1 0 1 0 1 -1/270
2 7 1 1 7 0 8 0 5 0 2 0 1 0 1 0 1 -1/270
2 7 1 4 6 1 7 1 8 0 1 0 1 0 1 0 1 -1/54
2 7 1 4 8 0 7 0 6 0 1 0 1 0 1 0 1 1/54
2 7 1 1 4 1 7 1 8 0 2 0 1 0 1 0 1 -1/270
2 7 1 1 4 0 8 0 5 0 7 0 1 0 1 0 1 -1/270
C.1. ORIGINAL EXPANSION ⋆aff mod ō(h̄7) 527
2 7 1 1 3 1 4 1 7 0 8 0 1 0 1 0 1 1/135
2 7 1 1 7 0 5 0 8 0 4 0 1 0 1 0 1 1/135
2 7 1 4 8 1 7 0 6 0 1 0 1 0 1 0 1 -1/54
2 7 1 3 8 1 7 0 6 0 1 0 1 0 1 0 1 1/54
2 7 1 1 7 1 8 0 5 0 3 0 1 0 1 0 1 1/270
2 7 1 1 3 1 5 0 7 0 8 0 1 0 1 0 1 1/270
2 7 1 1 5 1 8 0 2 0 7 0 1 0 1 0 1 1/45
2 7 1 1 4 1 8 0 3 0 7 0 1 0 1 0 1 1/45
2 7 1 1 4 1 8 0 5 0 7 0 1 0 1 0 1 1/270
2 7 1 1 3 1 8 0 2 0 7 0 1 0 1 0 1 1/270
2 7 1 1 7 1 6 1 8 0 4 0 1 0 1 0 1 -1/108
2 7 1 1 5 0 7 0 6 0 8 0 1 0 1 0 1 -1/108
2 7 1 1 5 1 8 1 7 0 6 0 1 0 1 0 1 -1/108
2 7 1 1 6 0 8 0 7 0 2 0 1 0 1 0 1 -1/108
2 7 1 1 5 1 7 0 8 0 6 0 1 0 1 0 1 1/54
2 7 1 1 7 1 8 0 3 0 6 0 1 0 1 0 1 1/54
2 7 1 5 8 1 4 1 2 0 7 0 1 0 1 0 1 -3*zeta(3)^2/pi^6+11/12960
2 7 1 3 8 1 7 0 5 0 2 0 1 0 1 0 1 -3*zeta(3)^2/pi^6+11/12960
2 7 1 4 8 1 5 1 7 0 2 0 1 0 1 0 1 -3*zeta(3)^2/pi^6-13/6480
2 7 1 5 8 1 2 0 3 0 7 0 1 0 1 0 1 -3*zeta(3)^2/pi^6-13/6480
2 7 1 4 5 1 2 1 7 0 8 0 1 0 1 0 1 -59/12960
2 7 1 3 5 1 8 0 2 0 7 0 1 0 1 0 1 59/12960
2 7 1 4 8 1 2 1 7 0 3 0 1 0 1 0 1 -3*zeta(3)^2/pi^6-11/4320
2 7 1 5 8 1 4 0 2 0 7 0 1 0 1 0 1 -3*zeta(3)^2/pi^6-11/4320
2 7 1 3 8 4 6 1 7 0 1 0 1 0 1 0 1 19/12960
2 7 1 3 8 4 6 0 7 0 1 0 1 0 1 0 1 -19/12960
2 7 1 5 6 1 2 1 7 1 8 0 1 0 1 0 1 1/270
2 7 1 5 6 0 7 0 2 0 8 0 1 0 1 0 1 1/270
2 7 1 4 7 1 5 1 8 1 2 0 1 0 1 0 1 -1/270
2 7 1 4 7 0 5 0 8 0 2 0 1 0 1 0 1 -1/270
2 7 1 4 7 1 8 1 5 1 2 0 1 0 1 0 1 -1/270
2 7 1 4 7 0 8 0 5 0 2 0 1 0 1 0 1 -1/270
2 7 1 5 8 1 2 1 6 0 7 0 1 0 1 0 1 11/4320
2 7 1 3 8 1 7 0 2 0 6 0 1 0 1 0 1 11/4320
2 7 1 4 6 1 7 1 8 0 2 0 1 0 1 0 1 -11/4320
2 7 1 5 8 1 2 0 7 0 6 0 1 0 1 0 1 -11/4320
2 7 1 4 8 1 2 1 6 0 7 0 1 0 1 0 1 -1/270
2 7 1 5 8 1 7 0 2 0 6 0 1 0 1 0 1 -1/270
2 7 1 4 5 1 2 1 8 0 3 0 1 0 1 0 1 -1/480
2 7 1 3 5 1 4 0 2 0 8 0 1 0 1 0 1 1/480
2 7 1 3 4 1 5 1 8 0 2 0 1 0 1 0 1 1/2880
2 7 1 4 5 1 2 0 3 0 8 0 1 0 1 0 1 1/2880
2 7 1 3 5 1 4 1 2 0 8 0 1 0 1 0 1 1/360
2 7 1 3 5 1 8 0 2 0 4 0 1 0 1 0 1 -1/360
2 7 1 5 8 1 2 1 7 0 3 0 1 0 1 0 1 1/432
2 7 1 3 8 1 4 0 2 0 7 0 1 0 1 0 1 1/432
2 7 1 3 8 1 5 1 7 0 2 0 1 0 1 0 1 -1/432
2 7 1 4 8 1 2 0 3 0 7 0 1 0 1 0 1 -1/432
2 7 1 3 8 1 4 1 2 0 7 0 1 0 1 0 1 1/270
2 7 1 5 8 1 7 0 2 0 4 0 1 0 1 0 1 1/270
2 7 1 1 6 1 8 1 5 1 3 1 4 0 1 0 1 -2/945
2 7 1 0 6 0 8 0 5 0 3 0 4 0 1 0 1 2/945
2 7 1 1 6 1 4 1 5 1 8 0 3 0 1 0 1 -1/210
2 7 1 1 4 0 2 0 5 0 6 0 8 0 1 0 1 1/210
2 7 1 3 7 1 5 1 8 1 4 0 1 0 1 0 1 -1/270
2 7 1 3 7 0 5 0 8 0 4 0 1 0 1 0 1 -1/270
2 7 1 4 8 1 2 1 5 1 6 1 7 0 1 0 1 1/945
2 7 1 4 8 0 2 0 5 0 6 0 7 0 1 0 1 -1/945
2 7 1 3 4 1 8 1 5 1 7 0 1 0 1 0 1 -1/270
2 7 1 3 4 0 8 0 5 0 7 0 1 0 1 0 1 -1/270
2 7 1 1 5 1 8 1 3 1 4 0 2 0 1 0 1 -1/945
2 7 1 1 3 0 6 0 8 0 4 0 5 0 1 0 1 1/945
2 7 1 1 5 1 8 1 6 0 4 0 3 0 1 0 1 27/8*zeta(3)^2/pi^6-53/3024
2 7 1 1 5 1 6 0 3 0 8 0 2 0 1 0 1 -27/8*zeta(3)^2/pi^6+53/3024
2 7 1 1 3 1 5 1 6 0 4 0 8 0 1 0 1 -3*zeta(3)^2/pi^6+377/90720
2 7 1 1 8 1 5 0 6 0 2 0 3 0 1 0 1 3*zeta(3)^2/pi^6-377/90720
2 7 1 3 8 1 4 1 5 0 7 0 1 0 1 0 1 -9/8*zeta(3)^2/pi^6-89/12960
2 7 1 5 8 1 7 0 3 0 4 0 1 0 1 0 1 -9/8*zeta(3)^2/pi^6-89/12960
2 7 1 3 5 1 4 1 7 0 8 0 1 0 1 0 1 -3/4*zeta(3)^2/pi^6-13/6480
2 7 1 3 4 1 8 0 5 0 7 0 1 0 1 0 1 3/4*zeta(3)^2/pi^6+13/6480
2 7 1 1 3 1 8 1 2 0 6 0 4 0 1 0 1 3/2*zeta(3)^2/pi^6-53/45360
2 7 1 1 3 1 4 0 6 0 8 0 5 0 1 0 1 -3/2*zeta(3)^2/pi^6+53/45360
2 7 1 1 3 1 5 1 2 0 8 0 4 0 1 0 1 -3/8*zeta(3)^2/pi^6+11/2268
528 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 7 1 1 4 1 8 0 6 0 3 0 5 0 1 0 1 3/8*zeta(3)^2/pi^6-11/2268
2 7 1 3 4 1 5 1 7 0 8 0 1 0 1 0 1 -3/4*zeta(3)^2/pi^6+1/360
2 7 1 4 5 1 8 0 3 0 7 0 1 0 1 0 1 -3/4*zeta(3)^2/pi^6+1/360
2 7 1 5 8 1 4 1 7 0 3 0 1 0 1 0 1 -9/8*zeta(3)^2/pi^6+7/6480
2 7 1 3 8 1 4 0 5 0 7 0 1 0 1 0 1 -9/8*zeta(3)^2/pi^6+7/6480
2 7 1 4 5 1 7 1 8 0 3 0 1 0 1 0 1 3/4*zeta(3)^2/pi^6-91/6480
2 7 1 3 5 1 4 0 8 0 7 0 1 0 1 0 1 -3/4*zeta(3)^2/pi^6+91/6480
2 7 1 1 6 1 8 1 2 0 4 0 3 0 1 0 1 9/8*zeta(3)^2/pi^6-173/30240
2 7 1 1 5 1 4 0 8 0 6 0 3 0 1 0 1 -9/8*zeta(3)^2/pi^6+173/30240
2 7 1 1 8 1 2 1 6 0 4 0 3 0 1 0 1 -27/8*zeta(3)^2/pi^6+377/90720
2 7 1 1 5 1 4 0 6 0 3 0 8 0 1 0 1 27/8*zeta(3)^2/pi^6-377/90720
2 7 1 3 8 1 5 1 7 0 4 0 1 0 1 0 1 -9/8*zeta(3)^2/pi^6-13/1440
2 7 1 4 8 1 5 0 3 0 7 0 1 0 1 0 1 -9/8*zeta(3)^2/pi^6-13/1440
2 7 1 1 8 1 5 1 2 0 6 0 4 0 1 0 1 -9/8*zeta(3)^2/pi^6+11/18144
2 7 1 1 5 1 2 0 3 0 6 0 8 0 1 0 1 9/8*zeta(3)^2/pi^6-11/18144
2 7 1 1 3 1 5 1 8 0 6 0 4 0 1 0 1 -3/2*zeta(3)^2/pi^6+11/18144
2 7 1 1 5 1 2 0 6 0 8 0 3 0 1 0 1 3/2*zeta(3)^2/pi^6-11/18144
2 7 1 1 3 1 4 1 5 0 6 0 8 0 1 0 1 3/2*zeta(3)^2/pi^6-53/45360
2 7 1 1 8 1 2 0 6 0 3 0 5 0 1 0 1 -3/2*zeta(3)^2/pi^6+53/45360
2 7 1 1 6 1 5 1 3 0 8 0 4 0 1 0 1 15/8*zeta(3)^2/pi^6-173/30240
2 7 1 1 8 1 5 0 3 0 6 0 2 0 1 0 1 -15/8*zeta(3)^2/pi^6+173/30240
2 7 1 1 4 1 8 1 5 1 6 0 3 0 1 0 1 -1/210
2 7 1 1 4 0 5 0 8 0 6 0 2 0 1 0 1 1/210
2 7 1 1 8 1 4 1 5 1 6 0 3 0 1 0 1 -1/270
2 7 1 1 4 0 8 0 5 0 6 0 2 0 1 0 1 1/270
2 7 1 1 4 1 8 1 5 1 7 0 3 0 1 0 1 -1/270
2 7 1 1 4 0 5 0 8 0 6 0 7 0 1 0 1 1/270
2 7 1 3 4 1 8 1 5 1 6 1 2 0 1 0 1 1/945
2 7 1 3 4 0 8 0 5 0 6 0 2 0 1 0 1 -1/945
2 7 1 3 8 1 6 1 2 1 4 1 5 0 1 0 1 -1/945
2 7 1 3 8 0 6 0 2 0 4 0 5 0 1 0 1 1/945
2 7 1 4 7 1 5 1 8 1 6 1 2 0 1 0 1 1/945
2 7 1 4 7 0 5 0 8 0 6 0 2 0 1 0 1 -1/945
2 7 1 1 5 1 6 1 3 1 4 0 8 0 1 0 1 -1/945
2 7 1 1 8 0 6 0 2 0 4 0 5 0 1 0 1 1/945
2 7 1 1 4 1 6 1 5 1 7 0 8 0 1 0 1 -1/270
2 7 1 1 8 0 5 0 2 0 6 0 7 0 1 0 1 1/270
2 7 1 1 6 1 7 1 8 0 3 0 4 0 1 0 1 -1/90
2 7 1 1 6 1 5 0 3 0 7 0 8 0 1 0 1 1/90
2 7 1 1 3 1 5 1 7 0 8 0 4 0 1 0 1 -1/540
2 7 1 1 5 1 7 0 6 0 8 0 3 0 1 0 1 1/540
2 7 1 4 6 1 7 1 8 0 3 0 1 0 1 0 1 -1/108
2 7 1 5 8 1 4 0 7 0 6 0 1 0 1 0 1 -1/108
2 7 1 1 7 1 8 1 2 0 3 0 4 0 1 0 1 -1/540
2 7 1 1 4 1 5 0 8 0 6 0 7 0 1 0 1 1/540
2 7 1 1 5 1 6 1 7 0 8 0 4 0 1 0 1 -1/90
2 7 1 1 7 1 5 0 3 0 8 0 2 0 1 0 1 1/90
2 7 1 1 4 1 5 1 6 0 7 0 8 0 1 0 1 -1/540
2 7 1 1 7 1 8 0 6 0 2 0 3 0 1 0 1 1/540
2 7 1 4 8 1 5 1 6 0 7 0 1 0 1 0 1 -1/108
2 7 1 5 8 1 7 0 3 0 6 0 1 0 1 0 1 -1/108
2 7 1 1 7 1 5 1 2 0 8 0 4 0 1 0 1 -1/540
2 7 1 1 5 1 8 0 3 0 6 0 7 0 1 0 1 1/540
2 7 1 1 5 1 2 1 6 0 4 0 3 0 1 0 1 -1/960
2 7 1 1 4 1 5 0 6 0 2 0 3 0 1 0 1 1/960
2 7 1 1 3 1 5 1 2 0 6 0 4 0 1 0 1 1/960
2 7 1 1 4 1 2 0 6 0 3 0 5 0 1 0 1 -1/960
2 7 1 3 8 4 7 1 5 1 2 0 1 0 1 0 1 -41/4*zeta(3)^2/pi^6+163/11340
2 7 1 3 8 4 7 0 5 0 2 0 1 0 1 0 1 -41/4*zeta(3)^2/pi^6+163/11340
2 7 1 5 7 1 8 1 2 1 6 0 3 0 1 0 1 3/16*zeta(3)^2/pi^6+943/362880
2 7 1 6 7 1 4 0 8 0 2 0 3 0 1 0 1 -3/16*zeta(3)^2/pi^6-943/362880
2 7 1 3 8 4 7 1 5 0 2 0 1 0 1 0 1 -115/16*zeta(3)^2/pi^6+487/45360
2 7 1 3 8 5 7 1 2 0 4 0 1 0 1 0 1 -115/16*zeta(3)^2/pi^6+487/45360
2 7 1 6 8 1 5 1 2 1 4 0 3 0 1 0 1 -7*zeta(3)^2/pi^6+1963/181440
2 7 1 3 8 1 4 0 6 0 2 0 5 0 1 0 1 7*zeta(3)^2/pi^6-1963/181440
2 7 1 6 8 1 5 1 2 1 7 0 3 0 1 0 1 1/16*zeta(3)^2/pi^6+257/362880
2 7 1 3 8 1 4 0 6 0 2 0 7 0 1 0 1 -1/16*zeta(3)^2/pi^6-257/362880
2 7 1 3 8 1 5 1 6 1 4 0 2 0 1 0 1 13/2*zeta(3)^2/pi^6-31/2880
2 7 1 4 8 1 2 0 6 0 3 0 5 0 1 0 1 -13/2*zeta(3)^2/pi^6+31/2880
2 7 1 1 7 1 5 1 6 1 8 0 3 0 4 0 1 65/16*zeta(3)^2/pi^6-437/72576
2 7 1 1 5 1 6 0 3 0 7 0 2 0 8 0 1 65/16*zeta(3)^2/pi^6-437/72576
2 7 1 6 7 4 5 1 8 1 2 1 3 0 1 0 1 149/16*zeta(3)^2/pi^6-5239/362880
2 7 1 6 7 4 5 0 8 0 2 0 3 0 1 0 1 -149/16*zeta(3)^2/pi^6+5239/362880
C.1. ORIGINAL EXPANSION ⋆aff mod ō(h̄7) 529
2 7 1 3 4 1 6 1 7 1 2 1 8 0 5 0 1 -99/16*zeta(3)^2/pi^6+53/5760
2 7 1 5 6 1 4 0 2 0 7 0 3 0 8 0 1 -99/16*zeta(3)^2/pi^6+53/5760
2 7 1 1 7 1 6 1 8 1 4 1 5 0 3 0 2 27/16*zeta(3)^2/pi^6-289/120960
2 7 1 1 4 1 5 0 3 0 8 0 2 0 6 0 7 -27/16*zeta(3)^2/pi^6+289/120960
2 7 1 3 7 1 5 1 8 1 4 0 2 0 1 0 1 -31/16*zeta(3)^2/pi^6+131/51840
2 7 1 4 7 1 2 0 6 0 8 0 5 0 1 0 1 31/16*zeta(3)^2/pi^6-131/51840
2 7 1 3 7 1 4 1 5 0 8 0 2 0 1 0 1 -7/8*zeta(3)^2/pi^6-11/45360
2 7 1 5 8 1 2 1 7 0 6 0 4 0 1 0 1 7/8*zeta(3)^2/pi^6+11/45360
2 7 1 6 7 1 8 1 3 0 2 0 4 0 1 0 1 3/16*zeta(3)^2/pi^6+173/362880
2 7 1 4 8 1 2 1 5 0 6 0 7 0 1 0 1 -3/16*zeta(3)^2/pi^6-173/362880
2 7 1 3 8 1 6 1 7 0 2 0 4 0 1 0 1 -2*zeta(3)^2/pi^6+17/6048
2 7 1 6 7 1 2 1 5 0 8 0 4 0 1 0 1 2*zeta(3)^2/pi^6-17/6048
2 7 1 5 8 1 2 1 6 0 4 0 3 0 1 0 1 -379/32*zeta(3)^2/pi^6+3463/181440
2 7 1 3 8 1 6 1 5 0 2 0 4 0 1 0 1 379/32*zeta(3)^2/pi^6-3463/181440
2 7 1 3 8 1 4 1 5 0 6 0 2 0 1 0 1 -151/32*zeta(3)^2/pi^6+191/25920
2 7 1 5 8 1 2 1 3 0 6 0 4 0 1 0 1 151/32*zeta(3)^2/pi^6-191/25920
2 7 1 5 8 1 6 1 3 0 4 0 2 0 1 0 1 -1/4*zeta(3)^2/pi^6+41/36288
2 7 1 3 8 1 5 1 2 0 6 0 4 0 1 0 1 1/4*zeta(3)^2/pi^6-41/36288
2 7 1 1 8 1 6 1 5 0 3 0 4 0 1 0 1 -1/216
2 7 1 1 6 1 5 0 8 0 2 0 3 0 1 0 1 1/216
2 7 1 1 3 1 6 1 2 0 8 0 4 0 1 0 1 1/270
2 7 1 1 8 1 4 0 6 0 3 0 5 0 1 0 1 -1/270
2 7 1 1 8 1 5 1 3 0 6 0 4 0 1 0 1 -1/1080
2 7 1 1 5 1 2 0 8 0 6 0 3 0 1 0 1 1/1080
2 7 1 4 7 1 8 1 5 1 6 0 1 0 1 0 1 1/270
2 7 1 4 7 0 8 0 5 0 6 0 1 0 1 0 1 1/270
2 7 1 4 8 1 5 1 6 1 2 1 7 0 1 0 1 1/945
2 7 1 4 8 0 6 0 5 0 7 0 2 0 1 0 1 -1/945
2 7 1 1 8 1 4 1 5 1 7 0 3 0 1 0 1 -1/270
2 7 1 1 4 0 8 0 5 0 6 0 7 0 1 0 1 1/270
2 7 1 1 6 1 7 1 5 1 3 1 4 0 2 0 1 -1/945
2 7 1 1 3 0 7 0 2 0 6 0 4 0 5 0 1 -1/945
2 7 1 3 4 1 6 1 5 1 2 1 8 0 1 0 1 1/945
2 7 1 3 4 0 6 0 5 0 8 0 2 0 1 0 1 1/945
2 7 1 1 6 1 7 1 5 1 3 1 4 0 8 0 1 -1/945
2 7 1 1 8 0 7 0 2 0 6 0 4 0 5 0 1 -1/945
2 7 1 1 8 1 4 1 5 1 6 0 7 0 1 0 1 -1/270
2 7 1 1 7 0 8 0 5 0 6 0 2 0 1 0 1 1/270
2 7 1 1 4 1 5 1 6 1 8 0 3 0 1 0 1 -1/945
2 7 1 1 4 0 6 0 5 0 8 0 2 0 1 0 1 1/945
2 7 1 5 6 1 7 1 8 0 4 0 1 0 1 0 1 7/1080
2 7 1 3 8 1 5 0 7 0 6 0 1 0 1 0 1 7/1080
2 7 1 1 7 1 8 1 6 0 4 0 3 0 1 0 1 -43/4320
2 7 1 1 6 1 5 0 7 0 8 0 3 0 1 0 1 43/4320
2 7 1 3 8 1 4 1 6 0 7 0 1 0 1 0 1 -1/270
2 7 1 4 8 1 7 0 5 0 6 0 1 0 1 0 1 -1/270
2 7 1 1 3 1 7 1 2 0 8 0 4 0 1 0 1 1/270
2 7 1 1 4 1 7 0 6 0 8 0 5 0 1 0 1 -1/270
2 7 1 1 8 1 4 1 5 0 6 0 7 0 1 0 1 -1/2160
2 7 1 1 7 1 2 0 8 0 6 0 3 0 1 0 1 1/2160
2 7 1 1 8 1 5 1 7 0 6 0 4 0 1 0 1 -19/4320
2 7 1 1 6 1 2 0 3 0 7 0 8 0 1 0 1 19/4320
2 7 1 3 8 1 5 1 6 0 7 0 1 0 1 0 1 -7/1080
2 7 1 4 8 1 7 0 3 0 6 0 1 0 1 0 1 -7/1080
2 7 1 1 7 1 5 1 3 0 8 0 4 0 1 0 1 -19/4320
2 7 1 1 5 1 7 0 8 0 6 0 3 0 1 0 1 19/4320
2 7 1 1 7 1 8 1 3 0 6 0 4 0 1 0 1 -1/2160
2 7 1 1 3 1 5 0 8 0 6 0 7 0 1 0 1 1/2160
2 7 1 4 5 1 7 1 8 0 6 0 1 0 1 0 1 -23/2160
2 7 1 3 5 1 8 0 7 0 6 0 1 0 1 0 1 23/2160
2 7 1 1 3 1 4 1 6 0 7 0 8 0 1 0 1 1/270
2 7 1 1 8 1 7 0 6 0 3 0 5 0 1 0 1 -1/270
2 7 1 1 7 1 5 1 6 0 4 0 8 0 1 0 1 -43/4320
2 7 1 1 7 1 6 0 3 0 8 0 2 0 1 0 1 43/4320
2 7 1 5 6 1 2 1 3 0 8 0 4 0 1 0 1 -59/32*zeta(3)^2/pi^6+779/362880
2 7 1 3 4 1 8 1 5 0 6 0 2 0 1 0 1 59/32*zeta(3)^2/pi^6-779/362880
2 7 1 3 6 1 4 1 5 0 2 0 8 0 1 0 1 -25/32*zeta(3)^2/pi^6+13/13440
2 7 1 3 5 1 8 1 2 0 6 0 4 0 1 0 1 -25/32*zeta(3)^2/pi^6+13/13440
2 7 1 3 5 1 6 1 2 0 8 0 4 0 1 0 1 -17/16*zeta(3)^2/pi^6-37/90720
2 7 1 3 6 1 8 1 5 0 2 0 4 0 1 0 1 -17/16*zeta(3)^2/pi^6-37/90720
2 7 1 4 8 1 6 1 5 1 7 0 2 0 1 0 1 -35/8*zeta(3)^2/pi^6+121/15120
2 7 1 5 8 1 2 0 3 0 6 0 7 0 1 0 1 35/8*zeta(3)^2/pi^6-121/15120
2 7 1 1 4 1 7 1 5 1 6 0 3 0 2 0 1 -33/2*zeta(3)^2/pi^6+103/4536
530 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 7 1 1 5 1 4 0 6 0 3 0 7 0 2 0 1 -33/2*zeta(3)^2/pi^6+103/4536
2 7 1 4 6 1 2 1 5 1 8 0 3 0 1 0 1 11/8*zeta(3)^2/pi^6-53/30240
2 7 1 3 5 1 4 0 2 0 6 0 8 0 1 0 1 11/8*zeta(3)^2/pi^6-53/30240
2 7 1 4 6 1 2 1 5 1 7 0 8 0 1 0 1 -5/4*zeta(3)^2/pi^6+131/72576
2 7 1 3 5 1 8 0 2 0 6 0 7 0 1 0 1 -5/4*zeta(3)^2/pi^6+131/72576
2 7 1 3 4 1 5 1 6 1 8 0 2 0 1 0 1 -11/8*zeta(3)^2/pi^6+137/45360
2 7 1 4 5 1 2 0 6 0 3 0 8 0 1 0 1 11/8*zeta(3)^2/pi^6-137/45360
2 7 1 1 7 1 4 1 5 1 6 0 8 0 3 0 1 -27/16*zeta(3)^2/pi^6+43/120960
2 7 1 1 8 1 5 0 3 0 6 0 7 0 2 0 1 -27/16*zeta(3)^2/pi^6+43/120960
2 7 1 5 6 1 2 1 7 1 8 0 4 0 1 0 1 1/2*zeta(3)^2/pi^6+1/4480
2 7 1 3 6 1 4 0 7 0 2 0 8 0 1 0 1 1/2*zeta(3)^2/pi^6+1/4480
2 7 1 1 7 1 5 1 8 1 6 0 4 0 3 0 1 -17/16*zeta(3)^2/pi^6+9/4480
2 7 1 1 5 1 6 0 2 0 7 0 8 0 3 0 1 -17/16*zeta(3)^2/pi^6+9/4480
2 7 1 4 6 1 5 1 8 1 2 0 3 0 1 0 1 -17/16*zeta(3)^2/pi^6+131/181440
2 7 1 3 5 1 4 0 6 0 8 0 2 0 1 0 1 -17/16*zeta(3)^2/pi^6+131/181440
2 7 1 3 4 1 8 1 5 1 6 0 2 0 1 0 1 -35/16*zeta(3)^2/pi^6+37/12096
2 7 1 4 5 1 2 0 8 0 6 0 3 0 1 0 1 35/16*zeta(3)^2/pi^6-37/12096
2 7 1 4 7 1 5 1 8 1 6 0 2 0 1 0 1 1/4*zeta(3)^2/pi^6+1/12960
2 7 1 5 7 1 2 0 6 0 8 0 3 0 1 0 1 -1/4*zeta(3)^2/pi^6-1/12960
2 7 1 1 5 1 8 1 7 1 6 0 4 0 2 0 1 1/1440
2 7 1 1 5 1 6 0 8 0 7 0 2 0 3 0 1 1/1440
2 7 1 4 7 5 8 1 6 1 2 1 3 0 1 0 1 -49/16*zeta(3)^2/pi^6+1849/362880
2 7 1 4 7 5 8 0 6 0 2 0 3 0 1 0 1 49/16*zeta(3)^2/pi^6-1849/362880
2 7 1 5 7 4 8 1 6 1 3 0 1 0 1 0 1 -31/8*zeta(3)^2/pi^6+893/90720
2 7 1 4 7 5 8 0 3 0 6 0 1 0 1 0 1 -31/8*zeta(3)^2/pi^6+893/90720
2 7 1 3 8 1 5 1 2 1 6 1 7 0 4 0 1 143/16*zeta(3)^2/pi^6-53/4032
2 7 1 5 8 1 4 0 2 0 6 0 7 0 3 0 1 143/16*zeta(3)^2/pi^6-53/4032
2 7 1 3 8 1 7 1 5 1 6 1 2 0 4 0 1 71/16*zeta(3)^2/pi^6-1/160
2 7 1 5 8 1 4 0 6 0 3 0 7 0 2 0 1 71/16*zeta(3)^2/pi^6-1/160
2 7 1 4 8 1 2 1 5 1 7 0 3 0 1 0 1 53/8*zeta(3)^2/pi^6-163/18144
2 7 1 5 8 1 4 0 2 0 6 0 7 0 1 0 1 -53/8*zeta(3)^2/pi^6+163/18144
2 7 1 3 8 4 7 1 5 1 6 0 1 0 1 0 1 29/8*zeta(3)^2/pi^6-41/9072
2 7 1 3 8 4 6 0 5 0 7 0 1 0 1 0 1 29/8*zeta(3)^2/pi^6-41/9072
2 7 1 3 4 1 8 1 5 1 7 0 2 0 1 0 1 5/4*zeta(3)^2/pi^6-1019/362880
2 7 1 4 5 1 2 0 8 0 6 0 7 0 1 0 1 -5/4*zeta(3)^2/pi^6+1019/362880
2 7 1 1 5 1 6 1 7 1 8 1 4 0 2 0 3 -zeta(3)^2/pi^6+293/181440
2 7 1 1 4 1 5 0 7 0 8 0 2 0 3 0 6 zeta(3)^2/pi^6-293/181440
2 7 1 3 4 1 6 1 8 1 2 1 7 0 5 0 1 95/8*zeta(3)^2/pi^6-1301/72576
2 7 1 5 6 1 4 0 2 0 7 0 8 0 3 0 1 95/8*zeta(3)^2/pi^6-1301/72576
2 7 1 4 7 1 5 1 8 1 2 0 3 0 1 0 1 -5*zeta(3)^2/pi^6+71/10080
2 7 1 5 7 1 4 0 6 0 8 0 2 0 1 0 1 5*zeta(3)^2/pi^6-71/10080
2 7 1 3 4 1 8 1 7 1 6 1 2 0 5 0 1 179/16*zeta(3)^2/pi^6-2029/120960
2 7 1 5 6 1 4 0 7 0 8 0 3 0 2 0 1 179/16*zeta(3)^2/pi^6-2029/120960
2 7 1 3 5 4 7 1 8 1 6 0 1 0 1 0 1 -3/8*zeta(3)^2/pi^6-109/60480
2 7 1 3 5 4 7 0 8 0 6 0 1 0 1 0 1 -3/8*zeta(3)^2/pi^6-109/60480
2 7 1 3 8 1 4 1 7 0 6 0 2 0 1 0 1 9/4*zeta(3)^2/pi^6-29/11340
2 7 1 5 8 1 4 1 2 0 6 0 7 0 1 0 1 -9/4*zeta(3)^2/pi^6+29/11340
2 7 1 3 6 1 4 1 7 0 2 0 8 0 1 0 1 1/4*zeta(3)^2/pi^6-11/13440
2 7 1 4 5 1 2 1 8 0 6 0 7 0 1 0 1 1/4*zeta(3)^2/pi^6-11/13440
2 7 1 6 7 1 2 1 3 0 4 0 8 0 1 0 1 -57/16*zeta(3)^2/pi^6+85/18144
2 7 1 3 7 1 8 1 5 0 6 0 2 0 1 0 1 57/16*zeta(3)^2/pi^6-85/18144
2 7 1 5 7 4 8 1 6 0 3 0 1 0 1 0 1 1/8*zeta(3)^2/pi^6+331/90720
2 7 1 4 7 5 8 1 3 0 6 0 1 0 1 0 1 1/8*zeta(3)^2/pi^6+331/90720
2 7 1 6 7 1 8 1 2 0 4 0 3 0 1 0 1 17/4*zeta(3)^2/pi^6-121/25920
2 7 1 4 7 1 6 1 5 0 8 0 2 0 1 0 1 -17/4*zeta(3)^2/pi^6+121/25920
2 7 1 3 6 1 8 1 7 0 2 0 4 0 1 0 1 5/4*zeta(3)^2/pi^6-227/51840
2 7 1 4 6 1 2 1 5 0 7 0 8 0 1 0 1 5/4*zeta(3)^2/pi^6-227/51840
2 7 1 3 7 1 8 1 6 0 4 0 2 0 1 0 1 -35/16*zeta(3)^2/pi^6+11/7560
2 7 1 5 7 1 6 1 2 0 8 0 4 0 1 0 1 35/16*zeta(3)^2/pi^6-11/7560
2 7 1 3 8 5 6 1 7 0 4 0 1 0 1 0 1 1/2*zeta(3)^2/pi^6+353/181440
2 7 1 3 8 4 6 1 5 0 7 0 1 0 1 0 1 1/2*zeta(3)^2/pi^6+353/181440
2 7 1 3 7 1 6 1 2 0 4 0 8 0 1 0 1 17/4*zeta(3)^2/pi^6-1709/181440
2 7 1 5 8 1 6 1 7 0 4 0 2 0 1 0 1 -17/4*zeta(3)^2/pi^6+1709/181440
2 7 1 3 4 1 7 1 5 0 8 0 2 0 1 0 1 zeta(3)^2/pi^6-43/17280
2 7 1 5 6 1 2 1 7 0 8 0 4 0 1 0 1 -zeta(3)^2/pi^6+43/17280
2 7 1 3 7 1 8 1 2 0 6 0 4 0 1 0 1 -15/8*zeta(3)^2/pi^6+13/5040
2 7 1 6 7 1 4 1 5 0 2 0 8 0 1 0 1 15/8*zeta(3)^2/pi^6-13/5040
2 7 1 3 5 4 6 1 8 0 7 0 1 0 1 0 1 3/16*zeta(3)^2/pi^6-23/13440
2 7 1 3 4 5 6 1 8 0 7 0 1 0 1 0 1 3/16*zeta(3)^2/pi^6-23/13440
2 7 1 6 7 1 5 1 8 1 2 0 4 0 1 0 1 -3/4*zeta(3)^2/pi^6+7/2592
2 7 1 3 7 1 5 0 6 0 8 0 2 0 1 0 1 3/4*zeta(3)^2/pi^6-7/2592
2 7 1 6 7 1 8 1 5 1 2 0 4 0 1 0 1 1/4320
2 7 1 3 7 1 5 0 8 0 6 0 2 0 1 0 1 -1/4320
C.1. ORIGINAL EXPANSION ⋆ mod ō(h̄7aff ) 531
2 7 1 4 7 1 8 1 5 1 6 0 2 0 1 0 1 1/2160
2 7 1 5 7 1 2 0 8 0 6 0 3 0 1 0 1 -1/2160
2 7 1 1 5 1 8 1 7 1 6 0 4 0 3 0 1 -7/4*zeta(3)^2/pi^6+17/6720
2 7 1 1 5 1 6 0 7 0 8 0 2 0 3 0 1 -7/4*zeta(3)^2/pi^6+17/6720
2 7 1 4 6 1 8 1 5 1 2 0 3 0 1 0 1 -1/420
2 7 1 3 5 1 4 0 8 0 6 0 2 0 1 0 1 -1/420
2 7 1 3 4 1 5 1 7 1 6 1 2 0 8 0 1 -1/1890
2 7 1 4 5 1 8 0 6 0 3 0 7 0 2 0 1 -1/1890
2 7 1 4 7 1 8 1 5 1 2 0 3 0 1 0 1 -1/540
2 7 1 5 7 1 4 0 8 0 6 0 2 0 1 0 1 1/540
2 7 1 6 7 1 2 1 3 0 8 0 4 0 1 0 1 -1/4320
2 7 1 4 7 1 8 1 5 0 6 0 2 0 1 0 1 1/4320
2 7 1 3 7 1 4 1 6 0 8 0 2 0 1 0 1 -1/2160
2 7 1 5 7 1 8 1 2 0 6 0 4 0 1 0 1 1/2160
2 7 1 3 7 1 6 1 2 0 8 0 4 0 1 0 1 -1/720
2 7 1 6 7 1 8 1 5 0 2 0 4 0 1 0 1 1/720
2 7 1 3 5 1 6 1 2 0 4 0 8 0 1 0 1 11/4*zeta(3)^2/pi^6-31/5040
2 7 1 3 5 1 6 1 8 0 4 0 2 0 1 0 1 11/4*zeta(3)^2/pi^6-31/5040
2 7 1 3 4 1 5 1 6 0 8 0 2 0 1 0 1 -2*zeta(3)^2/pi^6+299/90720
2 7 1 5 6 1 8 1 2 0 3 0 4 0 1 0 1 2*zeta(3)^2/pi^6-299/90720
2 7 1 3 5 1 4 1 2 0 6 0 8 0 1 0 1 3/4*zeta(3)^2/pi^6-53/90720
2 7 1 3 5 1 4 1 8 0 6 0 2 0 1 0 1 3/4*zeta(3)^2/pi^6-53/90720
2 7 1 3 6 1 5 1 2 1 4 0 8 0 1 0 1 -1/1890
2 7 1 3 4 1 8 0 6 0 2 0 5 0 1 0 1 -1/1890
2 7 1 1 7 1 4 1 5 1 6 0 3 0 8 0 1 -1/540
2 7 1 1 8 1 4 0 6 0 2 0 7 0 3 0 1 -1/540
2 7 1 6 8 1 5 1 2 1 4 0 7 0 1 0 1 1/2*zeta(3)^2/pi^6-1/45360
2 7 1 3 8 1 7 0 6 0 2 0 5 0 1 0 1 -1/2*zeta(3)^2/pi^6+1/45360
2 7 1 1 4 1 7 1 5 1 6 0 3 0 8 0 1 1/2*zeta(3)^2/pi^6-89/30240
2 7 1 1 8 1 5 0 6 0 2 0 7 0 3 0 1 1/2*zeta(3)^2/pi^6-89/30240
2 7 1 6 7 1 8 1 2 0 3 0 4 0 1 0 1 1/864
2 7 1 4 7 1 5 1 6 0 8 0 2 0 1 0 1 -1/864
2 7 1 3 8 1 6 1 7 0 4 0 2 0 1 0 1 -1/864
2 7 1 5 7 1 6 1 2 0 4 0 8 0 1 0 1 1/864
2 7 1 3 8 4 5 1 7 1 2 0 1 0 1 0 1 37/16*zeta(3)^2/pi^6-701/181440
2 7 1 4 5 2 8 0 3 0 7 0 1 0 1 0 1 -37/16*zeta(3)^2/pi^6+701/181440
2 7 1 3 5 4 8 1 2 1 7 0 1 0 1 0 1 -1/16*zeta(3)^2/pi^6+43/181440
2 7 1 3 5 4 8 0 2 0 7 0 1 0 1 0 1 -1/16*zeta(3)^2/pi^6+43/181440
2 7 1 4 8 1 2 1 5 1 6 0 7 0 1 0 1 -45/16*zeta(3)^2/pi^6+1723/362880
2 7 1 5 8 1 7 0 2 0 6 0 3 0 1 0 1 45/16*zeta(3)^2/pi^6-1723/362880
2 7 1 4 5 1 2 1 8 1 6 0 7 0 1 0 1 -199/362880
2 7 1 5 6 1 7 0 2 0 8 0 3 0 1 0 1 199/362880
2 7 1 4 5 2 7 1 8 0 3 0 1 0 1 0 1 -11/32*zeta(3)^2/pi^6+811/362880
2 7 1 4 5 2 7 1 3 0 8 0 1 0 1 0 1 11/32*zeta(3)^2/pi^6-811/362880
2 7 1 3 4 5 7 1 8 0 2 0 1 0 1 0 1 -19/32*zeta(3)^2/pi^6-307/362880
2 7 1 3 5 4 7 1 2 0 8 0 1 0 1 0 1 -19/32*zeta(3)^2/pi^6-307/362880
2 7 1 5 7 1 8 1 2 1 6 0 4 0 1 0 1 1/720
2 7 1 6 7 1 5 0 8 0 2 0 3 0 1 0 1 -1/720
2 7 1 4 5 1 2 1 8 1 6 0 3 0 1 0 1 4*zeta(3)^2/pi^6-43/7560
2 7 1 5 6 1 4 0 2 0 8 0 3 0 1 0 1 -4*zeta(3)^2/pi^6+43/7560
2 7 1 4 8 1 5 1 6 1 2 0 3 0 1 0 1 287/16*zeta(3)^2/pi^6-1013/36288
2 7 1 5 8 1 4 0 6 0 3 0 2 0 1 0 1 -287/16*zeta(3)^2/pi^6+1013/36288
2 7 1 3 8 4 6 1 7 1 2 0 1 0 1 0 1 13/8*zeta(3)^2/pi^6-1/45360
2 7 1 3 8 5 6 0 2 0 7 0 1 0 1 0 1 13/8*zeta(3)^2/pi^6-1/45360
2 7 1 4 8 1 2 1 5 1 6 0 3 0 1 0 1 -273/16*zeta(3)^2/pi^6+4703/181440
2 7 1 5 8 1 4 0 2 0 6 0 3 0 1 0 1 273/16*zeta(3)^2/pi^6-4703/181440
2 7 1 5 7 1 6 1 2 1 8 0 4 0 1 0 1 23/4*zeta(3)^2/pi^6-29/4536
2 7 1 6 7 1 5 0 3 0 2 0 8 0 1 0 1 -23/4*zeta(3)^2/pi^6+29/4536
2 7 1 3 8 5 6 1 2 0 7 0 1 0 1 0 1 7/16*zeta(3)^2/pi^6+29/15120
2 7 1 3 8 4 6 1 7 0 2 0 1 0 1 0 1 7/16*zeta(3)^2/pi^6+29/15120
2 7 1 4 8 1 5 1 6 1 2 0 7 0 1 0 1 3/4*zeta(3)^2/pi^6-17/7560
2 7 1 5 8 1 7 0 6 0 3 0 2 0 1 0 1 -3/4*zeta(3)^2/pi^6+17/7560
2 7 1 3 8 5 6 1 7 1 2 0 1 0 1 0 1 1/864
2 7 1 3 8 4 6 0 2 0 7 0 1 0 1 0 1 1/864
2 7 1 4 8 1 6 1 5 1 2 0 7 0 1 0 1 -1/540
2 7 1 5 8 1 7 0 3 0 6 0 2 0 1 0 1 1/540
2 7 1 3 4 1 5 1 6 1 2 0 8 0 1 0 1 -1/1890
2 7 1 4 5 1 8 0 6 0 3 0 2 0 1 0 1 1/1890
2 7 1 3 8 5 6 1 7 0 2 0 1 0 1 0 1 -1/864
2 7 1 3 8 4 6 1 2 0 7 0 1 0 1 0 1 -1/864
2 7 1 6 7 5 8 1 3 1 2 0 4 0 1 0 1 443/32*zeta(3)^2/pi^6-7627/362880
2 7 1 4 8 6 7 1 5 0 3 0 2 0 1 0 1 -443/32*zeta(3)^2/pi^6+7627/362880
2 7 1 4 6 5 7 3 8 1 2 0 1 0 1 0 1 39/16*zeta(3)^2/pi^6-479/181440
532 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 7 1 4 6 5 7 3 8 0 2 0 1 0 1 0 1 39/16*zeta(3)^2/pi^6-479/181440
2 7 1 5 7 4 8 1 6 1 3 0 2 0 1 0 1 -569/32*zeta(3)^2/pi^6+827/30240
2 7 1 6 8 5 7 1 2 0 4 0 3 0 1 0 1 569/32*zeta(3)^2/pi^6-827/30240
2 7 1 3 8 1 7 1 2 1 6 0 4 0 5 0 1 319/16*zeta(3)^2/pi^6-97/3360
2 7 1 5 8 1 6 1 7 0 4 0 2 0 3 0 1 319/16*zeta(3)^2/pi^6-97/3360
2 7 1 5 7 6 8 1 3 1 4 0 2 0 1 0 1 -21/8*zeta(3)^2/pi^6+493/120960
2 7 1 5 8 4 7 1 2 0 6 0 3 0 1 0 1 21/8*zeta(3)^2/pi^6-493/120960
2 7 1 5 8 4 6 1 7 1 3 0 2 0 1 0 1 143/32*zeta(3)^2/pi^6-4901/725760
2 7 1 6 7 4 5 1 2 0 8 0 3 0 1 0 1 143/32*zeta(3)^2/pi^6-4901/725760
2 7 1 6 8 4 5 1 7 1 2 0 3 0 1 0 1 203/32*zeta(3)^2/pi^6-7157/725760
2 7 1 5 6 4 7 1 2 0 8 0 3 0 1 0 1 -203/32*zeta(3)^2/pi^6+7157/725760
2 7 1 1 8 1 5 1 7 1 6 0 2 0 3 0 4 29/16*zeta(3)^2/pi^6-281/120960
2 7 1 1 6 1 5 1 7 0 8 0 3 0 2 0 4 -29/16*zeta(3)^2/pi^6+281/120960
2 7 1 4 8 5 6 1 3 1 2 0 7 0 1 0 1 29/16*zeta(3)^2/pi^6-53/20736
2 7 1 5 8 4 6 1 7 0 3 0 2 0 1 0 1 29/16*zeta(3)^2/pi^6-53/20736
2 7 1 4 5 1 2 1 7 1 6 0 8 0 3 0 1 1/4*zeta(3)^2/pi^6-481/725760
2 7 1 6 7 1 8 1 5 0 2 0 4 0 3 0 1 1/4*zeta(3)^2/pi^6-481/725760
2 7 1 1 4 1 8 1 5 1 7 0 3 0 2 0 1 -1/540
2 7 1 1 5 1 4 0 6 0 8 0 7 0 3 0 1 -1/540
2 7 1 1 8 1 7 1 6 0 2 0 3 0 4 0 1 -1/432
2 7 1 1 5 1 7 1 6 0 8 0 3 0 4 0 1 -1/432
2 7 1 1 5 1 2 1 8 0 7 0 4 0 3 0 1 -1/2160
2 7 1 1 6 1 5 1 3 0 7 0 8 0 4 0 1 -1/2160
2 7 1 1 7 1 6 1 5 0 3 0 2 0 4 0 1 -105/16*zeta(3)^2/pi^6+731/90720
2 7 1 1 3 1 5 1 2 0 7 0 4 0 6 0 1 -31/16*zeta(3)^2/pi^6+47/18144
2 7 1 1 6 1 2 1 5 0 7 0 4 0 3 0 1 17/4*zeta(3)^2/pi^6-643/90720
2 7 1 1 5 1 2 1 6 0 7 0 3 0 4 0 1 17/4*zeta(3)^2/pi^6-643/90720
2 7 1 3 7 1 6 1 8 1 4 1 5 0 1 0 1 1/945
2 7 1 3 7 0 6 0 8 0 4 0 5 0 1 0 1 -1/945
2 7 1 1 6 1 8 1 5 1 3 1 4 0 2 0 1 -1/945
2 7 1 1 3 0 7 0 8 0 6 0 4 0 5 0 1 -1/945
2 7 1 4 5 2 7 1 8 1 6 1 3 0 1 0 1 -55/8*zeta(3)^2/pi^6+1021/103680
2 7 1 4 5 2 7 0 8 0 6 0 3 0 1 0 1 55/8*zeta(3)^2/pi^6-1021/103680
2 7 1 5 7 2 4 1 8 1 6 1 3 0 1 0 1 57/16*zeta(3)^2/pi^6-3659/725760
2 7 1 5 7 2 4 0 8 0 6 0 3 0 1 0 1 -57/16*zeta(3)^2/pi^6+3659/725760
2 7 1 4 7 1 8 1 5 1 6 0 3 0 1 0 1 5/8*zeta(3)^2/pi^6+199/90720
2 7 1 5 7 1 4 0 8 0 6 0 3 0 1 0 1 -5/8*zeta(3)^2/pi^6-199/90720
2 7 1 3 5 1 6 1 7 1 8 0 4 0 1 0 1 1/16*zeta(3)^2/pi^6+41/181440
2 7 1 5 6 1 4 0 8 0 7 0 3 0 1 0 1 1/16*zeta(3)^2/pi^6+41/181440
2 7 1 7 8 1 6 1 2 1 4 1 5 0 3 0 1 -7/16*zeta(3)^2/pi^6+263/362880
2 7 1 3 8 1 4 0 7 0 2 0 5 0 6 0 1 -7/16*zeta(3)^2/pi^6+263/362880
2 7 1 6 7 1 5 1 8 1 4 0 3 0 1 0 1 2*zeta(3)^2/pi^6-37/22680
2 7 1 3 7 1 4 0 6 0 8 0 5 0 1 0 1 -2*zeta(3)^2/pi^6+37/22680
2 7 1 3 8 1 6 1 7 1 4 1 5 0 2 0 1 -25/8*zeta(3)^2/pi^6+25/5184
2 7 1 4 8 1 2 0 7 0 3 0 5 0 6 0 1 -25/8*zeta(3)^2/pi^6+25/5184
2 7 1 1 4 1 7 1 5 1 8 0 3 0 2 0 1 53/16*zeta(3)^2/pi^6-853/120960
2 7 1 1 5 1 4 0 6 0 3 0 7 0 8 0 1 53/16*zeta(3)^2/pi^6-853/120960
2 7 1 3 7 1 5 1 6 1 8 1 2 0 4 0 1 3*zeta(3)^2/pi^6-3131/725760
2 7 1 3 5 1 4 0 7 0 6 0 8 0 2 0 1 -3*zeta(3)^2/pi^6+3131/725760
2 7 1 3 4 1 6 1 5 1 8 1 7 0 2 0 1 -47/16*zeta(3)^2/pi^6+3229/725760
2 7 1 4 5 1 2 0 7 0 6 0 8 0 3 0 1 -47/16*zeta(3)^2/pi^6+3229/725760
2 7 1 4 6 1 8 1 5 1 7 0 3 0 1 0 1 67/45360
2 7 1 3 5 1 4 0 8 0 6 0 7 0 1 0 1 67/45360
2 7 1 1 4 1 8 1 5 1 7 0 2 0 3 0 1 21/16*zeta(3)^2/pi^6-1483/362880
2 7 1 1 5 1 4 0 8 0 6 0 7 0 3 0 1 21/16*zeta(3)^2/pi^6-1483/362880
2 7 1 1 4 1 8 1 5 1 6 0 7 0 3 0 1 1/2*zeta(3)^2/pi^6-157/90720
2 7 1 1 5 1 2 0 6 0 8 0 7 0 3 0 1 1/2*zeta(3)^2/pi^6-157/90720
2 7 1 4 8 1 5 1 6 1 7 0 3 0 1 0 1 -19/8*zeta(3)^2/pi^6+199/45360
2 7 1 5 8 1 4 0 6 0 3 0 7 0 1 0 1 19/8*zeta(3)^2/pi^6-199/45360
2 7 1 4 8 2 7 1 6 1 3 1 5 0 1 0 1 3*zeta(3)^2/pi^6-7/1728
2 7 1 4 8 2 7 0 6 0 3 0 5 0 1 0 1 -3*zeta(3)^2/pi^6+7/1728
2 7 1 4 6 1 5 1 8 1 7 0 3 0 1 0 1 -7/16*zeta(3)^2/pi^6+31/45360
2 7 1 3 5 1 4 0 6 0 8 0 7 0 1 0 1 -7/16*zeta(3)^2/pi^6+31/45360
2 7 1 3 7 1 8 1 5 1 6 1 2 0 4 0 1 -27/8*zeta(3)^2/pi^6+3961/725760
2 7 1 3 5 1 4 0 6 0 8 0 7 0 2 0 1 27/8*zeta(3)^2/pi^6-3961/725760
2 7 1 1 5 1 4 1 8 1 7 0 2 0 3 0 1 -65/16*zeta(3)^2/pi^6+2123/362880
2 7 1 1 5 1 4 0 7 0 6 0 3 0 8 0 1 -65/16*zeta(3)^2/pi^6+2123/362880
2 7 1 3 4 1 5 1 8 1 6 1 7 0 2 0 1 -37/8*zeta(3)^2/pi^6+5011/725760
2 7 1 4 5 1 2 0 6 0 8 0 7 0 3 0 1 -37/8*zeta(3)^2/pi^6+5011/725760
2 7 1 1 5 1 7 1 3 1 4 0 2 0 6 0 1 5*zeta(3)^2/pi^6-239/30240
2 7 1 1 4 1 2 0 7 0 3 0 5 0 6 0 1 5*zeta(3)^2/pi^6-239/30240
2 7 1 1 7 1 4 1 5 1 8 0 3 0 6 0 1 -35/16*zeta(3)^2/pi^6+37/17280
2 7 1 1 5 1 2 0 3 0 6 0 7 0 8 0 1 -35/16*zeta(3)^2/pi^6+37/17280
C.1. ORIGINAL EXPANSION ⋆aff mod ō(h̄7) 533
2 7 1 5 8 2 4 1 6 1 3 1 7 0 1 0 1 -31/16*zeta(3)^2/pi^6+461/145152
2 7 1 5 8 2 4 0 6 0 3 0 7 0 1 0 1 31/16*zeta(3)^2/pi^6-461/145152
2 7 1 4 5 2 8 1 6 1 3 1 7 0 1 0 1 -29/8*zeta(3)^2/pi^6+733/145152
2 7 1 4 5 2 8 0 6 0 3 0 7 0 1 0 1 29/8*zeta(3)^2/pi^6-733/145152
2 7 1 3 4 1 5 1 6 1 7 0 8 0 1 0 1 3/4*zeta(3)^2/pi^6+1/1890
2 7 1 4 5 1 8 0 6 0 3 0 7 0 1 0 1 -3/4*zeta(3)^2/pi^6-1/1890
2 7 1 3 8 1 5 1 6 1 4 0 7 0 1 0 1 5/4*zeta(3)^2/pi^6-13/4320
2 7 1 4 8 1 7 0 6 0 3 0 5 0 1 0 1 -5/4*zeta(3)^2/pi^6+13/4320
2 7 1 3 7 1 5 1 2 1 6 1 8 0 4 0 1 -7/4*zeta(3)^2/pi^6+13/4480
2 7 1 3 5 1 4 0 2 0 6 0 7 0 8 0 1 7/4*zeta(3)^2/pi^6-13/4480
2 7 1 3 4 1 5 1 7 1 6 1 8 0 2 0 1 5/4*zeta(3)^2/pi^6-17/9072
2 7 1 4 5 1 2 0 6 0 3 0 7 0 8 0 1 5/4*zeta(3)^2/pi^6-17/9072
2 7 1 3 6 1 5 1 8 1 4 0 7 0 1 0 1 13/16*zeta(3)^2/pi^6-149/45360
2 7 1 3 4 1 7 0 6 0 8 0 5 0 1 0 1 13/16*zeta(3)^2/pi^6-149/45360
2 7 1 1 5 1 7 1 3 1 4 0 2 0 8 0 1 7/16*zeta(3)^2/pi^6-97/51840
2 7 1 1 4 1 8 0 7 0 3 0 5 0 6 0 1 7/16*zeta(3)^2/pi^6-97/51840
2 7 1 1 5 1 8 1 3 1 4 0 7 0 2 0 1 -5/8*zeta(3)^2/pi^6+7/8640
2 7 1 1 3 1 4 0 7 0 8 0 5 0 6 0 1 -5/8*zeta(3)^2/pi^6+7/8640
2 7 1 1 6 1 8 1 5 1 3 1 4 0 2 0 7 -11/16*zeta(3)^2/pi^6+281/362880
2 7 1 1 4 1 2 0 8 0 3 0 7 0 5 0 6 11/16*zeta(3)^2/pi^6-281/362880
2 7 1 4 6 2 5 1 3 1 7 1 8 0 1 0 1 3/16*zeta(3)^2/pi^6-71/362880
2 7 1 5 6 2 4 0 8 0 3 0 7 0 1 0 1 -3/16*zeta(3)^2/pi^6+71/362880
2 7 1 3 4 1 8 1 5 1 6 0 7 0 1 0 1 -9/16*zeta(3)^2/pi^6-11/8640
2 7 1 4 5 1 7 0 8 0 6 0 3 0 1 0 1 9/16*zeta(3)^2/pi^6+11/8640
2 7 1 1 8 1 6 1 7 0 4 0 2 0 3 0 1 25/8*zeta(3)^2/pi^6-131/12960
2 7 1 1 5 1 6 1 7 0 4 0 8 0 3 0 1 25/8*zeta(3)^2/pi^6-131/12960
2 7 1 1 5 1 2 1 8 0 7 0 3 0 4 0 1 -15/8*zeta(3)^2/pi^6+37/25920
2 7 1 1 5 1 6 1 3 0 7 0 8 0 4 0 1 -15/8*zeta(3)^2/pi^6+37/25920
2 7 1 1 8 1 2 1 6 0 4 0 7 0 3 0 1 -11/16*zeta(3)^2/pi^6+55/72576
2 7 1 1 6 1 5 1 3 0 7 0 4 0 8 0 1 -11/16*zeta(3)^2/pi^6+55/72576
2 7 1 3 8 1 4 1 5 0 6 0 7 0 1 0 1 -3/8*zeta(3)^2/pi^6-19/60480
2 7 1 5 7 1 8 1 3 0 6 0 4 0 1 0 1 3/8*zeta(3)^2/pi^6+19/60480
2 7 1 1 3 1 5 1 2 0 7 0 4 0 8 0 1 19/16*zeta(3)^2/pi^6-199/120960
2 7 1 1 5 1 8 1 3 0 7 0 4 0 6 0 1 19/16*zeta(3)^2/pi^6-199/120960
2 7 1 1 3 1 8 1 2 0 7 0 4 0 6 0 1 zeta(3)^2/pi^6-191/90720
2 7 1 1 4 1 5 1 3 0 7 0 8 0 6 0 1 zeta(3)^2/pi^6-191/90720
2 7 1 5 7 1 6 1 3 0 4 0 8 0 1 0 1 1/8*zeta(3)^2/pi^6+37/181440
2 7 1 3 8 1 5 1 7 0 6 0 4 0 1 0 1 -1/8*zeta(3)^2/pi^6-37/181440
2 7 1 3 5 1 6 1 7 0 4 0 8 0 1 0 1 -13/2520
2 7 1 1 6 1 2 1 7 0 4 0 8 0 3 0 1 7/4*zeta(3)^2/pi^6-163/36288
2 7 1 1 6 1 8 1 5 0 7 0 4 0 3 0 1 7/4*zeta(3)^2/pi^6-163/36288
2 7 1 1 6 1 2 1 8 0 7 0 4 0 3 0 1 3/8*zeta(3)^2/pi^6-61/45360
2 7 1 1 3 1 5 1 6 0 7 0 8 0 4 0 1 3/8*zeta(3)^2/pi^6-61/45360
2 7 1 3 7 1 6 1 5 0 8 0 4 0 1 0 1 3/8*zeta(3)^2/pi^6-149/30240
2 7 1 5 7 1 8 1 6 0 4 0 3 0 1 0 1 -3/8*zeta(3)^2/pi^6+149/30240
2 7 1 1 5 1 2 1 6 0 4 0 8 0 3 0 1 -5/2*zeta(3)^2/pi^6+13/7560
2 7 1 1 5 1 8 1 6 0 7 0 3 0 4 0 1 -5/2*zeta(3)^2/pi^6+13/7560
2 7 1 1 8 1 2 1 6 0 7 0 3 0 4 0 1 -2*zeta(3)^2/pi^6+59/20160
2 7 1 1 3 1 5 1 6 0 4 0 7 0 8 0 1 -2*zeta(3)^2/pi^6+59/20160
2 7 1 3 6 1 7 1 5 0 8 0 4 0 1 0 1 -1/8*zeta(3)^2/pi^6-703/120960
2 7 1 3 5 1 6 1 7 0 8 0 4 0 1 0 1 -1/8*zeta(3)^2/pi^6-703/120960
2 7 1 5 6 1 7 1 3 0 8 0 4 0 1 0 1 -1/16*zeta(3)^2/pi^6+47/90720
2 7 1 3 4 1 8 1 5 0 6 0 7 0 1 0 1 1/16*zeta(3)^2/pi^6-47/90720
2 7 1 1 3 1 6 1 2 0 7 0 8 0 4 0 1 -3/8*zeta(3)^2/pi^6+1/1440
2 7 1 1 3 1 5 1 8 0 7 0 4 0 6 0 1 -3/8*zeta(3)^2/pi^6+1/1440
2 7 1 3 6 1 4 1 5 0 7 0 8 0 1 0 1 -1/4*zeta(3)^2/pi^6-967/362880
2 7 1 3 5 1 8 1 7 0 6 0 4 0 1 0 1 -1/4*zeta(3)^2/pi^6-967/362880
2 7 1 3 8 4 7 1 6 1 2 0 5 0 1 0 1 57/16*zeta(3)^2/pi^6-599/120960
2 7 1 6 8 2 7 1 5 0 3 0 4 0 1 0 1 -57/16*zeta(3)^2/pi^6+599/120960
2 7 1 3 6 1 5 1 7 1 8 0 4 0 2 0 1 -7/8*zeta(3)^2/pi^6+1091/725760
2 7 1 3 5 1 6 1 2 0 7 0 4 0 8 0 1 7/8*zeta(3)^2/pi^6-1091/725760
2 7 1 6 8 1 5 1 7 1 4 0 3 0 2 0 1 65/8*zeta(3)^2/pi^6-4343/362880
2 7 1 3 8 1 5 1 2 0 7 0 4 0 6 0 1 65/8*zeta(3)^2/pi^6-4343/362880
2 7 1 6 7 1 5 1 8 1 2 0 3 0 4 0 1 -15/32*zeta(3)^2/pi^6+97/290304
2 7 1 3 4 1 5 1 6 0 7 0 8 0 2 0 1 -15/32*zeta(3)^2/pi^6+97/290304
2 7 1 4 7 1 8 1 5 1 6 0 2 0 3 0 1 -75/32*zeta(3)^2/pi^6+5129/1451520
2 7 1 3 5 1 6 1 2 0 7 0 8 0 4 0 1 75/32*zeta(3)^2/pi^6-5129/1451520
2 7 1 3 6 1 5 1 2 1 4 0 7 0 8 0 1 -1/4*zeta(3)^2/pi^6+13/45360
2 7 1 3 5 1 4 1 8 0 7 0 2 0 6 0 1 1/4*zeta(3)^2/pi^6-13/45360
2 7 1 6 8 1 5 1 2 1 4 0 7 0 3 0 1 -9/16*zeta(3)^2/pi^6+17/16128
2 7 1 3 8 1 4 1 5 0 7 0 2 0 6 0 1 -9/16*zeta(3)^2/pi^6+17/16128
2 7 1 4 6 1 2 1 5 1 8 0 7 0 3 0 1 45/32*zeta(3)^2/pi^6-3019/1451520
2 7 1 3 6 1 4 1 5 0 2 0 7 0 8 0 1 -45/32*zeta(3)^2/pi^6+3019/1451520
534 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 7 1 3 8 1 5 1 7 1 4 0 2 0 6 0 1 5/16*zeta(3)^2/pi^6-379/725760
2 7 1 5 8 1 2 1 3 0 7 0 4 0 6 0 1 5/16*zeta(3)^2/pi^6-379/725760
2 7 1 3 8 4 6 1 5 1 7 0 2 0 1 0 1 9/32*zeta(3)^2/pi^6-205/290304
2 7 1 3 8 4 5 1 2 0 6 0 7 0 1 0 1 9/32*zeta(3)^2/pi^6-205/290304
2 7 1 3 4 1 7 1 5 1 8 0 2 0 6 0 1 25/32*zeta(3)^2/pi^6-181/161280
2 7 1 5 6 1 2 1 3 0 7 0 4 0 8 0 1 -25/32*zeta(3)^2/pi^6+181/161280
2 7 1 3 5 6 7 1 8 1 4 0 2 0 1 0 1 -1/32*zeta(3)^2/pi^6+169/483840
2 7 1 4 8 2 5 1 3 0 6 0 7 0 1 0 1 1/32*zeta(3)^2/pi^6-169/483840
2 7 1 1 8 1 4 1 5 1 6 0 2 0 3 0 7 -3/16*zeta(3)^2/pi^6-23/362880
2 7 1 1 5 1 2 1 6 0 7 0 3 0 8 0 4 3/16*zeta(3)^2/pi^6+23/362880
2 7 1 1 8 1 4 1 5 1 6 0 7 0 2 0 3 17/16*zeta(3)^2/pi^6-19/10368
2 7 1 1 6 1 2 1 5 0 7 0 4 0 8 0 3 -17/16*zeta(3)^2/pi^6+19/10368
2 7 1 3 8 4 6 1 5 1 2 0 7 0 1 0 1 -29/32*zeta(3)^2/pi^6+1573/1451520
2 7 1 4 5 2 7 1 8 0 6 0 3 0 1 0 1 -29/32*zeta(3)^2/pi^6+1573/1451520
2 7 1 3 6 4 7 1 5 1 2 0 8 0 1 0 1 -87/32*zeta(3)^2/pi^6+751/207360
2 7 1 5 8 2 4 1 7 0 6 0 3 0 1 0 1 87/32*zeta(3)^2/pi^6-751/207360
2 7 1 3 8 5 6 1 7 1 2 0 4 0 1 0 1 61/32*zeta(3)^2/pi^6-2939/1451520
2 7 1 4 6 2 7 1 5 0 8 0 3 0 1 0 1 61/32*zeta(3)^2/pi^6-2939/1451520
2 7 1 4 6 1 8 1 7 1 2 0 5 0 3 0 1 -23/16*zeta(3)^2/pi^6+451/145152
2 7 1 3 6 1 7 1 5 0 8 0 4 0 2 0 1 23/16*zeta(3)^2/pi^6-451/145152
2 7 1 3 6 1 7 1 5 1 2 0 4 0 8 0 1 9/8*zeta(3)^2/pi^6-2689/1451520
2 7 1 3 5 1 6 1 8 0 4 0 7 0 2 0 1 -9/8*zeta(3)^2/pi^6+2689/1451520
2 7 1 3 4 1 7 1 5 1 6 0 2 0 8 0 1 -5/8*zeta(3)^2/pi^6+271/290304
2 7 1 5 6 1 8 1 2 0 7 0 3 0 4 0 1 5/8*zeta(3)^2/pi^6-271/290304
2 7 1 3 6 5 7 1 8 1 2 0 4 0 1 0 1 -13/32*zeta(3)^2/pi^6+29/23040
2 7 1 6 7 2 4 1 5 0 8 0 3 0 1 0 1 13/32*zeta(3)^2/pi^6-29/23040
2 7 1 4 5 1 8 1 7 1 6 0 2 0 3 0 1 19/16*zeta(3)^2/pi^6-1243/725760
2 7 1 6 7 1 2 1 5 0 8 0 4 0 3 0 1 19/16*zeta(3)^2/pi^6-1243/725760
2 7 1 3 5 4 7 1 2 1 6 0 8 0 1 0 1 -3/8*zeta(3)^2/pi^6+19/32256
2 7 1 6 8 2 5 1 7 0 4 0 3 0 1 0 1 3/8*zeta(3)^2/pi^6-19/32256
2 7 1 3 8 4 5 1 6 1 2 0 7 0 1 0 1 3/8*zeta(3)^2/pi^6-1241/1451520
2 7 1 5 6 2 7 1 8 0 3 0 4 0 1 0 1 3/8*zeta(3)^2/pi^6-1241/1451520
2 7 1 4 6 1 2 1 7 1 8 0 5 0 3 0 1 45/16*zeta(3)^2/pi^6-899/241920
2 7 1 3 6 1 7 1 5 0 2 0 4 0 8 0 1 -45/16*zeta(3)^2/pi^6+899/241920
2 7 1 1 5 1 4 1 7 1 6 0 3 0 2 0 1 23/2*zeta(3)^2/pi^6-1583/90720
2 7 1 1 5 1 4 0 7 0 6 0 3 0 2 0 1 23/2*zeta(3)^2/pi^6-1583/90720
2 7 1 3 8 1 6 1 5 1 2 1 7 0 4 0 1 -39/4*zeta(3)^2/pi^6+1807/120960
2 7 1 5 8 1 4 0 7 0 6 0 3 0 2 0 1 -39/4*zeta(3)^2/pi^6+1807/120960
2 7 1 3 6 1 7 1 2 0 4 0 8 0 1 0 1 -3/2*zeta(3)^2/pi^6-479/362880
2 7 1 3 6 1 7 1 5 0 2 0 8 0 1 0 1 -3/2*zeta(3)^2/pi^6-479/362880
2 7 1 4 6 1 5 1 8 1 2 0 7 0 1 0 1 -409/181440
2 7 1 3 5 1 7 0 6 0 8 0 2 0 1 0 1 -409/181440
2 7 1 5 6 1 2 1 7 1 8 0 3 0 1 0 1 1/720
2 7 1 3 6 1 5 0 7 0 2 0 8 0 1 0 1 1/720
2 7 1 3 5 1 6 1 7 1 8 0 2 0 1 0 1 -1/4320
2 7 1 5 6 1 2 0 8 0 7 0 3 0 1 0 1 -1/4320
2 7 1 1 5 1 4 1 7 1 6 0 8 0 2 0 1 5/4*zeta(3)^2/pi^6-149/60480
2 7 1 1 8 1 4 0 7 0 6 0 3 0 2 0 1 5/4*zeta(3)^2/pi^6-149/60480
2 7 1 4 6 1 8 1 5 1 2 0 7 0 1 0 1 -1/540
2 7 1 3 5 1 7 0 8 0 6 0 2 0 1 0 1 -1/540
2 7 1 3 4 1 6 1 5 1 2 1 7 0 8 0 1 1/1890
2 7 1 4 5 1 8 0 7 0 6 0 3 0 2 0 1 -1/1890
2 7 1 3 6 1 7 1 2 0 8 0 4 0 1 0 1 -1/720
2 7 1 4 6 1 7 1 5 0 2 0 8 0 1 0 1 -1/720
2 7 1 3 4 1 7 1 6 0 8 0 2 0 1 0 1 -1/4320
2 7 1 5 6 1 7 1 2 0 8 0 4 0 1 0 1 1/4320
2 7 1 3 6 1 4 1 2 0 7 0 8 0 1 0 1 1/540
2 7 1 4 5 1 7 1 8 0 6 0 2 0 1 0 1 1/540
2 7 1 1 5 1 4 1 8 1 6 0 7 0 3 0 1 1/2*zeta(3)^2/pi^6-29/36288
2 7 1 1 5 1 2 0 6 0 7 0 3 0 8 0 1 1/2*zeta(3)^2/pi^6-29/36288
2 7 1 1 4 1 8 1 5 1 6 0 7 0 2 0 1 -1/1440
2 7 1 1 5 1 2 0 8 0 6 0 7 0 3 0 1 -1/1440
2 7 1 1 5 1 2 1 7 0 4 0 8 0 3 0 1 -1/1440
2 7 1 1 8 1 5 1 6 0 7 0 3 0 4 0 1 -1/1440
2 7 1 1 3 1 5 1 2 0 7 0 8 0 4 0 1 1/1440
2 7 1 1 5 1 8 1 2 0 7 0 4 0 6 0 1 1/1440
2 7 1 3 5 4 8 1 6 1 7 0 2 0 1 0 1 -11/16*zeta(3)^2/pi^6+1349/1451520
2 7 1 3 6 5 7 1 2 0 4 0 8 0 1 0 1 11/16*zeta(3)^2/pi^6-1349/1451520
2 7 1 4 8 1 5 1 7 1 6 0 2 0 3 0 1 -199/16*zeta(3)^2/pi^6+1961/103680
2 7 1 6 8 1 2 1 5 0 7 0 4 0 3 0 1 -199/16*zeta(3)^2/pi^6+1961/103680
2 7 1 3 8 4 6 1 7 1 2 0 5 0 1 0 1 -45/8*zeta(3)^2/pi^6+4237/483840
2 7 1 4 6 2 7 1 5 0 3 0 8 0 1 0 1 -45/8*zeta(3)^2/pi^6+4237/483840
2 7 1 4 6 1 2 1 8 1 7 0 5 0 3 0 1 -197/32*zeta(3)^2/pi^6+14659/1451520
C.1. ORIGINAL EXPANSION ⋆aff mod ō(h̄7) 535
2 7 1 3 6 1 7 1 5 0 2 0 8 0 4 0 1 197/32*zeta(3)^2/pi^6-14659/1451520
2 7 1 7 8 1 5 1 2 1 6 0 4 0 3 0 1 -37/4*zeta(3)^2/pi^6+10153/725760
2 7 1 3 8 1 6 1 5 0 2 0 7 0 4 0 1 -37/4*zeta(3)^2/pi^6+10153/725760
2 7 1 3 4 1 7 1 8 1 6 0 2 0 5 0 1 -187/32*zeta(3)^2/pi^6+2495/290304
2 7 1 5 6 1 2 1 7 0 4 0 8 0 3 0 1 -187/32*zeta(3)^2/pi^6+2495/290304
2 7 1 3 8 6 7 1 5 1 2 0 4 0 1 0 1 -175/32*zeta(3)^2/pi^6+593/72576
2 7 1 4 8 2 7 1 5 0 6 0 3 0 1 0 1 175/32*zeta(3)^2/pi^6-593/72576
2 7 1 3 8 4 7 1 5 1 6 0 2 0 1 0 1 289/32*zeta(3)^2/pi^6-707/51840
2 7 1 3 8 5 7 1 2 0 6 0 4 0 1 0 1 -289/32*zeta(3)^2/pi^6+707/51840
2 7 1 3 5 6 8 1 2 1 7 0 4 0 1 0 1 153/32*zeta(3)^2/pi^6-9907/1451520
2 7 1 4 7 2 5 1 6 0 8 0 3 0 1 0 1 -153/32*zeta(3)^2/pi^6+9907/1451520
2 7 1 6 8 1 5 1 7 1 2 0 4 0 3 0 1 71/16*zeta(3)^2/pi^6-4519/725760
2 7 1 3 8 1 6 1 5 0 7 0 4 0 2 0 1 71/16*zeta(3)^2/pi^6-4519/725760
2 7 1 3 8 4 5 1 6 1 7 0 2 0 1 0 1 105/32*zeta(3)^2/pi^6-6407/1451520
2 7 1 3 7 5 6 1 2 0 8 0 4 0 1 0 1 105/32*zeta(3)^2/pi^6-6407/1451520
2 7 1 3 8 1 5 1 7 1 6 0 4 0 2 0 1 -15/16*zeta(3)^2/pi^6+479/241920
2 7 1 5 8 1 2 1 6 0 7 0 3 0 4 0 1 -15/16*zeta(3)^2/pi^6+479/241920
2 7 1 3 6 4 5 1 8 1 2 0 7 0 1 0 1 45/32*zeta(3)^2/pi^6-37/15120
2 7 1 5 6 2 4 1 7 0 8 0 3 0 1 0 1 -45/32*zeta(3)^2/pi^6+37/15120
2 7 1 3 4 1 7 1 5 1 6 0 8 0 2 0 1 -17/16*zeta(3)^2/pi^6+787/725760
2 7 1 5 6 1 8 1 2 0 7 0 4 0 3 0 1 17/16*zeta(3)^2/pi^6-787/725760
2 7 1 4 7 1 2 1 5 1 6 0 8 0 3 0 1 -29/16*zeta(3)^2/pi^6+2243/725760
2 7 1 3 5 1 6 1 8 0 7 0 2 0 4 0 1 29/16*zeta(3)^2/pi^6-2243/725760
2 7 1 3 5 4 6 1 2 1 7 0 8 0 1 0 1 15/32*zeta(3)^2/pi^6-17/40320
2 7 1 4 6 2 5 1 7 0 8 0 3 0 1 0 1 15/32*zeta(3)^2/pi^6-17/40320
2 7 1 3 5 4 6 1 8 1 7 0 2 0 1 0 1 1/8*zeta(3)^2/pi^6-299/362880
2 7 1 3 6 4 5 1 2 0 7 0 8 0 1 0 1 1/8*zeta(3)^2/pi^6-299/362880
2 7 1 1 5 1 4 1 6 1 7 0 8 0 3 0 2 -5/2*zeta(3)^2/pi^6+347/90720
2 7 1 1 6 1 2 1 5 0 8 0 7 0 4 0 3 5/2*zeta(3)^2/pi^6-347/90720
2 7 1 3 4 1 5 1 8 1 6 0 7 0 2 0 1 113/32*zeta(3)^2/pi^6-7801/1451520
2 7 1 5 6 1 2 1 3 0 7 0 8 0 4 0 1 113/32*zeta(3)^2/pi^6-7801/1451520
2 7 1 4 6 1 5 1 8 1 2 0 7 0 3 0 1 55/32*zeta(3)^2/pi^6-1303/483840
2 7 1 3 6 1 4 1 5 0 7 0 8 0 2 0 1 -55/32*zeta(3)^2/pi^6+1303/483840
2 7 1 3 6 1 5 1 7 1 2 0 4 0 8 0 1 1/4*zeta(3)^2/pi^6-89/60480
2 7 1 3 5 1 6 1 8 0 7 0 4 0 2 0 1 -1/4*zeta(3)^2/pi^6+89/60480
2 7 1 3 5 6 7 1 2 1 4 0 8 0 1 0 1 -1/4*zeta(3)^2/pi^6+1/90720
2 7 1 4 8 2 5 1 7 0 6 0 3 0 1 0 1 1/4*zeta(3)^2/pi^6-1/90720
2 7 1 4 6 1 8 1 5 1 2 0 7 0 3 0 1 1/4*zeta(3)^2/pi^6-157/181440
2 7 1 3 6 1 4 1 5 0 8 0 7 0 2 0 1 -1/4*zeta(3)^2/pi^6+157/181440
2 7 1 4 7 1 5 1 8 1 6 0 2 0 3 0 1 41/16*zeta(3)^2/pi^6-313/90720
2 7 1 3 5 1 6 1 2 0 8 0 7 0 4 0 1 -41/16*zeta(3)^2/pi^6+313/90720
2 7 1 3 6 4 7 1 8 1 2 0 5 0 1 0 1 21/8*zeta(3)^2/pi^6-23/6480
2 7 1 6 7 2 4 1 5 0 3 0 8 0 1 0 1 -21/8*zeta(3)^2/pi^6+23/6480
2 7 1 3 5 4 7 1 8 1 6 0 2 0 1 0 1 -19/8*zeta(3)^2/pi^6+73/24192
2 7 1 3 6 5 7 1 2 0 8 0 4 0 1 0 1 19/8*zeta(3)^2/pi^6-73/24192
2 7 1 4 6 1 2 1 5 1 7 0 8 0 3 0 1 -83/32*zeta(3)^2/pi^6+1427/362880
2 7 1 3 6 1 8 1 5 0 2 0 7 0 4 0 1 83/32*zeta(3)^2/pi^6-1427/362880
2 7 1 4 6 1 5 1 7 1 2 0 8 0 3 0 1 81/32*zeta(3)^2/pi^6-799/181440
2 7 1 3 6 1 8 1 5 0 7 0 4 0 2 0 1 -81/32*zeta(3)^2/pi^6+799/181440
2 7 1 6 7 1 5 1 2 1 4 0 3 0 8 0 1 -59/32*zeta(3)^2/pi^6+3701/1451520
2 7 1 3 4 1 5 1 8 0 7 0 2 0 6 0 1 -59/32*zeta(3)^2/pi^6+3701/1451520
2 7 1 6 7 1 5 1 2 1 8 0 3 0 4 0 1 19/16*zeta(3)^2/pi^6-19/11520
2 7 1 3 4 1 5 1 6 0 7 0 2 0 8 0 1 19/16*zeta(3)^2/pi^6-19/11520
2 7 1 3 6 1 7 1 5 1 8 0 4 0 2 0 1 19/16*zeta(3)^2/pi^6-419/241920
2 7 1 3 5 1 6 1 2 0 4 0 7 0 8 0 1 -19/16*zeta(3)^2/pi^6+419/241920
2 7 1 3 6 1 5 1 7 1 4 0 8 0 2 0 1 51/32*zeta(3)^2/pi^6-607/207360
2 7 1 3 5 1 8 1 2 0 7 0 4 0 6 0 1 -51/32*zeta(3)^2/pi^6+607/207360
2 7 1 1 5 1 8 1 3 1 4 0 7 0 2 0 6 -3/16*zeta(3)^2/pi^6+143/362880
2 7 1 1 3 1 5 1 2 0 8 0 4 0 6 0 7 3/16*zeta(3)^2/pi^6-143/362880
2 7 1 3 6 1 5 1 8 1 4 0 7 0 2 0 1 -1/2*zeta(3)^2/pi^6+191/181440
2 7 1 3 5 1 4 1 2 0 7 0 8 0 6 0 1 1/2*zeta(3)^2/pi^6-191/181440
2 7 1 4 5 2 7 3 8 1 6 0 1 0 1 0 1 45/16*zeta(3)^2/pi^6-1429/362880
2 7 1 3 5 4 8 2 6 0 7 0 1 0 1 0 1 45/16*zeta(3)^2/pi^6-1429/362880
2 7 1 4 8 1 2 1 5 1 6 0 7 0 3 0 1 -3/4*zeta(3)^2/pi^6+11/8640
2 7 1 5 8 1 6 1 3 0 7 0 2 0 4 0 1 -3/4*zeta(3)^2/pi^6+11/8640
2 7 1 3 4 1 7 1 8 1 2 0 5 0 6 0 1 -3/4*zeta(3)^2/pi^6+1/896
2 7 1 5 6 1 7 1 3 0 4 0 8 0 2 0 1 -3/4*zeta(3)^2/pi^6+1/896
2 7 1 3 8 1 7 1 5 1 2 0 4 0 6 0 1 -67/8*zeta(3)^2/pi^6+217/17280
2 7 1 5 8 1 6 1 3 0 4 0 7 0 2 0 1 -67/8*zeta(3)^2/pi^6+217/17280
2 7 1 1 5 1 7 1 3 1 4 0 8 0 6 0 1 -1/2*zeta(3)^2/pi^6+13/22680
2 7 1 1 8 1 2 0 7 0 3 0 5 0 6 0 1 -1/2*zeta(3)^2/pi^6+13/22680
2 7 1 5 7 2 4 1 6 1 8 1 3 0 1 0 1 -1/1890
2 7 1 5 7 2 4 0 6 0 8 0 3 0 1 0 1 1/1890
536 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 7 1 3 7 1 6 1 2 1 4 1 5 0 8 0 1 -1/1890
2 7 1 3 4 1 8 0 7 0 2 0 5 0 6 0 1 1/1890
2 7 1 3 4 1 7 1 5 1 2 0 8 0 6 0 1 1/4*zeta(3)^2/pi^6-13/45360
2 7 1 5 6 1 8 1 3 0 7 0 4 0 2 0 1 -1/4*zeta(3)^2/pi^6+13/45360
2 7 1 1 3 0 2 0 1 0 1 0 1 0 1 0 1 -1/720
2 7 1 1 4 1 8 0 2 0 1 0 1 0 1 0 1 1/108
2 7 1 1 3 0 2 0 8 0 1 0 1 0 1 0 1 -1/108
2 7 1 3 4 1 2 1 8 0 1 0 1 0 1 0 1 1/540
2 7 1 3 4 0 2 0 8 0 1 0 1 0 1 0 1 -1/540
2 7 1 3 4 1 2 0 8 0 1 0 1 0 1 0 1 1/1080
2 7 1 3 4 1 8 0 2 0 1 0 1 0 1 0 1 1/1080
2 7 1 3 8 2 7 0 1 0 1 0 1 0 1 0 1 1/1080
2 7 1 1 5 1 4 0 3 0 8 0 1 0 1 0 1 1/72
2 7 1 1 5 1 8 0 3 0 2 0 1 0 1 0 1 1/72
2 7 1 3 5 1 4 1 8 0 2 0 1 0 1 0 1 -1/2880
2 7 1 3 4 1 2 0 5 0 8 0 1 0 1 0 1 1/2880
2 7 1 3 4 2 7 1 8 0 1 0 1 0 1 0 1 -1/480
2 7 1 3 4 2 7 0 8 0 1 0 1 0 1 0 1 1/480
2 7 1 3 4 1 5 1 2 0 8 0 1 0 1 0 1 -1/360
2 7 1 4 5 1 8 0 3 0 2 0 1 0 1 0 1 -1/360
2 7 1 1 5 1 8 1 7 0 2 0 1 0 1 0 1 -1/108
2 7 1 1 5 0 8 0 7 0 2 0 1 0 1 0 1 -1/108
2 7 1 1 4 1 8 0 2 0 7 0 1 0 1 0 1 1/54
2 7 1 1 4 1 6 1 5 1 8 0 3 0 1 0 1 -1/270
2 7 1 1 4 0 5 0 2 0 6 0 8 0 1 0 1 1/270
2 7 1 3 5 1 2 1 7 1 8 0 1 0 1 0 1 -1/270
2 7 1 4 5 0 8 0 7 0 2 0 1 0 1 0 1 1/270
2 7 1 1 6 1 8 1 5 0 4 0 3 0 1 0 1 -1/90
2 7 1 1 6 1 5 0 3 0 8 0 2 0 1 0 1 1/90
2 7 1 3 5 1 2 1 7 0 8 0 1 0 1 0 1 -1/540
2 7 1 3 4 1 8 0 2 0 7 0 1 0 1 0 1 1/540
2 7 1 1 3 1 5 1 6 0 8 0 4 0 1 0 1 -1/540
2 7 1 1 8 1 4 0 3 0 6 0 2 0 1 0 1 1/540
2 7 1 4 5 1 7 1 8 0 2 0 1 0 1 0 1 -1/540
2 7 1 3 5 1 2 0 8 0 7 0 1 0 1 0 1 1/540
2 7 1 4 8 1 5 1 7 0 3 0 1 0 1 0 1 -1/108
2 7 1 5 8 1 4 0 3 0 7 0 1 0 1 0 1 -1/108
2 7 1 3 8 2 7 1 6 0 1 0 1 0 1 0 1 -1/540
2 7 1 3 8 2 7 0 6 0 1 0 1 0 1 0 1 1/540
2 7 1 1 8 1 2 1 5 0 4 0 3 0 1 0 1 -1/540
2 7 1 1 5 1 4 0 3 0 6 0 8 0 1 0 1 1/540
2 7 1 3 4 1 2 1 7 0 8 0 1 0 1 0 1 1/270
2 7 1 4 5 1 8 0 2 0 7 0 1 0 1 0 1 1/270
2 7 1 1 7 1 8 1 5 0 4 0 3 0 1 0 1 -1/108
2 7 1 1 6 1 4 0 3 0 7 0 8 0 1 0 1 1/108
2 7 1 1 7 1 5 1 6 0 8 0 4 0 1 0 1 -1/108
2 7 1 1 7 1 4 0 3 0 8 0 2 0 1 0 1 1/108
2 7 1 3 4 5 7 1 2 0 8 0 1 0 1 0 1 3/2*zeta(3)^2/pi^6-11/25920
2 7 1 3 5 4 7 1 8 0 2 0 1 0 1 0 1 3/2*zeta(3)^2/pi^6-11/25920
2 7 1 5 7 1 6 1 2 1 8 0 3 0 1 0 1 1/540
2 7 1 6 7 1 4 0 3 0 2 0 8 0 1 0 1 -1/540
2 7 1 3 5 4 7 1 8 1 2 0 1 0 1 0 1 1/540
2 7 1 3 4 5 8 0 2 0 7 0 1 0 1 0 1 1/540
2 7 1 4 8 1 6 1 5 1 2 0 3 0 1 0 1 -1/540
2 7 1 5 8 1 4 0 3 0 6 0 2 0 1 0 1 1/540
2 7 1 6 7 1 2 1 5 0 4 0 8 0 1 0 1 11/8640
2 7 1 3 7 1 8 1 5 0 4 0 2 0 1 0 1 -11/8640
2 7 1 6 8 1 2 1 5 0 4 0 3 0 1 0 1 1/864
2 7 1 3 8 1 6 1 5 0 4 0 2 0 1 0 1 -1/864
2 7 1 3 4 1 5 1 2 1 6 1 8 0 1 0 1 1/945
2 7 1 3 4 0 5 0 2 0 6 0 8 0 1 0 1 -1/945
2 7 1 3 6 1 5 1 8 1 4 0 2 0 1 0 1 -3/4*zeta(3)^2/pi^6+53/90720
2 7 1 3 4 1 2 0 6 0 8 0 5 0 1 0 1 -3/4*zeta(3)^2/pi^6+53/90720
2 7 1 3 8 2 5 1 7 1 4 0 1 0 1 0 1 -3/8*zeta(3)^2/pi^6+17/90720
2 7 1 3 4 2 8 0 5 0 7 0 1 0 1 0 1 -3/8*zeta(3)^2/pi^6+17/90720
2 7 1 3 5 2 4 1 7 1 8 0 1 0 1 0 1 -9/8*zeta(3)^2/pi^6+1/360
2 7 1 3 5 2 4 0 7 0 8 0 1 0 1 0 1 -9/8*zeta(3)^2/pi^6+1/360
2 7 1 3 6 1 2 1 5 0 8 0 4 0 1 0 1 -3/2*zeta(3)^2/pi^6+377/181440
2 7 1 3 5 1 6 1 8 0 2 0 4 0 1 0 1 -3/2*zeta(3)^2/pi^6+377/181440
2 7 1 3 4 2 7 1 5 0 8 0 1 0 1 0 1 -3/8*zeta(3)^2/pi^6-11/5040
2 7 1 3 5 2 7 1 8 0 4 0 1 0 1 0 1 -3/8*zeta(3)^2/pi^6-11/5040
2 7 1 3 5 2 4 1 8 0 7 0 1 0 1 0 1 -15/8*zeta(3)^2/pi^6+13/6480
2 7 1 3 5 1 2 1 8 0 6 0 4 0 1 0 1 -3/4*zeta(3)^2/pi^6+11/36288
C.1. ORIGINAL EXPANSION ⋆ 7aff mod ō(h̄ ) 537
2 7 1 3 6 1 4 1 5 0 8 0 2 0 1 0 1 -3/4*zeta(3)^2/pi^6+11/36288
2 7 1 3 4 1 2 1 5 0 6 0 8 0 1 0 1 3/4*zeta(3)^2/pi^6-53/90720
2 7 1 5 6 1 8 1 3 0 4 0 2 0 1 0 1 3/4*zeta(3)^2/pi^6-53/90720
2 7 1 4 5 1 8 1 6 1 2 0 3 0 1 0 1 1/420
2 7 1 5 6 1 4 0 8 0 3 0 2 0 1 0 1 -1/420
2 7 1 1 7 1 4 1 5 1 6 0 3 0 2 0 1 -1/540
2 7 1 1 4 1 5 0 6 0 3 0 7 0 2 0 1 -1/540
2 7 1 3 5 1 2 1 7 1 8 0 4 0 1 0 1 -1/540
2 7 1 5 6 1 4 0 8 0 7 0 2 0 1 0 1 -1/540
2 7 1 6 7 4 5 1 3 1 2 1 8 0 1 0 1 1/1890
2 7 1 6 7 4 5 0 3 0 2 0 8 0 1 0 1 -1/1890
2 7 1 3 4 1 5 1 2 1 6 0 8 0 1 0 1 1/1890
2 7 1 4 5 1 8 0 6 0 2 0 3 0 1 0 1 -1/1890
2 7 1 4 5 1 6 1 8 1 2 0 7 0 1 0 1 1/540
2 7 1 5 6 1 7 0 3 0 8 0 2 0 1 0 1 -1/540
2 7 1 3 6 1 8 1 7 0 4 0 2 0 1 0 1 -1/1080
2 7 1 3 6 1 2 1 5 0 7 0 8 0 1 0 1 -1/1080
2 7 1 3 8 2 7 1 6 0 4 0 1 0 1 0 1 -1/1080
2 7 1 3 8 2 7 1 5 0 6 0 1 0 1 0 1 -1/1080
2 7 1 3 6 1 7 1 5 0 8 0 2 0 1 0 1 -1/1080
2 7 1 3 6 1 2 1 7 0 4 0 8 0 1 0 1 -1/1080
2 7 1 1 7 1 5 1 6 0 3 0 2 0 4 0 1 -1/432
2 7 1 1 6 1 2 1 5 0 4 0 7 0 3 0 1 -1/2160
2 7 1 4 6 1 8 1 5 1 7 0 2 0 1 0 1 1/4320
2 7 1 3 5 1 2 0 8 0 6 0 7 0 1 0 1 1/4320
2 7 1 3 8 2 5 1 7 1 6 0 1 0 1 0 1 1/720
2 7 1 3 5 2 7 0 8 0 6 0 1 0 1 0 1 1/720
2 7 1 4 8 1 6 1 5 1 7 0 3 0 1 0 1 1/540
2 7 1 5 8 1 4 0 3 0 6 0 7 0 1 0 1 -1/540
2 7 1 5 8 2 4 1 3 1 6 1 7 0 1 0 1 -1/1890
2 7 1 5 8 2 4 0 3 0 6 0 7 0 1 0 1 1/1890
2 7 1 1 7 1 4 1 5 1 8 0 3 0 2 0 1 -1/540
2 7 1 1 5 1 4 0 3 0 6 0 7 0 8 0 1 -1/540
2 7 1 3 4 1 5 1 2 1 6 1 7 0 8 0 1 1/1890
2 7 1 4 5 1 8 0 6 0 2 0 7 0 3 0 1 1/1890
2 7 1 4 5 1 8 1 6 1 2 0 7 0 1 0 1 1/540
2 7 1 5 6 1 7 0 8 0 3 0 2 0 1 0 1 -1/540
2 7 1 1 7 1 4 1 5 1 6 0 8 0 2 0 1 -1/540
2 7 1 1 8 1 4 0 3 0 6 0 7 0 2 0 1 -1/540
2 7 1 4 6 1 5 1 2 1 8 0 3 0 1 0 1 -1/1890
2 7 1 3 5 1 4 0 6 0 2 0 8 0 1 0 1 -1/1890
2 7 1 6 7 1 8 1 5 0 4 0 3 0 1 0 1 7/2160
2 7 1 3 7 1 6 1 5 0 4 0 8 0 1 0 1 -7/2160
2 7 1 1 8 1 7 1 6 0 4 0 2 0 3 0 1 -43/8640
2 7 1 1 5 1 7 1 6 0 4 0 8 0 3 0 1 -43/8640
2 7 1 3 6 1 4 1 7 0 8 0 2 0 1 0 1 -1/4320
2 7 1 4 5 1 8 1 2 0 6 0 7 0 1 0 1 -1/4320
2 7 1 3 4 2 6 1 8 0 7 0 1 0 1 0 1 -1/720
2 7 1 3 5 2 6 1 8 0 7 0 1 0 1 0 1 -1/720
2 7 1 1 6 1 2 1 5 0 4 0 8 0 3 0 1 -19/8640
2 7 1 1 6 1 8 1 5 0 4 0 7 0 3 0 1 -19/8640
2 7 1 1 8 1 2 1 5 0 4 0 7 0 3 0 1 -1/4320
2 7 1 1 3 1 5 1 6 0 7 0 4 0 8 0 1 -1/4320
2 7 1 3 6 1 7 1 5 0 4 0 8 0 1 0 1 -23/4320
2 7 1 3 4 1 2 1 6 0 7 0 8 0 1 0 1 1/540
2 7 1 5 6 1 8 1 7 0 4 0 2 0 1 0 1 1/540
2 7 1 5 6 4 8 1 7 1 2 0 3 0 1 0 1 1/8*zeta(3)^2/pi^6+1/25920
2 7 1 6 7 4 5 1 2 0 3 0 8 0 1 0 1 1/8*zeta(3)^2/pi^6+1/25920
2 7 1 1 8 1 5 1 7 1 6 0 2 0 4 0 3 1/2880
2 7 1 1 6 1 5 1 7 0 8 0 2 0 3 0 4 -1/2880
2 7 1 5 6 4 8 1 7 1 3 0 2 0 1 0 1 15/16*zeta(3)^2/pi^6-13/10080
2 7 1 4 6 5 7 1 2 0 8 0 3 0 1 0 1 15/16*zeta(3)^2/pi^6-13/10080
2 7 1 5 7 4 8 2 3 1 6 0 1 0 1 0 1 25/16*zeta(3)^2/pi^6-7/1920
2 7 1 3 4 2 7 5 8 0 6 0 1 0 1 0 1 -25/16*zeta(3)^2/pi^6+7/1920
2 7 1 3 6 4 8 1 5 1 7 0 2 0 1 0 1 -9/8*zeta(3)^2/pi^6+29/22680
2 7 1 3 4 5 7 1 2 0 6 0 8 0 1 0 1 9/8*zeta(3)^2/pi^6-29/22680
2 7 1 3 5 6 7 1 8 1 2 0 4 0 1 0 1 3/8*zeta(3)^2/pi^6-7/5184
2 7 1 4 7 2 5 1 6 0 3 0 8 0 1 0 1 -3/8*zeta(3)^2/pi^6+7/5184
2 7 1 6 8 1 7 1 5 1 2 0 4 0 3 0 1 1/8640
2 7 1 3 8 1 6 1 5 0 4 0 7 0 2 0 1 1/8640
2 7 1 4 8 1 7 1 5 1 6 0 2 0 3 0 1 1/4320
2 7 1 6 8 1 2 1 5 0 4 0 7 0 3 0 1 1/4320
2 7 1 4 6 1 8 1 2 1 7 0 5 0 3 0 1 -7/8*zeta(3)^2/pi^6+17/13440
538 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 7 1 3 6 1 7 1 5 0 8 0 2 0 4 0 1 7/8*zeta(3)^2/pi^6-17/13440
2 7 1 6 8 4 5 1 2 1 3 0 7 0 1 0 1 -1/4*zeta(3)^2/pi^6+1/90720
2 7 1 4 8 5 6 1 7 0 3 0 2 0 1 0 1 -1/4*zeta(3)^2/pi^6+1/90720
2 7 1 3 4 1 7 1 2 1 6 0 8 0 5 0 1 -1/4*zeta(3)^2/pi^6+89/60480
2 7 1 5 6 1 8 1 7 0 4 0 2 0 3 0 1 -1/4*zeta(3)^2/pi^6+89/60480
2 7 1 3 8 1 7 1 2 1 6 0 5 0 4 0 1 1/1440
2 7 1 5 8 1 7 1 6 0 4 0 2 0 3 0 1 1/1440
2 7 1 3 5 4 7 1 6 1 2 0 8 0 1 0 1 -3/8*zeta(3)^2/pi^6+17/15120
2 7 1 6 8 2 5 1 7 0 3 0 4 0 1 0 1 3/8*zeta(3)^2/pi^6-17/15120
2 7 1 3 8 5 7 1 6 1 2 0 4 0 1 0 1 1/1728
2 7 1 6 8 2 7 1 5 0 4 0 3 0 1 0 1 -1/1728
2 7 1 3 7 1 6 1 8 1 4 1 5 0 2 0 1 5/16*zeta(3)^2/pi^6-7/17280
2 7 1 3 4 1 2 0 7 0 8 0 5 0 6 0 1 -5/16*zeta(3)^2/pi^6+7/17280
2 7 1 3 4 2 7 1 6 1 8 1 5 0 1 0 1 15/16*zeta(3)^2/pi^6-31/30240
2 7 1 3 4 2 7 0 6 0 8 0 5 0 1 0 1 -15/16*zeta(3)^2/pi^6+31/30240
2 7 1 3 4 2 5 1 8 1 6 1 7 0 1 0 1 11/16*zeta(3)^2/pi^6-179/181440
2 7 1 3 4 2 5 0 8 0 6 0 7 0 1 0 1 -11/16*zeta(3)^2/pi^6+179/181440
2 7 1 4 6 1 8 1 5 1 7 0 2 0 3 0 1 -3/16*zeta(3)^2/pi^6+61/90720
2 7 1 3 6 1 2 1 5 0 8 0 7 0 4 0 1 3/16*zeta(3)^2/pi^6-61/90720
2 7 1 3 5 2 6 1 8 1 7 0 4 0 1 0 1 3/8*zeta(3)^2/pi^6+23/181440
2 7 1 3 4 2 5 1 6 0 7 0 8 0 1 0 1 -3/8*zeta(3)^2/pi^6-23/181440
2 7 1 3 8 2 5 1 7 1 6 0 4 0 1 0 1 3/4*zeta(3)^2/pi^6-1/40320
2 7 1 3 7 2 5 1 6 0 4 0 8 0 1 0 1 -3/4*zeta(3)^2/pi^6+1/40320
2 7 1 6 7 1 5 1 8 1 4 0 3 0 2 0 1 -1/2*zeta(3)^2/pi^6+191/181440
2 7 1 3 4 1 5 1 2 0 7 0 8 0 6 0 1 -1/2*zeta(3)^2/pi^6+191/181440
2 7 1 3 4 1 7 1 2 1 8 0 5 0 6 0 1 -1/4*zeta(3)^2/pi^6+157/181440
2 7 1 5 6 1 7 1 3 0 4 0 2 0 8 0 1 -1/4*zeta(3)^2/pi^6+157/181440
2 7 1 3 8 2 6 1 5 1 7 0 4 0 1 0 1 -15/16*zeta(3)^2/pi^6+319/181440
2 7 1 3 8 2 4 1 5 0 6 0 7 0 1 0 1 15/16*zeta(3)^2/pi^6-319/181440
2 7 1 4 6 1 5 1 7 1 8 0 2 0 3 0 1 zeta(3)^2/pi^6-59/40320
2 7 1 3 6 1 2 1 5 0 7 0 4 0 8 0 1 -zeta(3)^2/pi^6+59/40320
2 7 1 3 5 2 6 1 8 1 4 0 7 0 1 0 1 3/16*zeta(3)^2/pi^6-11/24192
2 7 1 3 4 2 5 1 8 0 6 0 7 0 1 0 1 -3/16*zeta(3)^2/pi^6+11/24192
2 7 1 3 6 1 5 1 7 1 4 0 2 0 8 0 1 3/16*zeta(3)^2/pi^6-1/2880
2 7 1 3 5 1 2 1 8 0 7 0 4 0 6 0 1 -3/16*zeta(3)^2/pi^6+1/2880
2 7 1 3 5 2 4 1 6 1 7 0 8 0 1 0 1 -1/8*zeta(3)^2/pi^6+121/90720
2 7 1 3 5 2 6 1 7 0 8 0 4 0 1 0 1 1/8*zeta(3)^2/pi^6-121/90720
2 7 1 3 8 2 5 1 6 1 4 0 7 0 1 0 1 5/8*zeta(3)^2/pi^6-353/362880
2 7 1 3 8 2 5 1 7 0 6 0 4 0 1 0 1 -5/8*zeta(3)^2/pi^6+353/362880
2 7 1 3 5 4 6 1 8 1 2 0 7 0 1 0 1 409/362880
2 7 1 4 6 2 5 1 7 0 3 0 8 0 1 0 1 409/362880
2 7 1 4 6 1 7 1 8 1 2 0 5 0 3 0 1 1/1440
2 7 1 3 6 1 7 1 5 0 4 0 8 0 2 0 1 -1/1440
2 7 1 4 5 1 7 1 8 1 6 0 2 0 3 0 1 1/8640
2 7 1 6 7 1 2 1 5 0 4 0 8 0 3 0 1 1/8640
2 7 1 4 7 1 5 1 2 1 6 0 8 0 3 0 1 5/8*zeta(3)^2/pi^6-149/120960
2 7 1 3 5 1 6 1 8 0 2 0 7 0 4 0 1 -5/8*zeta(3)^2/pi^6+149/120960
2 7 1 4 6 1 7 1 5 1 2 0 8 0 3 0 1 -1/1080
2 7 1 3 6 1 8 1 5 0 4 0 7 0 2 0 1 1/1080
2 7 1 4 6 1 5 1 2 1 8 0 7 0 3 0 1 1/4*zeta(3)^2/pi^6-29/72576
2 7 1 3 6 1 4 1 5 0 7 0 2 0 8 0 1 -1/4*zeta(3)^2/pi^6+29/72576
2 7 1 1 8 1 4 1 5 1 6 0 7 0 3 0 2 -1/2880
2 7 1 1 6 1 2 1 5 0 4 0 7 0 8 0 3 1/2880
2 7 1 3 4 1 5 1 2 1 6 0 7 0 8 0 1 1/4*zeta(3)^2/pi^6-13/45360
2 7 1 5 6 1 8 1 3 0 7 0 2 0 4 0 1 1/4*zeta(3)^2/pi^6-13/45360
2 7 1 1 5 1 4 0 3 0 2 0 1 0 1 0 1 1/432
2 7 1 1 8 1 6 1 5 0 4 0 3 0 1 0 1 -1/216
2 7 1 1 6 1 5 0 8 0 3 0 2 0 1 0 1 1/216
2 7 1 4 5 1 6 1 8 1 2 0 3 0 1 0 1 1/540
2 7 1 5 6 1 4 0 3 0 8 0 2 0 1 0 1 -1/540
2 7 1 3 6 1 2 1 5 0 4 0 8 0 1 0 1 -1/1080
2 7 1 3 6 1 8 1 5 0 4 0 2 0 1 0 1 -1/1080
2 7 1 3 8 2 7 1 5 0 4 0 1 0 1 0 1 -1/1080
2 7 1 1 5 1 7 1 6 0 8 0 4 0 3 0 1 -1/432
2 7 1 1 8 1 7 1 6 0 2 0 4 0 3 0 1 -1/432
2 7 1 4 6 1 7 1 5 1 8 0 2 0 3 0 1 1/8640
2 7 1 3 6 1 2 1 5 0 4 0 7 0 8 0 1 -1/8640
2 7 1 3 8 2 5 1 6 1 7 0 4 0 1 0 1 1/1440
2 7 1 3 7 2 5 1 6 0 8 0 4 0 1 0 1 -1/1440
2 7 1 4 5 1 7 1 2 1 6 0 8 0 3 0 1 -1/1080
2 7 1 6 7 1 8 1 5 0 4 0 2 0 3 0 1 -1/1080
2 7 1 1 7 1 6 1 5 0 4 0 3 0 2 0 1 -1/1296
C.2. REDUCED EXPANSION ⋆redaff mod ō(h̄
7) 539
C.2 Reduced expansion ⋆redaff mod ō(h̄
7)
540 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
Encoding 2. In the format described in 2 6 1 1 4 1 7 1 3 0 6 0 2 0 5 -2/567
Chapter 11, Implementation 1: 2 6 1 3 5 1 7 1 6 0 4 0 2 0 1 -31/3780
2 6 1 1 7 1 4 1 5 0 6 0 3 0 2 -1/360
h^0: 2 6 1 3 5 1 2 1 6 0 4 0 7 0 1 -7/1620
2 0 1 1 2 6 1 4 5 1 6 1 2 0 7 0 3 0 1 -11/2160
h^1: 2 6 1 1 7 1 6 1 5 0 4 0 3 0 2 -1/1296
2 1 1 0 1 1 2 6 1 1 4 1 6 1 7 0 3 0 2 0 1 4/315
h^2: 2 6 1 4 6 1 7 1 5 0 3 0 1 0 1 -2/105
2 2 1 1 3 0 2 -1/6 2 6 1 1 6 1 4 1 7 0 3 0 5 0 1 -2/315
2 2 1 1 3 0 1 -1/3 2 6 1 1 4 1 7 1 5 0 6 0 2 0 1 -1/180
2 2 1 0 3 0 1 1/3 2 6 1 1 4 1 7 1 5 0 6 0 3 0 1 -4/105
2 2 1 0 1 0 1 1/2 2 6 1 1 4 1 6 1 3 0 7 0 2 0 1 1/270
h^3: 2 6 1 1 4 1 6 1 3 0 7 0 5 0 1 2/135
2 3 1 1 3 0 2 0 1 -1/6 2 6 1 4 6 1 5 1 7 0 3 0 1 0 1 -1/45
2 3 1 1 4 0 2 0 1 -1/3 2 6 1 3 7 2 6 1 5 0 1 0 1 0 1 -11/1080
2 3 1 1 4 0 1 0 1 -1/3 2 6 1 1 6 1 4 1 7 0 3 0 2 0 1 -2/135
2 3 1 0 4 0 1 0 1 1/3 2 6 1 1 7 1 6 1 5 0 4 0 3 0 1 -1/216
2 3 1 0 1 0 1 0 1 1/6 2 6 1 1 5 1 4 0 6 0 3 0 7 0 1 -4/315
h^4: 2 6 1 5 6 1 4 0 7 0 3 0 1 0 1 -2/105
2 4 1 1 5 1 4 0 3 0 2 1/72 2 6 1 1 5 1 2 0 3 0 6 0 7 0 1 2/315
2 4 1 1 3 1 4 0 5 0 2 -2/45 2 6 1 1 5 1 2 0 7 0 6 0 3 0 1 1/180
2 4 1 1 5 1 4 0 2 0 3 11/180 2 6 1 1 5 1 2 0 6 0 7 0 3 0 1 4/105
2 4 1 1 5 1 4 0 3 0 1 1/18 2 6 1 1 7 1 4 0 6 0 3 0 5 0 1 -1/270
2 4 1 1 3 1 5 0 2 0 1 -2/45 2 6 1 1 7 1 2 0 6 0 3 0 5 0 1 -2/135
2 4 1 1 3 1 4 0 5 0 1 2/15 2 6 1 5 6 1 4 0 3 0 7 0 1 0 1 -1/45
2 4 1 1 4 0 5 0 2 0 1 -1/18 2 6 1 3 7 2 6 0 5 0 1 0 1 0 1 11/1080
2 4 1 1 3 0 4 0 5 0 1 2/45 2 6 1 1 5 1 4 0 3 0 6 0 7 0 1 2/135
2 4 1 1 5 0 4 0 2 0 1 -2/15 2 6 1 1 6 1 5 0 7 0 3 0 2 0 1 1/216
2 4 1 1 3 0 2 0 1 0 1 -1/12 2 6 1 1 4 1 2 0 5 0 3 0 1 0 1 1/120
2 4 1 1 5 0 2 0 1 0 1 -1/3 2 6 1 3 4 1 7 0 6 0 1 0 1 0 1 2/45
2 4 1 1 5 0 4 0 1 0 1 -1/9 2 6 1 1 5 1 4 0 7 0 3 0 1 0 1 2/45
2 4 1 1 5 0 1 0 1 0 1 -1/6 2 6 1 1 7 1 5 0 3 0 2 0 1 0 1 2/45
2 4 1 0 5 0 1 0 1 0 1 1/6 2 6 1 1 5 1 6 0 3 0 7 0 1 0 1 1/18
2 4 1 0 1 0 1 0 1 0 1 1/24 2 6 1 3 7 1 6 0 5 0 1 0 1 0 1 2/45
2 4 1 1 5 1 4 0 1 0 1 1/18 2 6 1 4 7 1 6 0 5 0 1 0 1 0 1 -2/45
2 4 1 0 5 0 4 0 1 0 1 1/18 2 6 1 1 6 1 2 0 7 0 3 0 1 0 1 4/135
2 4 1 1 4 1 5 1 3 0 1 1/45 2 6 1 1 7 1 4 0 5 0 6 0 1 0 1 4/135
2 4 1 0 4 0 5 0 3 0 1 -1/45 2 6 1 3 7 2 6 0 1 0 1 0 1 0 1 11/720
2 4 1 1 4 1 5 1 3 0 2 1/90 2 6 1 1 7 1 4 0 3 0 2 0 1 0 1 1/18
2 4 1 1 3 0 5 0 2 0 4 1/90 2 6 1 1 6 1 5 0 7 0 3 0 1 0 1 1/54
h^5: 2 6 1 1 5 1 4 0 3 0 2 0 1 0 1 1/144
2 5 1 1 4 1 2 0 5 0 3 0 1 1/60 2 6 1 1 5 1 7 1 3 1 4 0 2 0 6 -1/378
2 5 1 3 4 1 6 0 5 0 1 0 1 2/15 2 6 1 3 6 1 5 1 2 1 4 0 7 0 1 -31/11340
2 5 1 1 6 1 2 0 5 0 3 0 1 2/45 2 6 1 1 7 1 4 1 5 1 6 0 3 0 2 -1/540
2 5 1 1 3 1 4 0 5 0 6 0 1 2/45 2 6 1 3 5 2 4 1 6 1 7 0 1 0 1 1/567
2 5 1 1 5 1 4 0 3 0 6 0 1 1/18 2 6 1 1 4 1 2 0 7 0 3 0 5 0 6 -1/378
2 5 1 3 4 1 6 0 2 0 1 0 1 11/360 2 6 1 3 4 1 7 0 6 0 2 0 5 0 1 -31/11340
2 5 1 1 5 1 4 0 3 0 2 0 1 1/72 2 6 1 1 4 1 5 0 6 0 3 0 7 0 2 -1/540
2 5 1 1 3 1 6 0 2 0 1 0 1 -2/45 2 6 1 3 5 2 4 0 6 0 7 0 1 0 1 1/567
2 5 1 1 3 1 4 0 6 0 1 0 1 2/15 2 6 1 1 7 0 6 0 1 0 1 0 1 0 1 -1/18
2 5 1 1 4 1 5 0 6 0 1 0 1 1/9 2 6 1 1 3 0 2 0 1 0 1 0 1 0 1 -1/144
2 5 1 1 4 1 6 0 2 0 1 0 1 1/18 2 6 1 1 3 0 7 0 1 0 1 0 1 0 1 -1/18
2 5 1 1 3 0 4 0 6 0 1 0 1 2/45 2 6 1 3 7 1 6 0 1 0 1 0 1 0 1 1/15
2 5 1 1 6 0 4 0 2 0 1 0 1 -2/15 2 6 1 1 3 1 7 0 2 0 1 0 1 0 1 2/45
2 5 1 1 5 0 6 0 2 0 1 0 1 -1/9 2 6 1 1 4 1 6 0 7 0 1 0 1 0 1 1/9
2 5 1 1 4 0 6 0 2 0 1 0 1 -1/18 2 6 1 1 7 1 6 0 5 0 1 0 1 0 1 1/54
2 5 1 1 6 0 2 0 1 0 1 0 1 -1/6 2 6 1 1 4 1 7 0 2 0 1 0 1 0 1 1/36
2 5 1 1 6 0 5 0 1 0 1 0 1 -1/9 2 6 1 3 7 0 6 0 1 0 1 0 1 0 1 1/15
2 5 1 1 3 0 2 0 1 0 1 0 1 -1/36 2 6 1 1 3 0 4 0 7 0 1 0 1 0 1 -2/45
2 5 1 1 4 1 5 1 3 0 2 0 1 1/90 2 6 1 1 7 0 2 0 6 0 1 0 1 0 1 -1/9
2 5 1 1 5 1 4 1 6 0 3 0 1 2/45 2 6 1 1 7 0 6 0 5 0 1 0 1 0 1 -1/54
2 5 1 1 4 1 6 1 5 0 3 0 1 -1/45 2 6 1 1 3 0 2 0 7 0 1 0 1 0 1 -1/36
2 5 1 1 3 0 5 0 2 0 4 0 1 1/90 2 6 1 1 4 1 5 1 3 0 2 0 1 0 1 1/180
2 5 1 1 4 0 5 0 6 0 2 0 1 -1/45 2 6 1 4 5 1 6 1 7 0 1 0 1 0 1 -2/45
2 5 1 1 4 0 2 0 5 0 6 0 1 2/45 2 6 1 1 7 1 4 1 6 0 3 0 1 0 1 -4/135
2 5 1 0 1 0 1 0 1 0 1 0 1 1/120 2 6 1 1 4 1 7 1 3 0 6 0 1 0 1 1/135
2 5 1 1 6 0 1 0 1 0 1 0 1 -1/18 2 6 1 1 5 1 7 1 6 0 2 0 1 0 1 -1/108
2 5 1 0 6 0 1 0 1 0 1 0 1 1/18 2 6 1 1 5 1 4 1 7 0 3 0 1 0 1 2/45
2 5 1 1 6 1 5 0 1 0 1 0 1 1/18 2 6 1 1 4 1 7 1 5 0 3 0 1 0 1 -1/45
2 5 1 0 6 0 5 0 1 0 1 0 1 1/18 2 6 1 1 3 0 5 0 2 0 4 0 1 0 1 1/180
2 5 1 1 4 1 6 1 3 0 1 0 1 1/45 2 6 1 4 7 0 6 0 5 0 1 0 1 0 1 2/45
2 5 1 0 4 0 6 0 3 0 1 0 1 -1/45 2 6 1 1 4 0 7 0 5 0 6 0 1 0 1 -4/135
h^6: 2 6 1 1 6 0 5 0 7 0 4 0 1 0 1 1/135
C.2. REDUCED EXPANSION ⋆redaff mod ō(h̄
7) 541
2 6 1 1 5 0 7 0 6 0 2 0 1 0 1 -1/108 2 7 1 4 8 1 5 1 7 0 3 0 1 0 1 0 1 -1/45
2 6 1 1 4 0 5 0 7 0 2 0 1 0 1 -1/45 2 7 1 3 8 2 7 1 6 0 1 0 1 0 1 0 1 -11/1080
2 6 1 1 4 0 2 0 5 0 7 0 1 0 1 2/45 2 7 1 1 8 1 2 1 5 0 4 0 3 0 1 0 1 -2/135
2 6 1 1 7 1 4 1 5 1 6 0 3 0 1 -1/270 2 7 1 1 7 1 5 1 6 0 8 0 4 0 1 0 1 -1/54
2 6 1 1 4 1 6 1 5 1 7 0 3 0 1 -1/270 2 7 1 1 8 1 6 1 5 0 4 0 3 0 1 0 1 -1/216
2 6 1 3 6 1 5 1 7 1 4 0 1 0 1 -2/315 2 7 1 1 7 1 6 1 2 0 8 0 4 0 1 0 1 2/135
2 6 1 1 5 1 4 1 7 1 6 0 3 0 1 -4/315 2 7 1 1 3 1 6 1 7 0 8 0 4 0 1 0 1 -1/135
2 6 1 1 4 0 7 0 5 0 6 0 2 0 1 1/270 2 7 1 1 5 1 4 0 6 0 3 0 8 0 1 0 1 -4/315
2 6 1 1 4 0 5 0 2 0 6 0 7 0 1 1/270 2 7 1 4 8 1 5 0 3 0 7 0 1 0 1 0 1 -2/105
2 6 1 3 6 0 5 0 7 0 4 0 1 0 1 -2/315 2 7 1 1 5 1 2 0 6 0 8 0 3 0 1 0 1 2/105
2 6 1 1 4 0 6 0 5 0 7 0 2 0 1 4/315 2 7 1 1 8 1 2 0 6 0 3 0 5 0 1 0 1 -8/945
2 6 1 0 1 0 1 0 1 0 1 0 1 0 1 1/720 2 7 1 1 8 1 5 0 3 0 6 0 2 0 1 0 1 2/105
2 6 1 1 7 0 1 0 1 0 1 0 1 0 1 -1/72 2 7 1 5 8 1 7 0 3 0 6 0 1 0 1 0 1 -2/45
2 6 1 0 7 0 1 0 1 0 1 0 1 0 1 1/72 2 7 1 1 5 1 8 0 3 0 6 0 7 0 1 0 1 4/135
2 6 1 1 7 1 6 0 1 0 1 0 1 0 1 1/36 2 7 1 1 8 1 4 0 6 0 3 0 5 0 1 0 1 -1/270
2 6 1 0 7 0 6 0 1 0 1 0 1 0 1 1/36 2 7 1 1 5 1 2 0 8 0 6 0 3 0 1 0 1 1/180
2 6 1 1 4 1 7 1 3 0 1 0 1 0 1 1/90 2 7 1 1 3 1 5 0 8 0 6 0 7 0 1 0 1 1/135
2 6 1 1 7 1 6 1 5 0 1 0 1 0 1 -1/162 2 7 1 3 5 1 8 0 7 0 6 0 1 0 1 0 1 1/45
2 6 1 0 4 0 7 0 3 0 1 0 1 0 1 -1/90 2 7 1 1 7 1 6 0 3 0 8 0 2 0 1 0 1 1/45
2 6 1 0 7 0 6 0 5 0 1 0 1 0 1 1/162 2 7 1 5 8 1 4 0 3 0 7 0 1 0 1 0 1 -1/45
2 6 1 1 6 1 7 1 5 1 3 1 4 0 1 -2/945 2 7 1 3 8 2 7 0 6 0 1 0 1 0 1 0 1 11/1080
2 6 1 0 6 0 7 0 5 0 3 0 4 0 1 2/945 2 7 1 1 5 1 4 0 3 0 6 0 8 0 1 0 1 2/135
2 6 1 1 7 1 4 1 5 1 6 0 1 0 1 -1/135 2 7 1 1 7 1 4 0 3 0 8 0 2 0 1 0 1 1/54
2 6 1 0 7 0 4 0 5 0 6 0 1 0 1 -1/135 2 7 1 1 6 1 5 0 8 0 3 0 2 0 1 0 1 1/216
2 6 1 1 6 1 7 1 5 1 3 1 4 0 2 -1/945 2 7 1 1 7 1 5 0 6 0 8 0 3 0 1 0 1 1/135
2 6 1 1 3 0 7 0 2 0 6 0 4 0 5 -1/945 2 7 1 1 8 1 5 0 3 0 6 0 7 0 1 0 1 -2/135
h^7: 2 7 1 1 4 1 8 1 5 1 7 0 3 0 2 0 1 -1/270
2 7 1 1 6 1 5 1 3 0 7 0 8 0 4 0 1 -1/180 2 7 1 1 5 1 7 1 3 1 4 0 2 0 6 0 1 11/7560
2 7 1 1 3 1 5 1 2 0 7 0 4 0 6 0 1 53/11340 2 7 1 1 7 1 4 1 5 1 8 0 3 0 6 0 1 -2/315
2 7 1 1 5 1 2 1 6 0 7 0 3 0 4 0 1 -31/3780 2 7 1 1 5 1 7 1 3 1 4 0 2 0 8 0 1 -4/315
2 7 1 1 3 1 5 1 6 0 4 0 7 0 8 0 1 2/135 2 7 1 3 4 1 8 1 5 1 6 0 7 0 1 0 1 -4/315
2 7 1 1 3 1 6 1 2 0 7 0 8 0 4 0 1 -8/945 2 7 1 1 5 1 4 1 7 1 6 0 3 0 2 0 1 -31/7560
2 7 1 1 3 1 5 1 8 0 7 0 4 0 6 0 1 -29/945 2 7 1 1 5 1 4 1 8 1 6 0 7 0 3 0 1 8/945
2 7 1 3 6 1 4 1 5 0 7 0 8 0 1 0 1 -46/945 2 7 1 1 5 1 7 1 3 1 4 0 8 0 6 0 1 -2/945
2 7 1 3 5 1 8 1 7 0 6 0 4 0 1 0 1 -32/945 2 7 1 1 7 1 4 1 5 1 6 0 3 0 2 0 1 -1/540
2 7 1 3 6 1 7 1 5 0 2 0 8 0 1 0 1 8/315 2 7 1 4 6 1 5 1 2 1 8 0 3 0 1 0 1 -37/15120
2 7 1 3 6 1 4 1 5 0 8 0 2 0 1 0 1 -191/22680 2 7 1 1 7 1 5 1 6 1 8 0 4 0 3 0 1 -1/135
2 7 1 3 6 1 2 1 7 0 4 0 8 0 1 0 1 -11/1080 2 7 1 1 7 1 5 1 8 1 6 0 4 0 2 0 1 1/270
2 7 1 1 6 1 2 1 5 0 4 0 7 0 3 0 1 -1/360 2 7 1 1 5 1 4 0 6 0 8 0 7 0 3 0 1 -1/270
2 7 1 1 8 1 2 1 5 0 4 0 7 0 3 0 1 -1/135 2 7 1 1 4 1 2 0 7 0 3 0 5 0 6 0 1 11/7560
2 7 1 1 3 1 5 1 6 0 7 0 4 0 8 0 1 -1/135 2 7 1 1 5 1 2 0 3 0 6 0 7 0 8 0 1 -2/315
2 7 1 3 6 1 7 1 5 0 4 0 8 0 1 0 1 -1/45 2 7 1 1 4 1 8 0 7 0 3 0 5 0 6 0 1 -4/315
2 7 1 3 8 2 7 1 5 0 4 0 1 0 1 0 1 -11/2160 2 7 1 4 5 1 7 0 8 0 6 0 3 0 1 0 1 4/315
2 7 1 1 8 1 7 1 6 0 2 0 4 0 3 0 1 -1/216 2 7 1 1 5 1 4 0 7 0 6 0 3 0 2 0 1 -31/7560
2 7 1 1 7 1 6 1 5 0 4 0 3 0 2 0 1 -1/1296 2 7 1 1 5 1 2 0 6 0 7 0 3 0 8 0 1 8/945
2 7 1 4 8 2 6 3 7 0 1 0 1 0 1 0 1 31/22680 2 7 1 1 8 1 2 0 7 0 3 0 5 0 6 0 1 -2/945
2 7 1 3 5 1 4 1 7 0 6 0 8 0 1 0 1 -16/945 2 7 1 1 4 1 5 0 6 0 3 0 7 0 2 0 1 -1/540
2 7 1 4 5 2 6 1 8 0 7 0 1 0 1 0 1 16/945 2 7 1 3 5 1 4 0 6 0 2 0 8 0 1 0 1 -37/15120
2 7 1 1 4 1 8 0 2 0 7 0 1 0 1 0 1 1/54 2 7 1 1 6 1 5 0 7 0 8 0 2 0 3 0 1 1/270
2 7 1 1 5 1 4 0 3 0 2 0 1 0 1 0 1 1/432 2 7 1 1 6 1 5 0 3 0 7 0 2 0 8 0 1 -1/135
2 7 1 1 5 1 4 0 2 0 3 0 1 0 1 0 1 1/360 2 7 1 1 8 0 2 0 1 0 1 0 1 0 1 0 1 -1/72
2 7 1 3 8 1 4 0 2 0 1 0 1 0 1 0 1 1/135 2 7 1 1 8 0 7 0 1 0 1 0 1 0 1 0 1 -1/54
2 7 1 1 5 1 8 0 3 0 4 0 1 0 1 0 1 1/15 2 7 1 1 3 0 2 0 1 0 1 0 1 0 1 0 1 -1/720
2 7 1 1 3 1 4 0 5 0 8 0 1 0 1 0 1 -1/45 2 7 1 3 8 1 7 0 1 0 1 0 1 0 1 0 1 1/45
2 7 1 1 5 1 8 0 3 0 7 0 1 0 1 0 1 1/18 2 7 1 1 3 1 8 0 2 0 1 0 1 0 1 0 1 2/135
2 7 1 1 7 1 8 0 5 0 3 0 1 0 1 0 1 4/135 2 7 1 1 4 1 7 0 8 0 1 0 1 0 1 0 1 1/18
2 7 1 4 8 1 7 0 6 0 1 0 1 0 1 0 1 -2/135 2 7 1 1 8 1 7 0 6 0 1 0 1 0 1 0 1 1/54
2 7 1 3 8 1 7 0 6 0 1 0 1 0 1 0 1 2/135 2 7 1 1 4 1 8 0 2 0 1 0 1 0 1 0 1 1/108
2 7 1 1 3 1 5 0 7 0 8 0 1 0 1 0 1 4/135 2 7 1 3 8 0 7 0 1 0 1 0 1 0 1 0 1 1/45
2 7 1 1 5 1 7 0 8 0 6 0 1 0 1 0 1 1/27 2 7 1 1 3 0 4 0 8 0 1 0 1 0 1 0 1 -2/135
2 7 1 1 5 1 4 0 3 0 8 0 1 0 1 0 1 1/36 2 7 1 1 8 0 2 0 7 0 1 0 1 0 1 0 1 -1/18
2 7 1 1 8 1 2 1 6 0 4 0 3 0 1 0 1 4/315 2 7 1 1 8 0 7 0 6 0 1 0 1 0 1 0 1 -1/54
2 7 1 3 8 1 5 1 7 0 4 0 1 0 1 0 1 -2/105 2 7 1 1 3 0 2 0 8 0 1 0 1 0 1 0 1 -1/108
2 7 1 1 3 1 5 1 8 0 6 0 4 0 1 0 1 -2/105 2 7 1 1 4 1 5 1 3 0 2 0 1 0 1 0 1 1/540
2 7 1 1 3 1 4 1 5 0 6 0 8 0 1 0 1 8/945 2 7 1 4 6 1 7 1 8 0 1 0 1 0 1 0 1 -2/45
2 7 1 1 6 1 5 1 3 0 8 0 4 0 1 0 1 -2/105 2 7 1 1 4 1 7 1 8 0 2 0 1 0 1 0 1 -4/135
2 7 1 4 8 1 5 1 6 0 7 0 1 0 1 0 1 -2/45 2 7 1 1 3 1 4 1 7 0 8 0 1 0 1 0 1 1/135
2 7 1 1 7 1 5 1 2 0 8 0 4 0 1 0 1 -4/135 2 7 1 1 5 1 8 1 7 0 6 0 1 0 1 0 1 -1/54
2 7 1 1 3 1 6 1 2 0 8 0 4 0 1 0 1 1/270 2 7 1 1 5 1 8 1 7 0 2 0 1 0 1 0 1 -1/108
2 7 1 1 8 1 5 1 3 0 6 0 4 0 1 0 1 -1/180 2 7 1 1 5 1 4 1 8 0 3 0 1 0 1 0 1 1/45
2 7 1 1 7 1 8 1 3 0 6 0 4 0 1 0 1 -1/135 2 7 1 1 3 1 5 1 8 0 4 0 1 0 1 0 1 -1/90
2 7 1 4 5 1 7 1 8 0 6 0 1 0 1 0 1 -1/45 2 7 1 1 3 0 5 0 2 0 4 0 1 0 1 0 1 1/540
2 7 1 1 7 1 5 1 6 0 4 0 8 0 1 0 1 -1/45 2 7 1 4 8 0 7 0 6 0 1 0 1 0 1 0 1 2/45
542 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
2 7 1 1 4 0 8 0 5 0 7 0 1 0 1 0 1 -4/135
2 7 1 1 7 0 5 0 8 0 4 0 1 0 1 0 1 1/135
2 7 1 1 6 0 8 0 7 0 2 0 1 0 1 0 1 -1/54
2 7 1 1 5 0 8 0 7 0 2 0 1 0 1 0 1 -1/108
2 7 1 1 4 0 5 0 8 0 2 0 1 0 1 0 1 -1/90
2 7 1 1 4 0 2 0 5 0 8 0 1 0 1 0 1 1/45
2 7 1 1 5 1 8 1 3 1 4 0 2 0 1 0 1 2/315
2 7 1 1 8 1 4 1 5 1 6 0 3 0 1 0 1 -1/270
2 7 1 1 4 1 6 1 5 1 7 0 8 0 1 0 1 -1/135
2 7 1 1 4 1 5 1 6 1 8 0 3 0 1 0 1 -2/105
2 7 1 1 4 1 6 1 5 1 8 0 3 0 1 0 1 -1/270
2 7 1 1 8 1 6 1 5 1 7 0 4 0 1 0 1 -2/135
2 7 1 1 5 1 7 1 8 1 6 0 4 0 1 0 1 1/135
2 7 1 4 8 1 2 1 5 1 7 0 1 0 1 0 1 2/315
2 7 1 1 3 0 6 0 8 0 4 0 5 0 1 0 1 -2/315
2 7 1 1 4 0 8 0 5 0 6 0 2 0 1 0 1 1/270
2 7 1 1 8 0 5 0 2 0 6 0 7 0 1 0 1 1/135
2 7 1 1 4 0 6 0 5 0 8 0 2 0 1 0 1 2/105
2 7 1 1 4 0 5 0 2 0 6 0 8 0 1 0 1 1/270
2 7 1 1 5 0 6 0 7 0 8 0 2 0 1 0 1 -1/135
2 7 1 1 5 0 8 0 2 0 6 0 7 0 1 0 1 2/135
2 7 1 4 8 0 2 0 5 0 7 0 1 0 1 0 1 2/315
2 7 1 1 6 1 7 1 5 1 3 1 4 0 2 0 1 -1/945
2 7 1 1 7 1 6 1 8 1 4 1 5 0 3 0 1 -4/945
2 7 1 3 4 1 8 1 5 1 6 1 7 0 1 0 1 2/945
2 7 1 1 6 1 4 1 8 1 7 1 5 0 3 0 1 4/945
2 7 1 1 8 1 6 1 7 1 4 1 5 0 2 0 1 -2/945
2 7 1 1 3 0 7 0 2 0 6 0 4 0 5 0 1 -1/945
2 7 1 1 4 0 2 0 7 0 8 0 5 0 6 0 1 -4/945
2 7 1 3 4 0 8 0 5 0 6 0 7 0 1 0 1 -2/945
2 7 1 1 3 0 8 0 7 0 2 0 5 0 6 0 1 -2/945
2 7 1 1 4 0 7 0 5 0 8 0 2 0 6 0 1 4/945
2 7 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1/5040
2 7 1 1 8 0 1 0 1 0 1 0 1 0 1 0 1 -1/360
2 7 1 0 8 0 1 0 1 0 1 0 1 0 1 0 1 1/360
2 7 1 1 8 1 7 0 1 0 1 0 1 0 1 0 1 1/108
2 7 1 0 8 0 7 0 1 0 1 0 1 0 1 0 1 1/108
2 7 1 1 3 1 4 1 8 0 1 0 1 0 1 0 1 1/270
2 7 1 1 8 1 7 1 6 0 1 0 1 0 1 0 1 -1/162
2 7 1 0 3 0 4 0 8 0 1 0 1 0 1 0 1 -1/270
2 7 1 0 8 0 7 0 6 0 1 0 1 0 1 0 1 1/162
2 7 1 1 8 1 4 1 5 1 7 0 1 0 1 0 1 -1/135
2 7 1 0 8 0 4 0 5 0 7 0 1 0 1 0 1 -1/135
2 7 1 1 6 1 8 1 5 1 3 1 4 0 1 0 1 -2/945
2 7 1 0 6 0 8 0 5 0 3 0 4 0 1 0 1 2/945
C.2. REDUCED EXPANSION ⋆redaff mod ō(h̄
7) 543
The formula for ⋆red mod ō(h̄7aff ) reads as follows:
f⋆red
(
aff g = fg+h̄P
ij∂)f∂ g(+h̄2 −1∂ ij k`i j `P ∂jP ∂if∂kg−1∂ P ijP k`∂ f∂ ∂ g+1∂ P ijP k`∂ ∂ f∂ g6 3 ` i k j 3 ` k i j
+ 1P ijP k`∂ ∂ f∂ ∂ g + h̄3 − 1P ij∂ P k`∂ Pmn∂ ∂ f∂ ∂ g− 1∂ P ijP k`∂ Pmn
2 k i ` j 6 n ` k i m j
∂
3 n ` k
∂if∂m∂jg
−1∂ ij
3 n
P ( )P k`Pmn∂k∂if∂m∂`∂ g+1∂ ij k` mnj nP P P ∂m∂k∂if∂`∂jg+1P ijP k`Pmn∂m∂k∂if∂n∂ ∂ g3 6 ` j
+ h̄4 1 ∂ P ij∂ P k`∂ Pmn∂ P pq` j q n ∂m∂if∂p∂kg − 2 ∂ ij k` mn pqqP ∂jP ∂`P ∂nP ∂k∂if∂p∂mg72 45
+ 11 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ ∂ f∂ ∂ g + 1 ∂ P ij∂ P k`∂ PmnP pqq j ` n m i p k ∂ ∂ f∂ ∂ ∂ g180 18 q n ` k i p m j
− 2 ∂ P ijP k`q ∂ mn`P ∂ P pq∂ 2 ij k` mn pq45 n k∂if∂p∂m∂jg + ∂`P ∂nP ∂qP P ∂k∂if∂p∂15 m∂jg
− 1 ∂ P ij` P k`∂ Pmn∂ P pqq n ∂m∂k∂if∂p∂jg + 2 ∂ P ij∂ P k`∂ Pmn` n q P pq∂18 45 m∂k∂if∂p∂jg
− 2 ∂ P ijP k`∂ Pmn∂ P pqq ` n ∂m∂k∂if∂p∂ 1jg − P ijP k`∂qPmn∂nP pq∂m∂k∂ f∂15 12 i p∂`∂jg
− 1∂ P ijP k`Pmnq ∂nP pq∂m∂k∂ f∂ ∂ ∂ g − 1∂ P ij∂ P k`PmnP pq3 i p ` j n q ∂9 m∂k∂if∂p∂`∂jg
− 1∂ P ijP k`PmnP pq∂ ∂ ∂ f∂ ∂ ∂ ∂ g + 1∂ P ijP k`Pmn pqq m k i p n ` j q P ∂p∂m∂k∂if∂n∂6 6 `∂jg
+ 1 P ijP k`PmnP pq∂p∂m∂k∂ f∂ ∂ ∂ ∂ g +
1 ∂ P ij∂ P k`Pmni q n ` j n q P
pq∂
24 18 k
∂if∂p∂m∂`∂jg
+ 1 ∂ P ij∂ P k`Pmnn q P
pq∂p∂m∂k∂if∂ ∂
1
` jg + ∂`P
ij∂nP
k`∂ mn pqqP P ∂if∂p∂m∂k∂jg18 45
− 1 ∂ P ij` ∂ k`nP ∂ Pmnq P pq∂p∂m∂45 k∂)if∂jg(+ 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ f∂ ∂ ∂ g90 q j ` n i p m k
+ 1 ∂ P ij∂ P k`∂ Pmnq j ` ∂nP
pq∂m∂k∂
5 1 ij k` mn pq rs
if∂pg +h̄ P ∂sP ∂`P ∂nP ∂qP ∂m∂k∂if∂r∂90 60 p∂jg
− 2 ∂ P ij∂ P k`∂ Pmn∂ P pqP rsn q r s ∂m∂k∂if∂p∂`∂jg+ 2 ∂ P ijP k`∂ Pmn∂ P pq∂ rs15 45 s ` n qP ∂m∂k∂if∂r∂p∂jg
+ 2 ∂ P ij∂ P k`∂ Pmn∂ P pqP rs∂ ∂ ∂ f∂ ∂ ∂ g+ 1 ∂ P ij∂ P k`∂ Pmn` n q s m k i r p j ` s q ∂nP
pqP rs∂m∂k∂if∂45 18 r∂p∂jg
− 11 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ ∂ ∂ f∂ ∂ 1 ij k` mn pq rs
360 s p n q m k i r `
∂jg+ P ∂72 nP ∂`P ∂sP ∂qP ∂p∂k∂if∂r∂m∂jg
− 2 ∂ P ijP k`Pmn∂ P pqs n ∂qP rs∂m∂k∂if∂r∂p∂`∂jg+ 2 ∂ P ijP k`n ∂qPmn∂ pq rs45 15 sP P ∂m∂k∂if∂r∂p∂`∂jg
+1∂ P ij∂ P k`∂ PmnP pqP rs∂ ∂ ∂ f∂ ∂ ∂ ∂ g+ 1 ∂ P ijn q s m k i r p ` j s P
k`∂qP
mn∂ P pqn P
rs∂
9 18 m
∂k∂if∂r∂p∂`∂jg
+ 2 ∂ P ijP k`∂ Pmn∂ P pqP rs∂ ∂ ∂ ∂ f∂ ∂ ∂ g− 2 ∂ P ijP k`Pmnn q s p m k i r ` j s ∂nP pq∂ rsqP ∂p∂m∂k∂if∂r∂`∂jg45 15
−1∂ P ij∂ P k`PmnP pq∂ P rsn s q ∂p∂m∂k∂if∂ 1 ij k` mn pq rs9 r∂`∂jg− ∂nP P P ∂sP ∂qP ∂p∂18 m∂k∂if∂r∂`∂jg
−1∂sP ijP k`PmnP pq∂ P rsq ∂p∂ ∂ 1 ij k` mn pq rs6 m k∂if∂r∂n∂`∂jg− ∂9 qP ∂sP P P P ∂p∂m∂k∂if∂r∂n∂`∂jg
− 1 P ijP k`Pmn∂ P pq∂ P rss q ∂p∂m∂k∂if∂r∂n∂`∂jg+ 1 P ij∂ P k`∂ Pmn∂ P pq rss ` n ∂qP ∂k∂if∂r∂p∂36 90 m∂jg
+ 2 ∂ P ij∂ P k`q s ∂
mn pq rs
`P ∂nP P ∂k∂if∂r∂p∂
1 ij
m∂jg− ∂nP ∂ P k`q ∂`Pmn∂ pq rssP P ∂k∂if∂r∂p∂45 45 m∂jg
+ 1 P ij∂ k` mn pq rssP ∂`P ∂nP ∂qP ∂p∂m∂k∂if∂r∂jg− 1 ∂`P ij∂sP k`Pmn∂ pq rs90 45 nP ∂qP ∂p∂m∂k∂if∂r∂jg
+ 2 ∂ P ij∂ k` mn pq rs` nP ∂sP P ∂qP ∂p∂m∂k∂ f∂ ∂ g+
1 P ijP k`PmnP pqP rs
45 i r j
∂ ∂
120 r p
∂m∂k∂if∂s∂q∂n∂`∂jg
− 1 ∂ P ijP k`PmnP pqs P rs∂p∂ 1 ij k` mn pq rs18 m∂k∂if∂r∂q∂n∂`∂jg+ ∂sP P P P P ∂r∂18 p∂m∂k∂if∂q∂n∂`∂jg
+ 1 ∂ P ij∂ P k`PmnP pqq s P
rs∂m∂k∂ f∂ ∂ ∂ ∂ ∂ g+
1 ∂ P ij∂ P k`PmnP pqP rsi r p n ` j q s ∂r∂p∂m∂18 18 k∂if∂n∂`∂jg)
+ 1 ∂ P ijn P
k`∂ Pmnq (∂sP pqP rs∂k∂if∂r∂p∂m∂`∂ g− 1j ∂nP ijP k`∂ mnqP ∂ P pqP rss ∂r∂p∂m∂k∂if∂45 45 `∂jg
+ h̄6 − 2 ∂ P iju ∂jP k`∂ Pmn∂ pq rs` nP ∂qP ∂sP tu∂m∂k∂567 if∂t∂r∂pg
+ 31 ∂ P iju ∂
k` mn
rP ∂`P ∂ P
pq∂ P rs∂ tu
3780 n q s
P ∂p∂k∂if∂t∂m∂jg
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tuq j ` n u s ∂r∂k∂if∂t∂p∂mg360
+ 7 ∂ P ij∂ P k`∂ Pmn∂ pq rs tu
1620 ` q t n
P ∂uP ∂sP ∂m∂k∂if∂r∂p∂jg
+ 11 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tu` t q n u s ∂m∂k∂if∂2160 r∂p∂jg
544 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tu∂ ∂ ∂ f∂ ∂ ∂ g
1296 ` j q n u s r m i t p k
+ 4 ∂ P iju P
k`∂ Pmn∂ P pq` n ∂qP
rs∂sP
tu∂
315 p
∂k∂if∂t∂r∂m∂jg
+ 2 ∂ ij k`qP ∂tP ∂sP
mn∂nP
pq∂ rs tuuP P ∂m∂k∂if∂r∂p∂`∂jg105
− 2 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rsP tus u ` n q ∂m∂k∂if∂t∂ ∂315 r p∂jg
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rsP tu∂
180 u s ` n q m
∂k∂if∂t∂r∂p∂jg
− 4 ∂ ij k`qP ∂sP ∂ Pmn∂ P pq∂ P rsP tu∂105 ` n u m∂k∂if∂t∂r∂p∂jg
+ 1 ∂ P ij` P
k`∂ mn pquP ∂nP ∂qP
rs∂sP
tu∂
270 m
∂k∂if∂t∂r∂p∂jg
+ 2 ∂`P
ij∂ P k`∂ Pmn∂ P pqn q s ∂uP
rsP tu∂m∂k∂135 if∂t∂r∂p∂jg
+ 1 ∂ ij k` mn pq rs tu
45 s
P ∂tP ∂qP ∂nP ∂uP P ∂m∂k∂if∂r∂p∂`∂jg
− 11 ∂ P ij∂ P k`∂ PmnP pq∂ rs tuq r t uP ∂sP ∂m∂k∂if∂p∂n∂`∂jg1080
− 2 ∂ P ij∂ P k`∂ PmnP pq∂ P rsu n ` q ∂sP tu∂p∂k∂if∂t∂r∂m∂135 jg
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs tuu n ` s q P ∂p∂k∂216 if∂t∂r∂m∂jg
− 4 ∂`P ij∂ k` mn pq rs tunP ∂sP ∂uP ∂qP P ∂p∂m∂k∂315 if∂t∂r∂jg
+ 2 ∂ ij k`nP ∂tP ∂ P
mn∂ P pq∂ P rs tus u q P ∂p∂m∂k∂105 if∂r∂`∂jg
+ 2 ∂ P ij∂ P k`∂ PmnP pq` n u ∂qP
rs∂ tusP ∂p∂315 m∂k∂if∂t∂r∂jg
+ 1 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tu` u n q s ∂p∂m∂k∂if∂t∂ ∂180 r jg
+ 4 ∂`P
ij∂ P k`Pmn∂ P pq∂ rs tuu n qP ∂sP ∂p∂m∂k∂if∂t∂r∂105 jg
− 1 ∂ P ij∂ P k`∂ Pmn pqu s ` ∂nP ∂qP rsP tu∂p∂m∂k∂if∂t∂r∂jg270
− 2 ∂ P ijP k`u ∂`Pmn∂ pq rs tu135 nP ∂qP ∂sP ∂p∂m∂k∂if∂t∂r∂jg
+ 1 ∂ P ij∂ P k`n t ∂uP
mn∂sP
pq∂qP
rsP tu∂p∂m∂k∂if∂45 r∂`∂jg
+ 11 ∂ P ij∂ P k`∂ PmnP pq∂ P rs∂ P tuq r t u s ∂ ∂1080 p m∂k∂if∂n∂`∂jg
+ 2 ∂ ij k` mn pq rs tu`P ∂nP ∂uP ∂sP ∂qP P ∂p∂m∂k∂if∂t∂r∂jg135
+ 1 ∂ P ijP k`∂ Pmn∂ P pq` q n ∂uP
rs∂sP
tu∂r∂m∂ ∂216 k if∂t∂p∂jg
+ 1 P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tu∂ ∂ ∂ ∂ f∂ ∂ ∂ ∂ g
120 u n q s p m k i t r ` j
− 2 ∂ ij k` mn pq rs tu
45 q
P ∂sP P ∂tP ∂uP P ∂p∂m∂k∂if∂r∂n∂`∂jg
+ 2 ∂ P ijP k`∂ Pmn∂ P pq∂ P rsP tun s u q ∂p∂m∂k∂if∂t∂r∂`∂jg45
+ 2 ∂ P ijP k`∂ Pmn∂ P pqP rs∂ P tuq u n s ∂r∂m∂k∂if∂t∂p∂`∂jg45
+ 1 ∂ P ij∂ P k`∂ PmnP pq∂ P rsP tun s u q ∂p∂m∂18 k∂if∂t∂r∂`∂jg
− 2 ∂ P ijq ∂sP k`∂ PmnP pq∂ P rst u P tu∂45 p∂m∂k∂if∂r∂n∂`∂jg
+ 2 ∂qP
ij∂ P k`s ∂tP
mn∂uP
pqP rsP tu∂p∂m∂k∂if∂r∂45 n∂`∂jg
+ 4 ∂ P ij∂ P k`PmnP pq∂ P rs∂ P tun u q s ∂p∂m∂k∂if∂t∂135 r∂`∂jg
+ 4 ∂ P ij∂ P k` mnn s ∂qP ∂uP
pqP rsP tu∂p∂m∂k∂if∂t∂135 r∂`∂jg
+ 11 ∂ P ij∂ P k`PmnP pqr t ∂
rs tu
720 u
P ∂sP ∂p∂m∂k∂if∂q∂n∂`∂jg
+ 1 ∂ P ijP k`Pmnq ∂
pq rs
nP ∂uP ∂sP
tu∂r∂m∂k∂if∂t∂p∂`∂18 jg
+ 1 ∂ P ij∂ P k`Pmn∂ P pq∂ P rsn u s q P
tu∂
54 p
∂m∂k∂if∂t∂r∂`∂jg
+ 1 P ijP k`∂ Pmn∂ P pqq n ∂uP
rs∂sP
tu∂
144 r
∂m∂k∂if∂t∂p∂`∂jg
C.2. REDUCED EXPANSION ⋆redaff mod ō(h̄
7) 545
− 1 ∂ P ij∂ P k`∂ Pmnu j ` ∂nP pq∂ P rsq ∂sP tu∂378 k∂if∂t∂r∂p∂mg
+ 31 ∂ ij k``P ∂tP ∂uP
mn∂nP
pq∂ rs tuqP ∂sP ∂k∂if∂11340 r∂p∂m∂jg
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pq` j u n ∂qP rs∂ P tus ∂m∂if∂t∂r∂p∂kg540
+ 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tu∂ ∂
567 n q r t u s k i
f∂p∂m∂`∂jg
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pqu j ` n ∂ P rs∂ tu378 q sP ∂p∂m∂k∂if∂t∂rg
+ 31 ∂ P ij∂ P k`∂ Pmnu r ` ∂nP
pq∂ rs tuqP ∂sP ∂p∂m∂k∂if∂t∂jg11340
− 1 ∂ P ij∂ P k`q j ∂ Pmn∂ P pq∂ P rs∂ P tu∂540 ` n u s r∂m∂k∂if∂t∂pg
+ 1 ∂ P ij∂ P k`∂ Pmnn q r ∂tP
pq∂ rs tu
567 u
P ∂sP ∂p∂m∂k∂if∂`∂jg
− 1 ∂ ijsP ∂ P k`u PmnP pqP rsP tu∂18 r∂p∂m∂k∂if∂t∂q∂n∂`∂jg
− 1 P ijP k`PmnP pq∂ rs tuuP ∂sP ∂r∂p∂m∂k∂if∂144 t∂q∂n∂`∂jg
− 1 ∂ P ijP k`PmnP pq∂ P rss u P tu∂r∂p∂m∂k∂if∂t∂q∂n∂`∂jg18
− 1 ∂ P ij∂ P k`PmnP pq∂ P rsP tus t u ∂p∂m∂k∂if∂15 r∂q∂n∂`∂jg
+ 2 ∂uP
ijP k`PmnP pq∂ rs tuqP ∂sP ∂p∂m∂k∂if∂t∂r∂n∂45 `∂jg
+ 1∂ P ij∂ P k`Pmnq s ∂uP
pqP rsP tu∂p∂m∂k∂if∂t∂r∂n∂`∂jg9
+ 1 ∂ P ij∂ P k`∂ PmnP pqP rsP tu∂ ∂ ∂ ∂ f∂ ∂ ∂ ∂ ∂ g
54 q s u p m k i t r n ` j
+ 1 ∂ ij k` mn pq rs tu
36 u
P P P ∂sP ∂qP P ∂p∂m∂k∂if∂t∂r∂n∂`∂jg
− 1 ∂ P ij∂ P k`PmnP pqs t ∂uP rsP tu∂15 r∂p∂m∂k∂if∂q∂n∂`∂jg
− 2 ∂ P ijP k`Pmn∂ P pq∂ P rsP tuq s u ∂r∂p∂m∂k∂if∂45 t∂n∂`∂jg
− 1∂ P ij∂ P k`PmnP pqP rsq u ∂ tu9 sP ∂r∂p∂m∂k∂if∂t∂n∂`∂jg
− 1 ∂ P ij∂ P k`∂ mn pq rs tu
54 q s u
P P P P ∂r∂p∂m∂k∂if∂t∂n∂`∂jg
− 1 ∂ P ijP k`PmnP pqq ∂uP rs∂ tu36 sP ∂r∂p∂m∂k∂if∂t∂n∂`∂jg
+ 1 P ijP k`∂ mnuP ∂nP
pq∂ P rsq ∂sP
tu∂m∂k∂if∂ ∂180 t r∂p∂`∂jg
+ 2 ∂ ij k` mn pq rs tu
45 q
P ∂sP ∂tP ∂uP P P ∂m∂k∂if∂r∂p∂n∂`∂jg
− 4 ∂ P ij∂ k` mn pq rs tu
135 s u
P P ∂nP ∂qP P ∂m∂k∂if∂t∂r∂p∂`∂jg
+ 1 ∂ P ij∂ P k`Pmn∂ P pq∂ P rsP tun q s u ∂m∂k∂if∂t∂r∂p∂`∂135 jg
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pqP rsP tus u q n ∂m∂k∂if∂ ∂108 t r∂p∂`∂jg
+ 2 ∂ P ijP k`∂ Pmn∂ P pq rs tus u n ∂qP P ∂m∂k∂if∂45 t∂r∂p∂`∂jg
− 1 ∂qP ijP k`∂ mn pq rs tu45 sP ∂nP ∂uP P ∂m∂k∂if∂t∂r∂p∂`∂jg
+ 1 P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tuu n q s ∂r∂p∂m∂k∂if∂t∂`∂jg180
− 2 ∂ P ij∂ P k`∂ Pmn∂ P pqP rs tuq s t u P ∂r∂p∂m∂k∂if∂n∂`∂jg45
− 4 ∂ P ijn ∂ k`qP ∂sPmnP pq∂ P rsu P tu∂135 r∂p∂m∂k∂if∂t∂`∂jg
+ 1 ∂ P ij∂ P k`∂ Pmnn u q ∂sP
pqP rsP tu∂r∂135 p∂m∂k∂if∂t∂`∂jg
− 1 ∂ P ij∂ P k`PmnP pq∂ P rs∂ P tun q u s ∂r∂p∂m∂k∂if∂t∂`∂108 jg
− 1 ∂ P ijP k`∂ PmnP pq∂ rs tun u qP ∂sP ∂r∂45 p∂m∂k∂if∂t∂`∂jg
+ 2 ∂ P ijP k`∂ Pmn∂ P pqP rs∂ P tu
45 n q u s
∂r∂p∂m∂k∂if∂t∂`∂jg
− 1 ∂ P ij∂ P k`u s ∂`Pmn∂ P pqn ∂ rs tuqP P ∂k∂if∂t∂ ∂270 r p∂m∂jg
546 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rsP tu
270 q n ` s u
∂k∂if∂t∂r∂p∂m∂jg
+ 2 ∂ P ij∂ P k`∂ Pmnn t q ∂sP
pq∂ rs tuuP P ∂k∂if∂r∂p∂ ∂315 m `∂jg
− 4 ∂ ij k` mn pqqP ∂sP ∂`P ∂nP ∂ rs tuuP P ∂k∂if∂315 t∂r∂p∂m∂jg
+ 1 ∂ ij k` mn pq rs tu`P P ∂uP ∂nP ∂qP ∂sP ∂r∂p∂m∂k∂if∂t∂270 jg
+ 1 ∂`P
ij∂ P k`n ∂qP
mnP pq∂ rsuP ∂sP
tu∂r∂p∂m∂k∂if∂t∂jg270
+ 2 ∂ P ijn ∂tP
k`∂ mn pqqP ∂sP ∂uP
rsP tu∂r∂p∂m∂k∂if∂`∂jg315
+ 4 ∂`P
ij∂nP
k`∂ mn pquP P ∂qP
rs∂ P tus ∂r∂p∂m∂k∂if∂315 t∂jg
+ 1 P ijP k`PmnP pqP rsP tu∂t∂r∂p∂m∂k∂if∂720 u∂s∂q∂n∂`∂jg
− 1 ∂ P ijP k`PmnP pqP rsP tuu ∂r∂p∂m∂k∂if∂t∂s∂72 q∂n∂`∂jg
+ 1 ∂ P ijP k`PmnP pqP rsu P
tu∂
72 t
∂r∂p∂m∂k∂if∂s∂q∂n∂`∂jg
+ 1 ∂ P ij∂ P k`PmnP pqP rsP tus u ∂p∂m∂k∂if∂t∂r∂q∂36 n∂`∂jg
+ 1 ∂ P ij∂ k` mn pq rs tu
36 s u
P P P P P ∂t∂r∂p∂m∂k∂if∂q∂n∂`∂jg
+ 1 ∂ P ijP k`Pmn∂ P pq∂ P rsP tu∂
90 q s u m
∂k∂if∂t∂r∂p∂n∂`∂jg
− 1 ∂qP ij∂ P k`s ∂uPmnP pqP rsP tu∂m∂k∂if∂t∂r∂p∂n∂`∂ g162 j
− 1 ∂ P ijP k`Pmn∂ P pq∂ P rsP tuq s u ∂t∂r∂p∂m∂k∂if∂n∂`∂90 jg
+ 1 ∂ P ij∂ P k`∂ PmnP pqP rsP tuq s u ∂t∂162 r∂p∂m∂k∂if∂n∂`∂jg
− 2 ∂ P ij∂ k` mn` nP ∂qP ∂ P pqs ∂ rs tu945 uP P ∂if∂t∂r∂p∂m∂k∂jg
+ 2 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs tu` n q s u P ∂t∂r∂p∂m∂k∂if∂jg945
− 1 ∂ P ijn ∂qP k`∂ mn pqsP P ∂ P rsu P tu∂k∂if∂t∂r∂p∂ ∂135 m `∂jg
− 1 ∂ P ij∂ k` mnn qP ∂sP P pq∂ P rsu P tu∂t∂r∂p∂ ∂135 m k∂if∂`∂jg
− 1 ∂ P iju ∂jP k`∂`Pmn∂ pqnP ∂ P rs∂ P tuq s ∂if∂t∂r∂p∂m∂945 kg
( )− 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tu∂ ∂ ∂ ∂ ∂ f∂ g945 u j ` n q s r p m k i t
+ h̄7 − 1 ∂ P ij∂ P k`∂ Pmn∂ P pq` w u n ∂qP rs∂ tu vwsP P ∂p∂m∂k∂if∂v∂t∂180 r∂jg
+ 53 P ij∂ P k`∂ Pmn∂ pq rs tuw ` nP ∂qP ∂sP ∂
vw
uP ∂p∂m∂k∂11340 if∂v∂t∂r∂jg
− 31 P ij∂ P k`∂ Pmnw ` ∂nP pq∂ rs tuqP ∂sP ∂ P vwu ∂r∂m∂k∂if∂v∂3780 t∂p∂jg
+ 2 ∂ P ij∂ P k`∂ Pmn` n s ∂uP
pq∂qP
rs∂ tu vwwP P ∂p∂m∂k∂if∂135 v∂t∂r∂jg
− 8 ∂ P ij` ∂wP k`Pmn∂ P pqn ∂ P rsq ∂ tu vw945 sP ∂uP ∂p∂m∂k∂if∂v∂t∂r∂jg
− 29 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tuP vws u ` n q w ∂p∂m∂k∂if∂v∂ ∂945 t r∂jg
+ 46 ∂ P ij∂ P k`∂ Pmnn q s ∂vP
pq∂ P rs∂ P tuP vwu w ∂p∂m∂k∂if∂t∂ ∂945 r `∂jg
+ 32 ∂ ij k` mn pqsP ∂uP ∂vP ∂nP ∂ P
rs∂ P tuP vwq w ∂p∂m∂k∂if∂t∂r∂`∂jg945
− 8 ∂ P ij∂ P k`∂ Pmn pq rs tu vw
315 n u r
∂wP ∂qP ∂sP P ∂p∂m∂k∂if∂v∂t∂`∂jg
+ 191 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tu∂ P vwn w r q s u ∂p∂m∂k∂if∂v∂t∂`∂ g22680 j
+ 11 ∂ P ij∂ P k`∂ PmnP pq∂ P rs∂ P tu vwn s v q w ∂uP ∂p∂m∂k∂if∂ ∂1080 t r∂`∂jg
− 1 P ij∂ P k`s ∂ Pmn∂ pq rs tu vw` nP ∂qP ∂wP ∂uP ∂t∂360 m∂k∂if∂v∂r∂p∂jg
− 1 ∂ P ijP k`s ∂ Pmn∂ P pq∂ P rs∂ P tu` n q w ∂ P vwu ∂t∂m∂k∂if∂v∂r∂p∂135 jg
− 1 ∂ P ij∂ P k`∂ Pmn` n u ∂sP pq∂qP rs∂ tu vwwP P ∂p∂m∂k∂if∂v∂135 t∂r∂jg
C.2. REDUCED EXPANSION ⋆redaff mod ō(h̄
7) 547
+ 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rsn u v s q ∂ P
tu
w P
vw∂
45 p
∂m∂k∂if∂t∂r∂`∂jg
− 11 ∂ P ij∂ P k`Pmn∂ P pqt v s ∂ P rsq ∂ tu vwwP ∂uP ∂p∂m∂k∂if∂r∂2160 n∂`∂jg
− 1 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tu vwn ` s q w ∂uP ∂t∂p∂k∂if∂ ∂216 v r∂m∂jg
− 1 P ij∂ P k`∂ Pmnn ` ∂sP pq∂ P rsq ∂ tu vw1296 wP ∂uP ∂t∂p∂k∂if∂v∂r∂m∂jg
− 31 ∂rP ij∂ P k`t ∂ mn pq rs tu vwvP P ∂wP ∂sP ∂uP ∂p∂m∂k∂if∂q∂n∂`∂jg22680
+ 16 ∂ ij k` mn pq rs tu vw
945 n
P ∂sP ∂qP ∂vP ∂uP ∂wP P ∂p∂m∂k∂if∂t∂r∂`∂jg
+ 16 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rsP tu vwq s t v w ∂uP ∂p∂m∂k∂if∂r∂945 n∂`∂jg
+ 1 ∂ ij k` mn pq rsqP ∂wP P P ∂uP ∂ P
tuP vws ∂r∂p∂54 m∂k∂if∂v∂t∂n∂`∂jg
+ 1 P ijP k`Pmn∂ P pqs ∂qP
rs∂ P tuw ∂
vw
uP ∂t∂p∂m∂k∂if∂v∂r∂ ∂432 n `∂jg
+ 1 P ijP k`Pmn∂wP
pq∂ rs tu vw
360 q
P ∂sP ∂uP ∂t∂p∂m∂k∂if∂v∂r∂n∂`∂jg
− 1 ∂ P ijP k`PmnP pqt ∂ rs tu vwwP ∂sP ∂uP ∂r∂p∂m∂k∂if∂v∂q∂n∂`∂jg135
+ 1 ∂ ij k` mn pq rs tu vw
15 u
P P P ∂wP ∂qP ∂sP P ∂r∂p∂m∂k∂if∂v∂t∂n∂`∂jg
− 1 ∂ P ijP k`Pmn∂ P pq∂ P rs tu vwq s u ∂wP P ∂r∂45 p∂m∂k∂if∂v∂t∂n∂`∂jg
+ 1 ∂ P ij∂ P k`Pmnq u ∂
pq rs tu
wP P ∂sP P
vw∂r∂p∂m∂k∂if∂18 v∂t∂n∂`∂jg
+ 4 ∂uP
ij∂wP
k`PmnP pq∂qP
rs∂ P tus P
vw∂
135 r
∂p∂m∂k∂if∂v∂t∂n∂`∂jg
+ 2 ∂ P ij∂ P k`∂ PmnP pq∂ P rsP tuP vw
135 s u v w
∂r∂p∂m∂k∂if∂t∂q∂n∂`∂jg
− 2 ∂ ij k` mn pq rssP ∂uP ∂vP P P ∂wP tuP vw∂ ∂135 r p∂m∂k∂if∂t∂q∂n∂`∂jg
+ 4 ∂ P ij∂ P k`Pmn∂ P pq rsq s u P ∂ P
tuP vww ∂r∂p∂m∂k∂if∂v∂ ∂135 t n∂`∂jg
+ 1 ∂ ij k` mn pq rs tu vw
27 q
P ∂sP ∂uP ∂wP P P P ∂r∂p∂m∂k∂if∂v∂t∂n∂`∂jg
+ 1 ∂ P ijP k`Pmn∂ P pq∂ P rs∂ P tuq w u s P
vw∂r∂p∂m∂k∂if∂v∂t∂n∂`∂36 jg
+ 4 ∂ P ijP k`Pmn∂ P pq∂ P rs∂ P tu∂ P vw∂ ∂
315 w n q s u r m
∂k∂if∂v∂t∂p∂`∂jg
+ 2 ∂ P ij∂ P k`Pmn∂ pq rss v uP ∂qP ∂wP
tuP vw∂p∂m∂k∂if∂ ∂105 t r∂n∂`∂jg
− 2 ∂ P ijP k`∂ Pmn∂ pq rs tu vw
105 s u n
P ∂qP ∂wP P ∂p∂m∂k∂if∂v∂t∂r∂`∂jg
+ 8 ∂ P ijn P
k`∂ mnqP ∂
pq
sP ∂uP
rs∂ tu vwwP P ∂p∂ ∂945 m k∂if∂v∂t∂r∂`∂jg
− 2 ∂ ij k` mn pq rs tu vw
105 n
P P ∂uP ∂wP ∂qP ∂sP P ∂p∂m∂k∂if∂v∂t∂r∂`∂jg
+ 2 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rsP tu vwq s v u w P ∂p∂m∂45 k∂if∂t∂r∂n∂`∂jg
− 4 ∂nP ij∂uP k`∂ Pmnw P pq∂ rs tu vw135 qP ∂sP P ∂p∂m∂k∂if∂v∂t∂r∂`∂jg
+ 1 ∂ P ijP k`Pmn∂ P pq∂ P rs∂ tu vwn w q sP ∂uP ∂p∂m∂k∂if∂v∂t∂r∂`∂jg270
− 1 ∂ P ijP k`w ∂uPmn∂ pqnP ∂qP rs∂ P tus P vw∂180 p∂m∂k∂if∂v∂t∂r∂`∂jg
− 1 ∂ ij k` mn pq rs tu vwuP ∂wP P ∂nP ∂qP ∂sP P ∂p∂m∂k∂if∂v∂t∂r∂`∂jg135
+ 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rsP tuq s u v w P
vw∂p∂45 m∂k∂if∂t∂r∂n∂`∂jg
− 1 ∂nP ij∂ P k`u ∂sPmn∂wP pq∂ rsqP P tuP vw∂45 p∂m∂k∂if∂v∂t∂r∂`∂jg
+ 1 ∂ ijuP ∂ P
k`
v P
mn∂sP
pq∂ P rs∂ P tuP vwq w ∂p∂m∂k∂if∂t∂r∂n∂`∂45 jg
− 11 ∂ P ij∂ P k`∂ PmnP pqP rs tu vws t v ∂wP ∂uP ∂p∂1080 m∂k∂if∂r∂q∂n∂`∂jg
− 2 ∂ P ijP k`∂ Pmn∂ P pqP rsw q n ∂sP tu∂ P vwu ∂r∂m∂k∂if∂v∂t∂p∂`∂jg135
− 1 ∂ P ij∂ k` mn pq rs tu vwn uP ∂wP ∂sP ∂qP P P ∂p∂m∂k∂if∂v∂54 t∂r∂`∂jg
− 1 ∂ P ijP k`∂ Pmn∂ P pqw q n ∂uP rs∂ tu vwsP P ∂r∂m∂k∂if∂v∂t∂p∂216 `∂jg
548 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
+ 2 ∂nP
ij∂uP
k`Pmn∂ P pqw ∂
rs tu vw
qP ∂sP P ∂p∂m∂k∂if∂ ∂135 v t∂r∂`∂jg
− 1 ∂ P ij∂ P k`Pmn∂ P pq∂ P rs∂ P tuP vwn s u q w ∂p∂m∂k∂if∂v∂t∂r∂`∂jg135
− 4 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tuP vwn q u w s ∂r∂p∂m∂k∂if∂v∂t∂`∂jg315
+ 2 ∂ P ij∂ P k`Pmn∂ P pq∂ P rs∂ P tuP vwq v u w s ∂r∂p∂m∂k∂if∂t∂n∂105 `∂jg
+ 2 ∂ P ijP k`∂ PmnP pq rs tu vwn w ∂qP ∂sP ∂uP ∂r∂p∂m∂k∂if∂v∂t∂`∂105 jg
− 8 ∂ P ijP k`Pmn∂ P pq∂ P rs∂ P tu∂ P vw∂ ∂ ∂ ∂ ∂ f∂ ∂ ∂ ∂ g
945 w n q s u r p m k i v t ` j
+ 2 ∂ P ijP k`∂ Pmn∂ pq rss w nP ∂qP P
tu∂ P vwu ∂t∂p∂m∂k∂if∂105 v∂r∂`∂jg
+ 2 ∂ P ij∂ P k`q u ∂
mn pq rs
vP ∂wP P ∂ P
tuP vws ∂r∂p∂45 m∂k∂if∂t∂n∂`∂jg
+ 4 ∂nP
ij∂ P k`∂ Pmn∂ P pqP rs∂ P tuP vwu q w s ∂r∂p∂m∂k∂if∂v∂135 t∂`∂jg
− 1 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tuP vw
270 w u n q s
∂r∂p∂m∂k∂if∂v∂t∂`∂jg
+ 1 ∂ ij k` mnnP P P ∂ P
pq∂ P rsw q ∂
tu vw
180 s
P ∂uP ∂r∂p∂m∂k∂if∂v∂t∂`∂jg
+ 1 ∂nP
ij∂qP
k`∂ mn pqsP P ∂uP
rs∂ P tuP vww ∂135 r∂p∂m∂k∂if∂v∂t∂`∂jg
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pqP rsq s u v ∂ P tuw P vw∂45 r∂p∂m∂k∂if∂t∂n∂`∂jg
+ 1 ∂ ijnP ∂ P
k`
s P
mn∂wP
pq∂ P rsP tuq ∂uP
vw∂
45 t
∂p∂m∂k∂if∂v∂r∂`∂jg
+ 1 ∂ P ij∂ P k`Pmn∂ P pq∂ P rs∂ P tu vwq v w u s P ∂r∂p∂m∂k∂if∂t∂n∂`∂jg45
+ 11 ∂sP
ij∂ P k`t ∂vP
mnP pqP rs∂wP
tu∂ vw
1080 u
P ∂r∂p∂m∂k∂if∂q∂n∂`∂jg
+ 2 ∂nP
ijP k`∂ Pmn∂ P pq∂ P rs tu vw
135 q w u
∂sP P ∂r∂p∂m∂k∂if∂v∂t∂`∂jg
+ 1 ∂ P ij∂ P k`PmnP pq∂ P rs∂ P tu∂ P vwn s q w u ∂t∂p∂m∂54 k∂if∂v∂r∂`∂jg
+ 1 ∂ P ijP k`Pmnn ∂sP
pq∂ rs tu vwqP ∂wP ∂uP ∂t∂p∂m∂k∂if∂v∂r∂`∂ g216 j
+ 1 ∂ P ij∂ P k`n w ∂
mn pq
uP P ∂ P
rs∂ P tuP vwq s ∂r∂p∂m∂k∂if∂v∂ ∂135 t `∂jg
− 2 ∂ P ij∂ k` mnn wP ∂qP ∂uP pqP rs∂ tu vwsP P ∂r∂p∂m∂k∂135 if∂v∂t∂`∂jg
− 1 ∂nP ijP k`∂`Pmn∂ pq rswP ∂qP ∂sP tu∂ vwuP ∂p∂k∂if∂v∂t∂r∂m∂jg270
+ 11 P ij∂wP
k`∂ Pmn` ∂
pq rs tu vw
nP ∂qP ∂sP ∂uP ∂m∂k∂if∂v∂t∂7560 r∂p∂jg
− 2 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tuP vwu w ` n q s ∂315 m∂k∂if∂v∂t∂r∂p∂jg
− 4 ∂ P ij∂ k` mn pq rs` wP P ∂nP ∂qP ∂sP tu∂ vwuP ∂m∂k∂if∂v∂t∂r∂ ∂315 p jg
− 4 ∂nP ij∂ P k`∂ Pmn∂ P pq∂ P rsq s v u ∂wP tuP vw∂m∂k∂if∂315 t∂r∂p∂`∂jg
− 31 P ij∂ k` mnwP ∂`P ∂ P pqn ∂ rs tuqP ∂sP ∂ P vwu ∂r∂k∂if∂v∂t∂p∂m∂jg7560
+ 8 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tus u ` n q w P
vw∂m∂k∂if∂v∂t∂r∂p∂945 jg
− 2 ∂ P ij` ∂ k`nP ∂qPmn∂ P pq∂ P rs∂ P tuP vw∂ ∂ ∂945 s u w m k if∂v∂t∂r∂p∂jg
− 1 P ij∂ P k`∂ Pmn∂ P pqn ` w ∂qP rs∂ tusP ∂ vwuP ∂p∂k∂if∂540 v∂t∂r∂m∂jg
+ 37 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tu∂ P vws v n q w u ∂m∂k∂if∂t∂r∂p∂15120 `∂jg
− 1 ∂ ij k` mn pq rs tu vw
135 u
P ∂nP ∂`P ∂wP ∂qP ∂sP P ∂p∂k∂if∂v∂t∂r∂m∂jg
+ 1 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rsn u ` s q ∂ P
tu
w P
vw∂
270 p
∂k∂if∂v∂t∂r∂m∂jg
− 1 ∂`P ij∂ k` mnwP ∂uP ∂ P pqn ∂ rs tu vwqP ∂sP P ∂r∂270 p∂m∂k∂if∂v∂t∂jg
+ 11 P ij∂ k` mnwP ∂`P ∂ P
pq
n ∂
rs tu
qP ∂sP ∂uP
vw∂ ∂
7560 r p
∂m∂k∂if∂v∂t∂jg
− 2 ∂ P ij` ∂nP k`∂ Pmn∂ P pqP rsq w ∂ tu vw315 sP ∂uP ∂r∂p∂m∂k∂if∂v∂t∂jg
− 4 ∂ P ij∂ k` mn pq rs tu vw
315 u w
P ∂`P ∂nP ∂qP ∂sP P ∂r∂p∂m∂k∂if∂v∂t∂jg
+ 4 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tuP vwn u v w q s ∂r∂p∂m∂k∂if∂t∂315 `∂jg
C.2. REDUCED EXPANSION ⋆red mod ō(h̄7aff ) 549
− 31 P ij∂ P k`w ∂`Pmn∂ pq rsnP ∂qP ∂sP tu∂ vw7560 uP ∂t∂r∂m∂k∂if∂v∂p∂jg
+ 8 ∂`P
ij∂ P k`n ∂
mn pq
wP P ∂qP
rs∂sP
tu∂uP
vw∂r∂p∂m∂k∂if∂v∂t∂jg945
− 2 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tu∂ P vw∂ ∂ ∂ ∂ ∂ f∂ ∂ ∂ g
945 w ` n q s u r p m k i v t j
− 1 P ij∂ P k`s ∂`Pmn∂ pqnP ∂qP rs∂ P tuw ∂uP vw∂ ∂540 t p∂m∂k∂if∂v∂r∂jg
+ 37 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tu∂ P vw∂ ∂ ∂ ∂ ∂ f∂ ∂ ∂ g
15120 n q w t s u r p m k i v ` j
+ 1 ∂ P ij∂ P k`` s P
mn∂ pqnP ∂qP
rs∂ P tuw ∂uP
vw∂ ∂
270 t p
∂m∂k∂if∂v∂r∂jg
− 1 ∂ P ij∂ P k`` n ∂sPmnP pq∂qP rs∂ P tuw ∂uP vw∂t∂p∂m∂k∂if∂v∂r∂jg135
− 1 ∂ ij k` mnwP P P P pqP rsP tu∂ vwuP ∂72 t∂r∂p∂m∂k∂if∂v∂s∂q∂n∂`∂jg
− 1 ∂ P ij∂ k` mn pqu wP P P P rsP tuP vw∂t∂54 r∂p∂m∂k∂if∂v∂s∂q∂n∂`∂jg
− 1 P ijP k`PmnP pqP rs∂ P tu∂ P vww u ∂t∂r∂p∂m∂k∂if∂v∂ ∂720 s q∂n∂`∂jg
− 1 ∂ P ij∂ P k`PmnP pqP rs∂ P tuP vw∂ ∂ ∂ ∂ ∂ f∂ ∂ ∂ ∂ ∂ ∂ g
45 u v w r p m k i t s q n ` j
+ 2 ∂ P ijP k`PmnP pqP rsw ∂
tu vw
sP ∂uP ∂r∂p∂m∂k∂if∂v∂135 t∂q∂n∂`∂jg
+ 1 ∂ P ij∂ P k`PmnP pq∂ P rsP tuP vws u w ∂r∂p∂m∂k∂if∂v∂18 t∂q∂n∂`∂jg
+ 1 ∂ ijsP ∂ P
k`
u ∂
mn
wP P
pqP rsP tuP vw∂r∂p∂m∂k∂if∂v∂t∂q∂54 n∂`∂jg
+ 1 ∂ P ijP k`PmnP pqw ∂uP
rs∂sP
tuP vw∂r∂p∂m∂k∂if∂v∂108 t∂q∂n∂`∂jg
− 1 ∂ P ij∂ P k`PmnP pqP rs∂ P tuP vwu v w ∂t∂r∂p∂m∂k∂if∂s∂q∂n∂`∂45 jg
− 2 ∂sP ijP k`PmnP pq∂ rsuP ∂ P tuP vww ∂t∂r∂135 p∂m∂k∂if∂v∂q∂n∂`∂jg
− 1 ∂ P ij∂ P k`PmnP pqs w P rsP tu∂ P vwu ∂t∂r∂p∂m∂k∂if∂v∂q∂n∂`∂jg18
− 1 ∂sP ij∂uP k`∂wPmnP pqP rsP tuP vw∂54 t∂r∂p∂m∂k∂if∂v∂q∂n∂`∂jg
− 1 ∂ P ijP k`PmnP pqP rs∂ P tu∂ P vws w u ∂t∂r∂p∂m∂k∂if∂v∂q∂n∂ ∂108 ` jg
+ 1 P ijP k`Pmn∂ pq rs tu vw
540 w
P ∂qP ∂sP ∂uP ∂p∂m∂k∂if∂v∂t∂r∂n∂`∂jg
+ 2 ∂ P ij∂ P k`∂ PmnP pq∂ P rsP tuP vws u v w ∂p∂m∂k∂if∂t∂r∂q∂n∂`∂jg45
− 4 ∂ P iju ∂wP k`PmnP pq∂ rsqP ∂sP tuP vw∂p∂m∂k∂if∂v∂t∂ ∂135 r n∂`∂jg
+ 1 ∂ P ij∂ P k`PmnP pq∂ P rs∂ P tu vwq s u w P ∂p∂135 m∂k∂if∂v∂t∂r∂n∂`∂jg
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pqP rsP tuP vwq s u w ∂p∂m∂k∂if∂v∂t∂r∂n∂`∂jg54
− 1 ∂uP ij∂ P k`Pmn∂ P pq∂ rs tu vw108 w s qP P P ∂p∂m∂k∂if∂v∂t∂r∂n∂`∂jg
+ 1 ∂ P ijP k`Pmn pqu ∂wP ∂qP
rs∂ P tuP vw
45 s
∂p∂m∂k∂if∂v∂t∂r∂n∂`∂jg
− 1 ∂ P ijP k`s Pmn∂uP pq∂ P rsq ∂wP tuP vw∂p∂m∂k∂if∂v∂t∂r∂90 n∂`∂jg
+ 1 P ijP k`Pmn∂ P pq∂ P rsw q ∂
tu vw
540 s
P ∂uP ∂t∂r∂p∂m∂k∂if∂v∂n∂`∂jg
− 2 ∂ P ij∂ P k`∂ PmnP pq∂ P rsP tuP vw
45 s u v w
∂t∂r∂p∂m∂k∂if∂q∂n∂`∂jg
− 4 ∂ P ij∂ P k`Pmnq s ∂ pq rs tu vw135 uP P ∂wP P ∂t∂r∂p∂m∂k∂if∂v∂n∂`∂jg
+ 1 ∂qP
ij∂ k`wP P
mn∂ P pq∂ P rsP tus u P
vw∂ ∂
135 t r
∂p∂m∂k∂if∂v∂n∂`∂jg
− 1 ∂ P ij∂ P k`∂ PmnP pqP rsP tu∂ P vwq s w u ∂54 t∂r∂p∂m∂k∂if∂v∂n∂`∂jg
− 1 ∂qP ij∂ P k`PmnP pqP rss ∂wP tu∂ P vwu ∂t∂r∂p∂m∂k∂if∂v∂ ∂108 n `∂jg
− 1 ∂ P ijP k`Pmn∂ pq rs tu vw
90 q w
P P ∂sP ∂uP ∂t∂r∂p∂m∂k∂if∂v∂n∂`∂jg
+ 1 ∂ P ijP k`Pmnq ∂
pq
sP ∂ P
rs
w P
tu∂uP
vw∂
45 t
∂r∂p∂m∂k∂if∂v∂n∂`∂jg
+ 2 ∂wP
ijP k`Pmn∂ P pqn ∂
rs tu vw
qP ∂sP ∂uP ∂m∂k∂if∂v∂t∂r∂ ∂315 p `∂jg
− 1 ∂ P ijP k`w ∂ mnuP ∂nP pq∂ P rsq ∂sP tuP vw∂270 m∂k∂if∂v∂t∂r∂p∂`∂jg
550 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
− 1 ∂ P ij∂ P k`∂ Pmn∂ P pqP rs∂ tu vwn q s u wP P ∂m∂k∂if∂v∂t∂r∂ ∂135 p `∂jg
− 2 ∂ P ijP k`s ∂ mnuP ∂nP pq∂qP rs∂wP tuP vw∂105 m∂k∂if∂v∂t∂r∂p∂`∂jg
− 1 ∂ P ijP k`∂ Pmn∂ P pq∂ P rs∂ P tuP vws q n u w ∂m∂k∂if∂v∂270 t∂r∂p∂`∂jg
− 2 ∂ P ij∂ P k`s u ∂wPmn∂nP pq∂ P rsP tuP vwq ∂m∂k∂if∂v∂t∂r∂ ∂135 p `∂jg
+ 1 ∂ ij k` mn pqqP ∂sP ∂uP ∂nP P
rs∂ P tuP vww ∂135 m∂k∂if∂v∂t∂r∂p∂`∂jg
− 2 ∂qP ij∂ P k`Pmn∂ P pq∂ P rsv s w P tu∂ vwuP ∂m∂k∂if∂t∂r∂p∂n∂`∂jg315
− 2 ∂ P ijP k`∂ mn pq rs tu vw
315 n q
P ∂sP ∂uP ∂wP P ∂t∂r∂p∂m∂k∂if∂v∂`∂jg
+ 1 ∂nP
ijP k`Pmn∂ P pq∂ rs tuw qP ∂sP ∂uP
vw∂t∂270 r∂p∂m∂k∂if∂v∂`∂jg
+ 1 ∂nP
ij∂wP
k`∂ mnqP ∂ P
pqP rsP tus ∂
vw
uP ∂t∂r∂p∂m∂k∂if∂v∂`∂jg135
+ 2 ∂ P ijP k`∂ Pmn∂ P pqP rs∂ tu vwn q w sP ∂uP ∂t∂r∂p∂m∂k∂105 if∂v∂`∂jg
+ 1 ∂nP
ijP k`∂ Pmn∂ pq rsq sP P ∂wP
tu∂ P vwu ∂t∂r∂p∂m∂k∂if∂v∂270 `∂jg
− 1 ∂ P ij∂ P k`∂ PmnP pqP rs∂ P tu∂ P vw∂ ∂ ∂ ∂ ∂ ∂ f∂ ∂ ∂ g
135 n q w s u t r p m k i v ` j
+ 2 ∂ ij k` mn pq rs tunP ∂qP ∂sP P ∂wP P ∂
vw
uP ∂t∂r∂p∂m∂k∂if∂v∂`∂jg135
− 2 ∂ P ij∂ P k`Pmn∂ P pq∂ P rsP tuq v s w ∂uP vw∂t∂r∂p∂315 m∂k∂if∂n∂`∂jg
− 1 P ij∂ P k`w ∂ mn pq rs tu vw945 `P ∂nP ∂qP ∂sP ∂uP ∂k∂if∂v∂t∂r∂p∂m∂jg
− 4 ∂ P ij∂ P k`∂ mn pqu w `P ∂nP ∂qP rs∂ P tus P vw∂k∂if∂v∂t∂r∂p∂945 m∂jg
+ 2 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ rs tu vwn q s v uP ∂wP P ∂k∂if∂t∂r∂p∂m∂`∂945 jg
+ 4 ∂ P ij∂ P k`∂ mn pq rs tu vw
945 q s `
P ∂nP ∂uP ∂wP P ∂k∂if∂v∂t∂r∂p∂m∂jg
− 2 ∂ ij k` mn pq rs tu vw
945 n
P ∂qP ∂`P ∂sP ∂uP ∂wP P ∂k∂if∂v∂t∂r∂p∂m∂jg
− 1 P ij∂ P k`w ∂`Pmn∂ P pqn ∂ rsqP ∂sP tu∂uP vw∂945 t∂r∂p∂m∂k∂if∂v∂jg
− 4 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rsP tu∂ P vw∂ ∂ ∂ ∂ ∂ ∂ f∂ ∂ g
945 ` n q s w u t r p m k i v j
− 2 ∂ P ij∂ P k`∂ Pmn∂ P pq∂ P rs∂ P tuP vw∂ ∂ ∂ ∂ ∂ ∂ f∂ ∂ g
945 n q s v u w t r p m k i ` j
− 2 ∂ P ij∂ k` mn pq rs tu vw
945 ` w
P P ∂nP ∂qP ∂sP ∂uP ∂t∂r∂p∂m∂k∂if∂v∂jg
+ 4 ∂ P ij∂ P k`∂ PmnP pq∂ P rs∂ P tu∂ P vw∂ ∂ ∂ ∂ ∂ ∂ f∂ ∂ g
945 ` n w q s u t r p m k i v j
+ 1 P ijP k`PmnP pqP rsP tuP vw∂v∂ ∂5040 t r∂p∂m∂k∂if∂w∂u∂s∂q∂n∂`∂jg
− 1 ∂ P ijP k`PmnP pqP rsP tuP vww ∂t∂r∂p∂m∂k∂360 if∂v∂u∂s∂q∂n∂`∂jg
+ 1 ∂ P ijP k`PmnP pqP rsP tuP vww ∂v∂t∂360 r∂p∂m∂k∂if∂u∂s∂q∂n∂`∂jg
+ 1 ∂uP
ij∂ P k`PmnP pqP rsP tuP vww ∂r∂p∂m∂k∂if∂v∂t∂s∂q∂n∂`∂108 jg
+ 1 ∂ P ij∂ P k`u w P
mnP pqP rsP tuP vw∂v∂t∂r∂p∂m∂k∂if∂ ∂108 s q∂n∂`∂jg
+ 1 ∂ P ijP k`PmnP pq∂ P rss u ∂
tu vw
270 w
P P ∂p∂m∂k∂if∂v∂t∂r∂q∂n∂`∂jg
− 1 ∂sP ij∂ P k`u ∂ PmnP pqw P rsP tuP vw∂p∂m∂k∂if∂162 v∂t∂r∂q∂n∂`∂jg
− 1 ∂ P ijP k`PmnP pq∂ P rs tus u ∂wP P vw∂v∂t∂r∂p∂m∂k∂270 if∂q∂n∂`∂jg
+ 1 ∂ P ij∂ P k`∂ PmnP pqs u w P
rsP tuP vw∂v∂162 t∂r∂p∂m∂k∂if∂q∂n∂`∂jg
− 1 ∂ P ij∂ P k`Pmn∂ P pqP rs∂ P tu vwq s u w P ∂m∂k∂if∂v∂t∂r∂p∂n∂`∂135 jg
− 1 ∂ P ijq ∂ k` mnsP P ∂ P pqP rsu ∂wP tuP vw∂v∂t∂r∂p∂m∂k∂if∂n∂`∂ g135 j
− 2 ∂ P ijP k`∂ Pmn∂ pq rsn q sP ∂uP ∂wP tuP vw∂k∂if∂v∂t∂r∂p∂m∂`∂jg945 )
+ 2 ∂ P ijP k`∂ Pmn∂ pq rsn q sP ∂uP ∂
tu
wP P
vw∂ 7
945 v
∂t∂r∂p∂m∂k∂if∂`∂jg + ō(h̄ )
C.3. ASSOCIATIVITY OF ⋆redaff mod ō(h̄
7) 551
C.3 Associativity of ⋆redaff mod ō(h̄
7)
We inspect that the reduced star product ⋆redaff mod ō(h̄
7) remains associative.
We read the star product:
[2]: from gcaops.graph.formality_graph_complex import FormalityGraphComplex
FGC = FormalityGraphComplex(SR, lazy=True); FGC
[2]: Formality graph complex over Symbolic Ring with Basis consisting of representatives of
isomorphism classes of formality graphs with no automorphisms that induce an odd
permutation on edges
[7]: affine_star7_reduced = FGC.element_from_kgs_encoding(open('data/affine_star7_reduced.
↪→txt').read().rstrip())
We calculate the associator:
[9]: %time affine_assoc7_reduced = affine_star7_reduced.insertion(0, affine_star7_reduced,␣
↪→max_num_aerial=7, max_aerial_in_degree=1) - affine_star7_reduced.insertion(1,␣
↪→affine_star7_reduced, max_num_aerial=7, max_aerial_in_degree=1)
CPU times: user 49.7 s, sys: 0 ns, total: 49.7 s
Wall time: 49.3 s
We prove the associativity:
[12]: from gcaops.graph.leibniz_graph_expansion import␣
↪→kontsevich_graph_sum_to_leibniz_graph_sum
from gcaops.graph.leibniz_graph_expansion import␣
↪→leibniz_graph_sum_to_kontsevich_graph_sum
for order in range(2,8):
print('h^{}:'.format(order))
assoc_part = affine_assoc7_reduced.homogeneous_part(3, order, 2*order)
diff_orders = list(assoc_part.differential_orders())
print('Number of differential orders:', len(diff_orders))
for diff_order in diff_orders:
print(diff_order, end=': ', flush=True)
component = assoc_part.part_of_differential_order(diff_order)
component_leibniz = kontsevich_graph_sum_to_leibniz_graph_sum(component,␣
↪→max_aerial_in_degree=1, verbose=True)
print(leibniz_graph_sum_to_kontsevich_graph_sum(component_leibniz,␣
↪→max_aerial_in_degree=1) == component)
h^2:
Number of differential orders: 1
(1, 1, 1): 3K -> +1L -> +0K
True
h^3:
Number of differential orders: 7
(1, 1, 2): 4K -> +3L -> +2K
True
(2, 1, 2): 3K -> +1L -> +0K
True
552 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
(1, 2, 2): 3K -> +1L -> +0K
True
(1, 1, 1): 2K -> +2L -> +2K
True
(2, 2, 1): 3K -> +1L -> +0K
True
(1, 2, 1): 4K -> +3L -> +2K
True
(2, 1, 1): 4K -> +3L -> +2K
True
h^4:
Number of differential orders: 22
(1, 1, 2): 4K -> +6L -> +5K
True
(1, 1, 3): 8K -> +6L -> +2K
True
(2, 1, 2): 18K -> +17L -> +8K
True
(1, 2, 2): 18K -> +18L -> +8K
True
(2, 1, 3): 7K -> +4L -> +2K
True
(1, 2, 3): 7K -> +4L -> +2K
True
(3, 1, 3): 3K -> +1L -> +0K
True
(2, 2, 3): 3K -> +1L -> +0K
True
(1, 3, 3): 3K -> +1L -> +0K
True
(3, 1, 2): 7K -> +4L -> +2K
True
(2, 2, 2): 16K -> +12L -> +8K
True
(1, 3, 2): 7K -> +4L -> +2K
True
(3, 1, 1): 8K -> +6L -> +2K
True
(2, 2, 1): 20K -> +18L -> +6K
True
(1, 3, 1): 9K -> +6L -> +1K
True
(2, 1, 1): 4K -> +6L -> +5K
True
(3, 2, 2): 3K -> +1L -> +0K
True
(2, 3, 2): 3K -> +1L -> +0K
True
(1, 2, 1): 2K -> +2L -> +4K -> +8L -> +6K
True
(2, 3, 1): 7K -> +4L -> +2K
True
(3, 2, 1): 7K -> +4L -> +2K
True
(3, 3, 1): 3K -> +1L -> +0K
True
h^5:
Number of differential orders: 50
C.3. ASSOCIATIVITY OF ⋆redaff mod ō(h̄
7) 553
(2, 1, 2): 8K -> +18L -> +18K
True
(1, 2, 2): 12K -> +30L -> +23K
True
(3, 1, 2): 40K -> +58L -> +35K
True
(2, 2, 2): 77K -> +110L -> +58K -> +37L -> +19K
True
(2, 1, 3): 40K -> +55L -> +33K
True
(1, 2, 3): 43K -> +61L -> +30K
True
(2, 2, 3): 65K -> +59L -> +23K
True
(3, 2, 2): 71K -> +62L -> +21K
True
(2, 3, 2): 68K -> +60L -> +20K
True
(1, 3, 2): 36K -> +50L -> +25K -> +11L -> +8K
True
(3, 1, 3): 31K -> +28L -> +13K
True
(1, 3, 3): 29K -> +28L -> +15K
True
(3, 2, 3): 22K -> +14L -> +8K
True
(2, 3, 3): 22K -> +14L -> +8K
True
(3, 1, 4): 7K -> +4L -> +2K
True
(2, 2, 4): 10K -> +5L -> +2K
True
(1, 3, 4): 7K -> +4L -> +2K
True
(4, 2, 3): 3K -> +1L -> +0K
True
(3, 3, 3): 3K -> +1L -> +0K
True
(4, 1, 3): 7K -> +4L -> +2K
True
(2, 4, 3): 3K -> +1L -> +0K
True
(1, 4, 3): 7K -> +4L -> +2K
True
(4, 1, 4): 3K -> +1L -> +0K
True
(3, 2, 4): 3K -> +1L -> +0K
True
(2, 3, 4): 3K -> +1L -> +0K
True
(1, 4, 4): 3K -> +1L -> +0K
True
(2, 1, 4): 14K -> +10L -> +4K
True
(1, 2, 4): 14K -> +10L -> +4K
True
(4, 2, 2): 10K -> +5L -> +2K
True
554 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
(3, 3, 2): 22K -> +14L -> +8K
True
(2, 4, 2): 10K -> +5L -> +2K
True
(4, 1, 2): 12K -> +9L -> +4K
True
(1, 4, 2): 13K -> +9L -> +3K
True
(1, 1, 4): 10K -> +9L -> +5K -> +1L -> +0K
True
(4, 2, 1): 12K -> +9L -> +4K
True
(4, 1, 1): 10K -> +9L -> +5K -> +1L -> +0K
True
(3, 3, 1): 33K -> +29L -> +11K
True
(3, 2, 1): 44K -> +61L -> +31K
True
(2, 4, 1): 13K -> +9L -> +3K
True
(2, 3, 1): 35K -> +50L -> +26K -> +11L -> +8K
True
(1, 4, 1): 8K -> +10L -> +7K
True
(1, 1, 3): 4K -> +10L -> +10K
True
(3, 1, 1): 4K -> +10L -> +10K
True
(2, 2, 1): 12K -> +30L -> +23K
True
(1, 3, 1): 6K -> +16L -> +12K
True
(4, 3, 2): 3K -> +1L -> +0K
True
(3, 4, 2): 3K -> +1L -> +0K
True
(3, 4, 1): 7K -> +4L -> +2K
True
(4, 3, 1): 7K -> +4L -> +2K
True
(4, 4, 1): 3K -> +1L -> +0K
True
h^6:
Number of differential orders: 95
(2, 2, 2): 46K -> +111L -> +104K -> +202L -> +130K
True
(1, 3, 2): 32K -> +80L -> +57K -> +96L -> +60K
True
(3, 1, 2): 27K -> +66L -> +74K -> +135L -> +66K
True
(3, 2, 2): 216K -> +423L -> +259K -> +205L -> +76K
True
(2, 3, 2): 225K -> +448L -> +239K
True
(4, 1, 2): 64K -> +109L -> +74K -> +53L -> +25K
True
(4, 2, 2): 164K -> +201L -> +90K -> +19L -> +10K
True
C.3. ASSOCIATIVITY OF ⋆redaff mod ō(h̄
7) 555
(3, 3, 2): 268K -> +362L -> +165K -> +55L -> +23K
True
(2, 2, 3): 219K -> +422L -> +251K -> +205L -> +83K
True
(1, 3, 3): 136K -> +246L -> +121K -> +73L -> +33K
True
(3, 1, 3): 113K -> +203L -> +141K -> +112L -> +36K
True
(3, 2, 3): 266K -> +358L -> +171K -> +57L -> +18K
True
(2, 3, 3): 270K -> +357L -> +160K -> +59L -> +26K
True
(4, 1, 3): 87K -> +106L -> +56K -> +18L -> +8K
True
(4, 2, 3): 99K -> +83L -> +30K
True
(3, 3, 3): 169K -> +139L -> +44K
True
(2, 4, 2): 147K -> +182L -> +78K -> +18L -> +11K
True
(1, 4, 2): 76K -> +126L -> +58K -> +32L -> +20K
True
(4, 3, 2): 99K -> +83L -> +30K
True
(3, 4, 2): 92K -> +78L -> +29K
True
(5, 1, 2): 24K -> +18L -> +8K -> +1L -> +0K
True
(5, 2, 2): 25K -> +17L -> +8K
True
(1, 4, 3): 74K -> +88L -> +41K -> +17L -> +10K
True
(2, 4, 3): 90K -> +77L -> +31K
True
(5, 2, 3): 14K -> +7L -> +2K
True
(4, 3, 3): 28K -> +16L -> +8K
True
(3, 4, 3): 28K -> +16L -> +8K
True
(3, 2, 4): 97K -> +81L -> +30K
True
(2, 3, 4): 95K -> +80L -> +32K
True
(1, 4, 4): 36K -> +31L -> +15K
True
(4, 1, 4): 41K -> +33L -> +14K
True
(4, 2, 4): 22K -> +14L -> +8K
True
(3, 3, 4): 28K -> +16L -> +8K
True
(2, 4, 4): 22K -> +14L -> +8K
True
(2, 5, 3): 10K -> +5L -> +2K
True
(1, 5, 3): 16K -> +10L -> +3K
True
556 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
(5, 1, 3): 18K -> +12L -> +5K
True
(5, 3, 3): 3K -> +1L -> +0K
True
(4, 4, 3): 3K -> +1L -> +0K
True
(3, 5, 3): 3K -> +1L -> +0K
True
(1, 5, 4): 7K -> +4L -> +2K
True
(5, 2, 4): 3K -> +1L -> +0K
True
(4, 3, 4): 3K -> +1L -> +0K
True
(3, 4, 4): 3K -> +1L -> +0K
True
(2, 5, 4): 3K -> +1L -> +0K
True
(2, 2, 4): 167K -> +202L -> +87K -> +18L -> +10K
True
(1, 3, 4): 91K -> +114L -> +52K
True
(3, 1, 4): 88K -> +106L -> +55K -> +18L -> +8K
True
(2, 5, 2): 20K -> +13L -> +5K
True
(1, 5, 2): 20K -> +17L -> +8K
True
(5, 3, 2): 14K -> +7L -> +2K
True
(4, 4, 2): 22K -> +14L -> +8K
True
(3, 5, 2): 10K -> +5L -> +2K
True
(1, 2, 4): 80K -> +129L -> +62K
True
(2, 1, 4): 64K -> +109L -> +74K -> +53L -> +25K
True
(4, 2, 1): 80K -> +129L -> +62K
True
(3, 3, 1): 134K -> +244L -> +125K -> +75L -> +30K
True
(2, 4, 1): 76K -> +126L -> +58K -> +32L -> +20K
True
(4, 3, 1): 91K -> +113L -> +52K
True
(3, 4, 1): 75K -> +92L -> +42K -> +13L -> +8K
True
(2, 5, 1): 20K -> +17L -> +8K
True
(1, 5, 1): 18K -> +15L -> +3K
True
(5, 2, 1): 24K -> +18L -> +8K -> +1L -> +0K
True
(5, 1, 1): 14K -> +15L -> +7K
True
(5, 3, 1): 18K -> +12L -> +5K
True
C.3. ASSOCIATIVITY OF ⋆redaff mod ō(h̄
7) 557
(4, 4, 1): 39K -> +32L -> +12K
True
(3, 5, 1): 16K -> +10L -> +3K
True
(1, 2, 3): 38K -> +92L -> +74K -> +112L -> +58K
True
(2, 1, 3): 28K -> +68L -> +75K -> +134L -> +65K
True
(3, 2, 1): 38K -> +93L -> +71K -> +109L -> +57K
True
(2, 3, 1): 32K -> +80L -> +57K -> +96L -> +60K
True
(1, 4, 1): 8K -> +18L -> +18K -> +34L -> +22K
True
(4, 1, 1): 12K -> +25L -> +19K
True
(5, 4, 2): 3K -> +1L -> +0K
True
(4, 5, 2): 3K -> +1L -> +0K
True
(1, 1, 4): 12K -> +25L -> +19K
True
(4, 1, 5): 9K -> +5L -> +2K
True
(3, 2, 5): 14K -> +7L -> +2K
True
(2, 3, 5): 14K -> +7L -> +2K
True
(1, 4, 5): 9K -> +5L -> +2K
True
(3, 1, 5): 20K -> +13L -> +5K
True
(2, 2, 5): 29K -> +19L -> +8K
True
(1, 3, 5): 20K -> +13L -> +5K
True
(2, 1, 5): 24K -> +18L -> +8K -> +1L -> +0K
True
(1, 2, 5): 24K -> +18L -> +8K -> +1L -> +0K
True
(1, 1, 5): 14K -> +15L -> +7K
True
(5, 1, 5): 3K -> +1L -> +0K
True
(4, 2, 5): 3K -> +1L -> +0K
True
(3, 3, 5): 3K -> +1L -> +0K
True
(2, 4, 5): 3K -> +1L -> +0K
True
(1, 5, 5): 3K -> +1L -> +0K
True
(5, 1, 4): 9K -> +5L -> +2K
True
(4, 5, 1): 7K -> +4L -> +2K
True
(5, 4, 1): 9K -> +5L -> +2K
True
558 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
(5, 5, 1): 3K -> +1L -> +0K
True
h^7:
Number of differential orders: 161
(3, 2, 2): 87K -> +314L -> +353K -> +915L -> +533K
True
(2, 3, 2): 107K -> +372L -> +341K -> +835L -> +468K
True
(3, 3, 2): 762K -> +1930L -> +1258K -> +1463L -> +475K
True
(2, 4, 2): 509K -> +1243L -> +699K -> +669L -> +269K
True
(4, 2, 2): 455K -> +1114L -> +766K -> +836L -> +295K
True
(4, 3, 2): 777K -> +1378L -> +672K -> +361L -> +125K
True
(3, 4, 2): 799K -> +1399L -> +640K -> +330L -> +113K
True
(5, 2, 2): 288K -> +438L -> +236K -> +109L -> +40K
True
(5, 3, 2): 264K -> +296L -> +133K -> +22L -> +5K
True
(4, 4, 2): 411K -> +506L -> +214K -> +61L -> +26K
True
(3, 2, 3): 638K -> +1612L -> +1293K -> +1732L -> +570K
True
(2, 3, 3): 760K -> +1919L -> +1251K -> +1457L -> +494K
True
(4, 2, 3): 736K -> +1333L -> +722K -> +421L -> +130K
True
(3, 3, 3): 1252K -> +2303L -> +1110K -> +658L -> +205K
True
(2, 4, 3): 795K -> +1392L -> +636K -> +336L -> +120K
True
(4, 3, 3): 632K -> +785L -> +334K -> +88L -> +30K
True
(3, 4, 3): 629K -> +781L -> +326K -> +86L -> +31K
True
(5, 2, 3): 262K -> +295L -> +135K -> +23L -> +5K
True
(5, 3, 3): 135K -> +107L -> +42K
True
(4, 4, 3): 208K -> +167L -> +57K
True
(3, 5, 2): 231K -> +259L -> +105K -> +21L -> +11K
True
(2, 5, 2): 297K -> +447L -> +196K -> +71L -> +33K
True
(5, 4, 2): 107K -> +86L -> +32K
True
(4, 5, 2): 101K -> +82L -> +30K
True
(6, 2, 2): 42K -> +30L -> +13K -> +1L -> +0K
True
(6, 3, 2): 28K -> +18L -> +8K
True
(2, 5, 3): 227K -> +254L -> +107K -> +26L -> +13K
True
C.3. ASSOCIATIVITY OF ⋆redaff mod ō(h̄
7) 559
(3, 5, 3): 117K -> +96L -> +40K
True
(5, 4, 3): 32K -> +18L -> +8K
True
(4, 5, 3): 28K -> +16L -> +8K
True
(4, 2, 4): 402K -> +494L -> +219K -> +63L -> +20K
True
(3, 3, 4): 641K -> +791L -> +330K -> +85L -> +29K
True
(2, 4, 4): 408K -> +499L -> +212K -> +68L -> +31K
True
(5, 2, 4): 103K -> +83L -> +32K
True
(4, 3, 4): 214K -> +169L -> +57K
True
(3, 4, 4): 210K -> +167L -> +57K
True
(2, 5, 4): 100K -> +81L -> +31K
True
(5, 3, 4): 32K -> +18L -> +8K
True
(4, 4, 4): 34K -> +18L -> +8K
True
(3, 5, 4): 28K -> +16L -> +8K
True
(3, 6, 3): 10K -> +5L -> +2K
True
(2, 6, 3): 23K -> +14L -> +5K
True
(6, 3, 3): 10K -> +5L -> +2K
True
(6, 2, 3): 28K -> +18L -> +8K
True
(6, 4, 3): 3K -> +1L -> +0K
True
(5, 5, 3): 3K -> +1L -> +0K
True
(4, 6, 3): 3K -> +1L -> +0K
True
(2, 6, 4): 10K -> +5L -> +2K
True
(6, 2, 4): 10K -> +5L -> +2K
True
(6, 3, 4): 3K -> +1L -> +0K
True
(5, 4, 4): 3K -> +1L -> +0K
True
(4, 5, 4): 3K -> +1L -> +0K
True
(3, 6, 4): 3K -> +1L -> +0K
True
(3, 2, 4): 733K -> +1340L -> +727K -> +411L -> +127K
True
(2, 3, 4): 779K -> +1389L -> +675K -> +352L -> +122K
True
(3, 6, 2): 23K -> +14L -> +5K
True
560 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
(2, 6, 2): 36K -> +27L -> +11K
True
(6, 4, 2): 10K -> +5L -> +2K
True
(5, 5, 2): 24K -> +15L -> +8K
True
(4, 6, 2): 10K -> +5L -> +2K
True
(2, 2, 4): 456K -> +1117L -> +772K -> +848L -> +289K
True
(5, 3, 1): 160K -> +243L -> +144K -> +78L -> +36K
True
(5, 2, 1): 106K -> +224L -> +152K -> +124L -> +58K
True
(4, 4, 1): 259K -> +452L -> +238K -> +124L -> +48K
True
(4, 3, 1): 289K -> +712L -> +422K -> +353L -> +134K
True
(3, 5, 1): 158K -> +240L -> +122K -> +52L -> +23K
True
(3, 4, 1): 280K -> +681L -> +392K -> +359L -> +142K
True
(2, 5, 1): 107K -> +226L -> +136K -> +94L -> +42K
True
(5, 4, 1): 114K -> +133L -> +63K
True
(4, 5, 1): 86K -> +99L -> +50K -> +17L -> +7K
True
(3, 6, 1): 24K -> +20L -> +10K
True
(2, 6, 1): 30K -> +28L -> +12K
True
(6, 3, 1): 28K -> +21L -> +10K -> +1L -> +0K
True
(6, 2, 1): 34K -> +32L -> +16K -> +2L -> +1K
True
(6, 4, 1): 15K -> +10L -> +4K
True
(5, 5, 1): 35K -> +30L -> +12K
True
(4, 6, 1): 16K -> +10L -> +3K
True
(2, 2, 3): 87K -> +318L -> +361K -> +922L -> +510K
True
(4, 2, 1): 42K -> +139L -> +115K -> +221L -> +128K
True
(3, 3, 1): 78K -> +276L -> +227K -> +453L -> +240K
True
(2, 4, 1): 49K -> +164L -> +125K -> +230L -> +135K
True
(1, 5, 2): 111K -> +233L -> +134K -> +87L -> +40K
True
(5, 1, 2): 94K -> +192L -> +162K -> +157L -> +60K
True
(1, 5, 3): 156K -> +237L -> +123K -> +55L -> +24K
True
(5, 1, 3): 147K -> +232L -> +160K -> +94L -> +42K
True
C.3. ASSOCIATIVITY OF ⋆red 7aff mod ō(h̄ ) 561
(6, 5, 2): 3K -> +1L -> +0K
True
(5, 6, 2): 3K -> +1L -> +0K
True
(1, 6, 2): 30K -> +28L -> +12K
True
(6, 1, 2): 34K -> +32L -> +16K -> +2L -> +1K
True
(1, 6, 3): 24K -> +20L -> +10K
True
(6, 1, 3): 28K -> +21L -> +10K -> +1L -> +0K
True
(3, 1, 4): 204K -> +481L -> +461K -> +600L -> +201K
True
(1, 3, 4): 287K -> +703L -> +423K -> +368L -> +135K
True
(4, 1, 5): 111K -> +127L -> +67K -> +15L -> +3K
True
(3, 2, 5): 261K -> +296L -> +135K -> +21L -> +4K
True
(2, 3, 5): 263K -> +297L -> +133K -> +20L -> +4K
True
(1, 4, 5): 113K -> +132L -> +65K
True
(3, 1, 5): 145K -> +231L -> +160K -> +94L -> +42K
True
(2, 2, 5): 296K -> +444L -> +234K -> +105L -> +38K
True
(1, 3, 5): 159K -> +243L -> +145K -> +77L -> +34K
True
(4, 1, 4): 231K -> +410L -> +265K -> +177L -> +72K
True
(1, 4, 4): 258K -> +451L -> +239K -> +125L -> +48K
True
(2, 1, 5): 93K -> +191L -> +162K -> +152L -> +55K
True
(1, 2, 5): 104K -> +223L -> +149K -> +119L -> +57K
True
(4, 1, 3): 202K -> +470L -> +456K -> +601L -> +205K
True
(1, 4, 3): 283K -> +693L -> +396K -> +355L -> +143K
True
(5, 1, 6): 9K -> +5L -> +2K
True
(4, 2, 6): 10K -> +5L -> +2K
True
(3, 3, 6): 10K -> +5L -> +2K
True
(2, 4, 6): 10K -> +5L -> +2K
True
(1, 5, 6): 9K -> +5L -> +2K
True
(4, 1, 6): 17K -> +11L -> +4K
True
(3, 2, 6): 32K -> +20L -> +8K
True
(2, 3, 6): 32K -> +20L -> +8K
True
562 APPENDIX C. KONTSEVICH’S AFFINE STAR PRODUCT ⋆ mod ō(h̄7)
(1, 4, 6): 17K -> +11L -> +4K
True
(5, 1, 5): 35K -> +29L -> +12K
True
(4, 2, 5): 104K -> +83L -> +29K
True
(3, 3, 5): 135K -> +108L -> +42K
True
(2, 4, 5): 107K -> +86L -> +30K
True
(1, 5, 5): 36K -> +30L -> +11K
True
(3, 1, 6): 28K -> +21L -> +10K -> +1L -> +0K
True
(2, 2, 6): 46K -> +32L -> +13K -> +1L -> +0K
True
(1, 3, 6): 28K -> +21L -> +10K -> +1L -> +0K
True
(5, 1, 4): 111K -> +127L -> +66K -> +14L -> +2K
True
(1, 5, 4): 86K -> +98L -> +48K -> +18L -> +9K
True
(2, 1, 6): 36K -> +33L -> +16K -> +2L -> +1K
True
(1, 2, 6): 36K -> +33L -> +16K -> +2L -> +1K
True
(1, 1, 6): 16K -> +18L -> +12K -> +3L -> +0K
True
(6, 1, 6): 3K -> +1L -> +0K
True
(5, 2, 6): 3K -> +1L -> +0K
True
(4, 3, 6): 3K -> +1L -> +0K
True
(3, 4, 6): 3K -> +1L -> +0K
True
(2, 5, 6): 3K -> +1L -> +0K
True
(1, 6, 6): 3K -> +1L -> +0K
True
(6, 1, 5): 9K -> +5L -> +2K
True
(5, 2, 5): 22K -> +14L -> +8K
True
(4, 3, 5): 32K -> +18L -> +8K
True
(3, 4, 5): 32K -> +18L -> +8K
True
(2, 5, 5): 24K -> +15L -> +8K
True
(1, 6, 5): 7K -> +4L -> +2K
True
(6, 1, 4): 15K -> +10L -> +4K
True
(1, 6, 4): 16K -> +10L -> +3K
True
(3, 1, 3): 32K -> +116L -> +161K -> +422L -> +272K
True
C.3. ASSOCIATIVITY OF ⋆redaff mod ō(h̄
7) 563
(1, 3, 3): 78K -> +277L -> +225K -> +442L -> +240K
True
(2, 1, 4): 21K -> +65L -> +72K -> +186L -> +137K
True
(1, 2, 4): 42K -> +139L -> +113K -> +219L -> +128K
True
(4, 1, 2): 21K -> +65L -> +72K -> +186L -> +137K
True
(1, 4, 2): 49K -> +164L -> +125K -> +226L -> +132K
True
(6, 2, 5): 3K -> +1L -> +0K
True
(5, 3, 5): 3K -> +1L -> +0K
True
(4, 4, 5): 3K -> +1L -> +0K
True
(3, 5, 5): 3K -> +1L -> +0K
True
(2, 6, 5): 3K -> +1L -> +0K
True
(6, 1, 1): 16K -> +18L -> +12K -> +3L -> +0K
True
(1, 6, 1): 12K -> +18L -> +15K
True
(1, 1, 5): 6K -> +16L -> +16K
True
(5, 1, 1): 6K -> +16L -> +16K
True
(1, 5, 1): 10K -> +28L -> +20K
True
(6, 6, 1): 3K -> +1L -> +0K
True
(5, 6, 1): 7K -> +4L -> +2K
True
(6, 5, 1): 9K -> +5L -> +2K
True
We conclude that the Kontsevich graph expansion of the reduced affine ⋆-product itself
is associative up to ō(h̄7); the analytic formula which one writes for f ⋆redaff g mod ō(h̄
7)
with arbitrary arguments f, g ∈ C∞(M) and any affine Poisson structure P in ⋆aff mod
ō(h̄7) is, we establish in this appendix, identically equal to the formula f ⋆aff g mod ō(h̄7):
all the coefficients of differential polynomials in f, g and P ij are rational numbers, and
ζ(3)2/π6 is not met at all.

Appendix D
Flows Qγ(P ) and factorizations
[[P,Qγ(P )]] = ♢γ(P, [[P, P ]])
Here are the flows Qγ(P ) and the respective solutions ♢γ(P, [[P, P ]]) of the Poisson cocycle
factorization problems [[P,Qγ(P )]] = ♢γ(P, [[P, P ]]) for the tetrahedral and pentagon-wheel
flows.
D.1 The tetrahedral flow Qγ3(P )
The tetrahedral flow Qγ3 :
[(2,3),(2,5),(3,4),(3,5),(4,1),(4,2),(5,0),(5,4)] -576
[(2,4),(2,5),(3,2),(3,5),(4,3),(4,5),(5,0),(5,1)] -192
[(2,3),(2,4),(3,4),(3,5),(4,1),(4,5),(5,0),(5,2)] -576
The Poisson differential [[P,Qγ3 ]]:
[(3,4),(3,5),(4,5),(4,6),(5,6),(5,7),(6,2),(6,3),(7,0),(7,1)] -2304
[(3,5),(3,6),(4,3),(4,6),(5,4),(5,6),(6,2),(6,7),(7,0),(7,1)] 768
[(3,5),(3,6),(4,3),(4,6),(5,4),(5,7),(6,2),(6,5),(7,0),(7,1)] -2304
[(3,5),(3,7),(4,3),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,5)] -2304
[(3,5),(3,7),(4,3),(4,7),(5,4),(5,7),(6,1),(6,2),(7,0),(7,6)] 768
[(3,4),(3,5),(4,5),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] 2304
[(3,4),(3,7),(4,6),(4,7),(5,2),(5,3),(6,1),(6,3),(7,0),(7,6)] 2304
[(3,4),(3,7),(4,6),(4,7),(5,2),(5,4),(6,1),(6,3),(7,0),(7,6)] 2304
[(3,4),(3,7),(4,6),(4,7),(5,2),(5,6),(6,1),(6,3),(7,0),(7,6)] 2304
[(3,4),(3,7),(4,6),(4,7),(5,2),(5,7),(6,1),(6,3),(7,0),(7,6)] 2304
[(3,5),(3,7),(4,3),(4,7),(5,4),(5,7),(6,2),(6,5),(7,0),(7,1)] 2304
[(3,5),(3,7),(4,3),(4,7),(5,4),(5,7),(6,2),(6,7),(7,0),(7,1)] 768
[(3,4),(3,6),(4,6),(4,7),(5,2),(5,3),(6,1),(6,7),(7,0),(7,3)] 2304
[(3,4),(3,6),(4,6),(4,7),(5,2),(5,4),(6,1),(6,7),(7,0),(7,3)] 2304
[(3,4),(3,6),(4,6),(4,7),(5,2),(5,6),(6,1),(6,7),(7,0),(7,3)] 2304
[(3,4),(3,6),(4,6),(4,7),(5,2),(5,7),(6,1),(6,7),(7,0),(7,3)] 2304
[(3,4),(3,6),(4,5),(4,6),(5,2),(5,3),(6,1),(6,5),(7,0),(7,3)] 2304
[(3,4),(3,6),(4,5),(4,6),(5,2),(5,3),(6,1),(6,5),(7,0),(7,4)] 2304
[(3,4),(3,6),(4,5),(4,6),(5,2),(5,3),(6,1),(6,5),(7,0),(7,5)] 2304
[(3,4),(3,6),(4,5),(4,6),(5,2),(5,3),(6,1),(6,5),(7,0),(7,6)] 2304
[(3,5),(3,6),(4,3),(4,6),(5,4),(5,6),(6,1),(6,2),(7,0),(7,5)] 2304
[(3,5),(3,6),(4,3),(4,6),(5,4),(5,6),(6,1),(6,2),(7,0),(7,6)] 768
[(3,4),(3,5),(4,5),(4,6),(5,2),(5,6),(6,1),(6,3),(7,0),(7,3)] 2304
[(3,4),(3,5),(4,5),(4,6),(5,2),(5,6),(6,1),(6,3),(7,0),(7,4)] 2304
[(3,4),(3,5),(4,5),(4,6),(5,2),(5,6),(6,1),(6,3),(7,0),(7,5)] 2304
[(3,4),(3,5),(4,5),(4,6),(5,2),(5,6),(6,1),(6,3),(7,0),(7,6)] 2304
[(3,4),(3,5),(4,5),(4,6),(5,6),(5,7),(6,1),(6,3),(7,0),(7,2)] 2304
[(3,5),(3,6),(4,3),(4,6),(5,4),(5,6),(6,1),(6,7),(7,0),(7,2)] -768
[(3,5),(3,6),(4,3),(4,6),(5,4),(5,7),(6,1),(6,5),(7,0),(7,2)] 2304
[(3,4),(3,7),(4,5),(4,7),(5,2),(5,3),(6,1),(6,3),(7,0),(7,5)] -2304
[(3,4),(3,7),(4,5),(4,7),(5,2),(5,3),(6,1),(6,4),(7,0),(7,5)] -2304
565
566 APPENDIX D. FLOWS Qγ(P )
[(3,4),(3,7),(4,5),(4,7),(5,2),(5,3),(6,1),(6,5),(7,0),(7,5)] -2304
[(3,4),(3,7),(4,5),(4,7),(5,2),(5,3),(6,1),(6,7),(7,0),(7,5)] -2304
[(3,5),(3,7),(4,3),(4,7),(5,4),(5,7),(6,1),(6,5),(7,0),(7,2)] -2304
[(3,5),(3,7),(4,3),(4,7),(5,4),(5,7),(6,1),(6,7),(7,0),(7,2)] -768
[(3,4),(3,5),(4,5),(4,7),(5,2),(5,7),(6,1),(6,3),(7,0),(7,3)] -2304
[(3,4),(3,5),(4,5),(4,7),(5,2),(5,7),(6,1),(6,4),(7,0),(7,3)] -2304
[(3,4),(3,5),(4,5),(4,7),(5,2),(5,7),(6,1),(6,5),(7,0),(7,3)] -2304
[(3,4),(3,5),(4,5),(4,7),(5,2),(5,7),(6,1),(6,7),(7,0),(7,3)] -2304
The Leibniz graph factorization ♢γ3(P, [[P, P ]]):
[(3,4),(3,6),(4,5),(4,6),(5,1),(5,2),(5,3),(6,0),(6,5)] -2304
[(3,5),(3,6),(4,3),(4,6),(5,4),(5,6),(6,0),(6,1),(6,2)] 768
[(3,4),(3,6),(4,5),(4,6),(5,2),(5,3),(6,0),(6,1),(6,5)] 2304
[(3,4),(3,5),(4,5),(4,6),(5,1),(5,2),(6,0),(6,3),(6,5)] -2304
[(3,4),(3,5),(4,2),(4,6),(5,1),(5,4),(6,0),(6,3),(6,5)] 2304
[(3,4),(3,5),(3,6),(4,5),(4,6),(5,1),(5,2),(6,0),(6,5)] 2304
[(3,4),(3,5),(3,6),(4,2),(4,5),(5,1),(5,6),(6,0),(6,4)] -2304
[(3,5),(3,6),(4,2),(4,3),(5,1),(5,4),(6,0),(6,4),(6,5)] -2304
[(3,4),(3,6),(4,5),(4,6),(5,1),(5,3),(6,0),(6,2),(6,5)] -2304
[(3,4),(3,5),(4,2),(4,5),(5,1),(5,6),(6,0),(6,3),(6,4)] 2304
[(3,4),(3,5),(3,6),(4,2),(4,6),(5,1),(5,4),(6,0),(6,5)] -2304
[(3,4),(3,6),(4,2),(4,5),(5,1),(5,3),(6,0),(6,4),(6,5)] -2304
[(3,4),(3,5),(4,5),(4,6),(5,1),(5,6),(6,0),(6,2),(6,3)] -2304
[(3,4),(3,6),(4,5),(4,6),(5,1),(5,3),(5,6),(6,0),(6,2)] -2304
[(3,4),(3,6),(4,2),(4,5),(5,1),(5,3),(5,6),(6,0),(6,4)] -2304
[(3,4),(3,5),(3,6),(4,5),(4,6),(5,1),(5,6),(6,0),(6,2)] -2304
[(3,5),(3,6),(4,2),(4,3),(5,1),(5,4),(5,6),(6,0),(6,4)] -2304
[(3,4),(3,5),(4,5),(4,6),(5,1),(5,2),(5,6),(6,0),(6,3)] -2304
[(3,4),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,0),(6,5)] -2304
[(3,4),(3,5),(4,2),(4,6),(5,1),(5,4),(5,6),(6,0),(6,3)] 2304
[(3,4),(3,5),(4,5),(4,6),(5,2),(5,6),(6,0),(6,1),(6,3)] 2304
[(3,4),(3,6),(4,5),(4,6),(5,2),(5,3),(5,6),(6,0),(6,1)] 2304
[(3,5),(3,6),(4,2),(4,3),(4,6),(5,1),(5,4),(6,0),(6,5)] 2304
[(3,4),(3,5),(3,6),(4,5),(4,6),(5,2),(5,6),(6,0),(6,1)] 2304
[(3,4),(3,6),(4,2),(4,5),(4,6),(5,1),(5,3),(6,0),(6,5)] 2304
[(3,5),(3,6),(4,2),(4,3),(4,5),(5,1),(5,6),(6,0),(6,4)] 2304
[(3,4),(3,5),(4,2),(4,5),(4,6),(5,1),(5,6),(6,0),(6,3)] 2304
D.2 The pentagon-wheel flow Qγ5(P )
The pentagon-wheel flow Qγ5 :
[(2,6),(2,7),(3,4),(3,6),(4,5),(4,7),(5,2),(5,3),(6,1),(6,5),(7,0),(7,5)] -7200
[(2,5),(2,7),(3,6),(3,7),(4,3),(4,6),(5,4),(5,6),(6,1),(6,2),(7,0),(7,6)] 7200
[(2,3),(2,5),(3,5),(3,7),(4,6),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,5)] -7200
[(2,6),(2,7),(3,4),(3,6),(4,2),(4,6),(5,3),(5,7),(6,5),(6,7),(7,0),(7,1)] 7200
[(2,5),(2,7),(3,2),(3,5),(4,3),(4,6),(5,4),(5,7),(6,1),(6,5),(7,0),(7,6)] 7200
[(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,3),(5,6),(6,1),(6,2),(7,0),(7,6)] -7200
[(2,5),(2,7),(3,5),(3,7),(4,3),(4,6),(5,4),(5,6),(6,1),(6,2),(7,0),(7,5)] 7200
[(2,5),(2,6),(3,2),(3,5),(4,3),(4,7),(5,4),(5,7),(6,1),(6,5),(7,0),(7,6)] -7200
[(2,6),(2,7),(3,6),(3,7),(4,3),(4,6),(5,2),(5,4),(6,5),(6,7),(7,0),(7,1)] -7200
[(2,5),(2,7),(3,5),(3,6),(4,3),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,5)] 7200
[(2,5),(2,6),(3,5),(3,7),(4,2),(4,3),(5,4),(5,7),(6,1),(6,5),(7,0),(7,6)] -7200
[(2,6),(2,7),(3,5),(3,7),(4,3),(4,7),(5,2),(5,7),(6,4),(6,7),(7,0),(7,1)] 1440
[(2,5),(2,6),(3,6),(3,7),(4,3),(4,6),(5,4),(5,6),(6,1),(6,7),(7,0),(7,2)] 7200
[(2,5),(2,7),(3,4),(3,5),(4,5),(4,6),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] -7200
[(2,4),(2,5),(3,5),(3,7),(4,3),(4,5),(5,6),(5,7),(6,1),(6,2),(7,0),(7,6)] 7200
[(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,3),(5,7),(6,1),(6,2),(7,0),(7,6)] -3600
[(2,6),(2,7),(3,6),(3,7),(4,3),(4,7),(5,2),(5,4),(6,5),(6,7),(7,0),(7,1)] -3600
[(2,5),(2,6),(3,2),(3,6),(4,3),(4,7),(5,4),(5,7),(6,1),(6,5),(7,0),(7,6)] -3600
[(2,4),(2,7),(3,5),(3,6),(4,3),(4,5),(5,6),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,3),(2,6),(3,5),(3,6),(4,2),(4,7),(5,4),(5,6),(6,1),(6,7),(7,0),(7,5)] -3600
[(2,5),(2,6),(3,2),(3,5),(4,3),(4,7),(5,6),(5,7),(6,4),(6,7),(7,0),(7,1)] 3600
[(2,6),(2,7),(3,4),(3,6),(4,5),(4,7),(5,2),(5,3),(6,1),(6,4),(7,0),(7,6)] 3600
[(2,3),(2,6),(3,5),(3,7),(4,2),(4,7),(5,4),(5,6),(6,1),(6,4),(7,0),(7,5)] -3600
[(2,3),(2,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,6),(2,7),(3,4),(3,6),(4,5),(4,6),(5,2),(5,3),(6,1),(6,7),(7,0),(7,4)] 3600
[(2,3),(2,6),(3,4),(3,7),(4,5),(4,7),(5,2),(5,6),(6,1),(6,4),(7,0),(7,5)] 3600
D.2. THE PENTAGON-WHEEL FLOW Qγ5(P ) 567
[(2,4),(2,5),(3,5),(3,7),(4,3),(4,6),(5,6),(5,7),(6,1),(6,2),(7,0),(7,2)] 3600
[(2,5),(2,6),(3,5),(3,7),(4,2),(4,3),(5,6),(5,7),(6,4),(6,7),(7,0),(7,1)] 3600
[(2,4),(2,7),(3,5),(3,6),(4,5),(4,6),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] 3600
[(2,5),(2,6),(3,5),(3,6),(4,3),(4,7),(5,4),(5,6),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,4),(2,5),(3,6),(3,7),(4,5),(4,7),(5,3),(5,6),(6,1),(6,4),(7,0),(7,2)] -3600
[(2,4),(2,5),(3,6),(3,7),(4,3),(4,5),(5,6),(5,7),(6,1),(6,4),(7,0),(7,2)] 3600
[(2,5),(2,6),(3,2),(3,5),(4,3),(4,6),(5,4),(5,7),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,4),(2,5),(3,2),(3,6),(4,3),(4,5),(5,6),(5,7),(6,1),(6,7),(7,0),(7,4)] 3600
[(2,6),(2,7),(3,5),(3,6),(4,3),(4,7),(5,2),(5,4),(6,1),(6,5),(7,0),(7,6)] -3600
[(2,6),(2,7),(3,4),(3,6),(4,5),(4,7),(5,2),(5,3),(6,1),(6,4),(7,0),(7,5)] -3600
[(2,4),(2,7),(3,6),(3,7),(4,5),(4,6),(5,3),(5,6),(6,1),(6,2),(7,0),(7,5)] 3600
[(2,5),(2,7),(3,5),(3,6),(4,3),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,3),(2,5),(3,5),(3,6),(4,6),(4,7),(5,4),(5,7),(6,1),(6,2),(7,0),(7,2)] -3600
[(2,6),(2,7),(3,5),(3,6),(4,3),(4,7),(5,2),(5,4),(6,1),(6,4),(7,0),(7,5)] -3600
[(2,5),(2,7),(3,2),(3,5),(4,3),(4,6),(5,4),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,5),(2,6),(3,5),(3,7),(4,2),(4,7),(5,6),(5,7),(6,3),(6,4),(7,0),(7,1)] 3600
[(2,5),(2,6),(3,5),(3,7),(4,2),(4,7),(5,4),(5,6),(6,3),(6,7),(7,0),(7,1)] 3600
[(2,4),(2,7),(3,6),(3,7),(4,5),(4,6),(5,3),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,6),(2,7),(3,4),(3,6),(4,2),(4,7),(5,3),(5,7),(6,5),(6,7),(7,0),(7,1)] 3600
[(2,6),(2,7),(3,2),(3,6),(4,3),(4,7),(5,4),(5,7),(6,5),(6,7),(7,0),(7,1)] 3600
[(2,5),(2,7),(3,6),(3,7),(4,3),(4,6),(5,4),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,4),(2,6),(3,5),(3,6),(4,3),(4,5),(5,6),(5,7),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,4),(2,5),(3,5),(3,6),(4,3),(4,6),(5,6),(5,7),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,6),(2,7),(3,5),(3,6),(4,3),(4,7),(5,2),(5,4),(6,1),(6,4),(7,0),(7,6)] 3600
[(2,4),(2,6),(3,2),(3,6),(4,5),(4,7),(5,3),(5,6),(6,1),(6,7),(7,0),(7,5)] 3600
[(2,4),(2,5),(3,6),(3,7),(4,3),(4,5),(5,6),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,4),(2,5),(3,5),(3,7),(4,3),(4,6),(5,6),(5,7),(6,1),(6,3),(7,0),(7,2)] -3600
[(2,4),(2,5),(3,5),(3,7),(4,3),(4,6),(5,6),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,6),(2,7),(3,5),(3,6),(4,3),(4,6),(5,2),(5,4),(6,1),(6,7),(7,0),(7,4)] 3600
[(2,3),(2,6),(3,4),(3,6),(4,5),(4,7),(5,2),(5,7),(6,1),(6,5),(7,0),(7,6)] 3600
[(2,3),(2,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,1),(6,4),(7,0),(7,2)] -3600
[(2,5),(2,7),(3,2),(3,6),(4,3),(4,5),(5,6),(5,7),(6,4),(6,7),(7,0),(7,1)] 3600
[(2,6),(2,7),(3,4),(3,5),(4,2),(4,5),(5,6),(5,7),(6,3),(6,7),(7,0),(7,1)] 3600
[(2,4),(2,5),(3,5),(3,6),(4,6),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] -3600
[(2,5),(2,6),(3,5),(3,7),(4,3),(4,6),(5,4),(5,6),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,5),(2,6),(3,6),(3,7),(4,3),(4,6),(5,4),(5,7),(6,1),(6,5),(7,0),(7,2)] -3600
[(2,4),(2,5),(3,4),(3,6),(4,5),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] -3600
[(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(5,4),(5,7),(6,1),(6,7),(7,0),(7,3)] -3600
[(2,5),(2,6),(3,2),(3,5),(4,3),(4,6),(5,4),(5,7),(6,1),(6,7),(7,0),(7,4)] -3600
[(2,3),(2,5),(3,4),(3,6),(4,2),(4,5),(5,6),(5,7),(6,1),(6,7),(7,0),(7,4)] 3600
[(2,6),(2,7),(3,4),(3,6),(4,5),(4,7),(5,2),(5,3),(6,1),(6,5),(7,0),(7,6)] -3600
[(2,5),(2,7),(3,6),(3,7),(4,3),(4,6),(5,4),(5,6),(6,1),(6,2),(7,0),(7,5)] 3600
[(2,3),(2,5),(3,5),(3,7),(4,6),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,6)] -3600
[(2,5),(2,6),(3,5),(3,7),(4,3),(4,6),(5,4),(5,7),(6,1),(6,3),(7,0),(7,2)] -3600
[(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,3),(5,6),(6,1),(6,2),(7,0),(7,5)] -3600
[(2,5),(2,7),(3,5),(3,7),(4,3),(4,6),(5,4),(5,6),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,3),(5,6),(6,1),(6,2),(7,0),(7,2)] -3600
[(2,5),(2,6),(3,5),(3,6),(4,3),(4,7),(5,4),(5,7),(6,1),(6,4),(7,0),(7,2)] -3600
[(2,5),(2,6),(3,5),(3,7),(4,2),(4,3),(5,4),(5,7),(6,1),(6,3),(7,0),(7,6)] -3600
[(2,5),(2,6),(3,2),(3,5),(4,3),(4,7),(5,4),(5,7),(6,1),(6,4),(7,0),(7,6)] -3600
[(2,6),(2,7),(3,5),(3,7),(4,2),(4,5),(5,6),(5,7),(6,3),(6,4),(7,0),(7,1)] -3600
[(2,6),(2,7),(3,5),(3,7),(4,2),(4,6),(5,4),(5,7),(6,3),(6,5),(7,0),(7,1)] -3600
[(2,5),(2,7),(3,5),(3,7),(4,3),(4,6),(5,4),(5,6),(6,2),(6,7),(7,0),(7,1)] 3600
[(2,4),(2,6),(3,6),(3,7),(4,5),(4,6),(5,3),(5,7),(6,1),(6,5),(7,0),(7,2)] -3600
[(2,3),(2,5),(3,5),(3,6),(4,6),(4,7),(5,4),(5,6),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] -3600
[(2,3),(2,5),(3,4),(3,5),(4,6),(4,7),(5,4),(5,6),(6,1),(6,7),(7,0),(7,2)] -3600
[(2,4),(2,6),(3,5),(3,7),(4,3),(4,5),(5,6),(5,7),(6,1),(6,4),(7,0),(7,2)] -3600
[(2,5),(2,6),(3,5),(3,7),(4,3),(4,7),(5,4),(5,6),(6,1),(6,4),(7,0),(7,2)] -3600
[(2,5),(2,6),(3,5),(3,7),(4,3),(4,7),(5,4),(5,6),(6,1),(6,3),(7,0),(7,2)] -3600
[(2,4),(2,5),(3,5),(3,6),(4,3),(4,6),(5,4),(5,7),(6,1),(6,7),(7,0),(7,2)] -3600
[(2,4),(2,5),(3,5),(3,6),(4,3),(4,7),(5,4),(5,6),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,5),(2,6),(3,4),(3,7),(4,2),(4,5),(5,3),(5,6),(6,1),(6,7),(7,0),(7,4)] -3600
[(2,5),(2,7),(3,4),(3,6),(4,2),(4,5),(5,3),(5,6),(6,1),(6,7),(7,0),(7,4)] 3600
[(2,5),(2,6),(3,5),(3,7),(4,2),(4,3),(5,4),(5,7),(6,1),(6,4),(7,0),(7,6)] -3600
[(2,6),(2,7),(3,5),(3,6),(4,3),(4,7),(5,2),(5,4),(6,1),(6,5),(7,0),(7,5)] -7200
[(2,5),(2,6),(3,6),(3,7),(4,3),(4,7),(5,4),(5,7),(6,1),(6,7),(7,0),(7,2)] 7200
[(2,3),(2,5),(3,5),(3,6),(4,6),(4,7),(5,4),(5,7),(6,1),(6,5),(7,0),(7,2)] -7200
[(2,5),(2,6),(3,2),(3,5),(4,3),(4,7),(5,4),(5,6),(6,1),(6,7),(7,0),(7,5)] -7200
[(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,3),(5,7),(6,1),(6,7),(7,0),(7,2)] 7200
[(2,5),(2,6),(3,5),(3,6),(4,3),(4,7),(5,4),(5,7),(6,1),(6,5),(7,0),(7,2)] -7200
[(2,5),(2,7),(3,2),(3,5),(4,3),(4,6),(5,4),(5,6),(6,1),(6,7),(7,0),(7,5)] 7200
[(2,5),(2,6),(3,5),(3,7),(4,3),(4,6),(5,4),(5,7),(6,1),(6,5),(7,0),(7,2)] -7200
568 APPENDIX D. FLOWS Qγ(P )
[(2,5),(2,6),(3,5),(3,7),(4,2),(4,3),(5,4),(5,6),(6,1),(6,7),(7,0),(7,5)] -7200
[(2,5),(2,7),(3,6),(3,7),(4,3),(4,7),(5,4),(5,7),(6,1),(6,2),(7,0),(7,6)] 7200
[(2,4),(2,5),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] -7200
[(2,4),(2,5),(3,5),(3,6),(4,3),(4,5),(5,6),(5,7),(6,1),(6,7),(7,0),(7,2)] 7200
[(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,3),(5,6),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,3),(2,6),(3,4),(3,7),(4,5),(4,7),(5,2),(5,6),(6,1),(6,7),(7,0),(7,5)] 3600
[(2,4),(2,6),(3,5),(3,7),(4,3),(4,5),(5,6),(5,7),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,4),(2,6),(3,5),(3,7),(4,3),(4,7),(5,2),(5,7),(6,1),(6,5),(7,0),(7,6)] 3600
[(2,6),(2,7),(3,4),(3,7),(4,5),(4,6),(5,2),(5,3),(6,1),(6,7),(7,0),(7,4)] 3600
[(2,4),(2,6),(3,2),(3,7),(4,5),(4,7),(5,3),(5,6),(6,1),(6,4),(7,0),(7,5)] 3600
[(2,3),(2,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,1),(6,7),(7,0),(7,2)] -3600
[(2,6),(2,7),(3,4),(3,7),(4,5),(4,7),(5,2),(5,3),(6,1),(6,4),(7,0),(7,6)] 3600
[(2,5),(2,6),(3,2),(3,7),(4,3),(4,7),(5,4),(5,6),(6,1),(6,4),(7,0),(7,5)] -3600
[(2,4),(2,5),(3,5),(3,6),(4,3),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,2)] 3600
[(2,5),(2,7),(3,4),(3,6),(4,5),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] 3600
[(2,5),(2,7),(3,5),(3,7),(4,3),(4,6),(5,4),(5,7),(6,1),(6,2),(7,0),(7,6)] -3600
[(2,4),(2,5),(3,6),(3,7),(4,5),(4,6),(5,3),(5,7),(6,1),(6,2),(7,0),(7,4)] -3600
[(2,4),(2,5),(3,6),(3,7),(4,3),(4,5),(5,6),(5,7),(6,1),(6,2),(7,0),(7,4)] -3600
[(2,5),(2,7),(3,2),(3,5),(4,3),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,6)] -3600
[(2,3),(2,5),(3,4),(3,5),(4,2),(4,7),(5,6),(5,7),(6,1),(6,3),(7,0),(7,6)] 3600
[(2,6),(2,7),(3,4),(3,6),(4,5),(4,7),(5,2),(5,3),(6,1),(6,7),(7,0),(7,5)] -3600
[(2,6),(2,7),(3,4),(3,7),(4,5),(4,6),(5,2),(5,3),(6,1),(6,5),(7,0),(7,4)] -3600
[(2,4),(2,6),(3,6),(3,7),(4,5),(4,7),(5,3),(5,7),(6,1),(6,5),(7,0),(7,2)] 3600
[(2,5),(2,6),(3,5),(3,7),(4,3),(4,6),(5,4),(5,7),(6,1),(6,7),(7,0),(7,2)] -3600
[(2,3),(2,5),(3,5),(3,7),(4,6),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,2)] -3600
[(2,6),(2,7),(3,5),(3,7),(4,3),(4,6),(5,2),(5,4),(6,1),(6,5),(7,0),(7,4)] -3600
[(2,5),(2,6),(3,2),(3,5),(4,3),(4,7),(5,4),(5,6),(6,1),(6,7),(7,0),(7,2)] -3600
[(2,5),(2,6),(3,6),(3,7),(4,3),(4,6),(5,4),(5,7),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,4),(2,6),(3,6),(3,7),(4,5),(4,6),(5,3),(5,7),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,4),(2,7),(3,5),(3,7),(4,3),(4,5),(5,6),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,4),(2,5),(3,5),(3,7),(4,3),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,6),(2,7),(3,5),(3,7),(4,3),(4,6),(5,2),(5,4),(6,1),(6,7),(7,0),(7,4)] 3600
[(2,5),(2,6),(3,4),(3,7),(4,2),(4,7),(5,3),(5,7),(6,1),(6,5),(7,0),(7,6)] -3600
[(2,4),(2,5),(3,6),(3,7),(4,3),(4,5),(5,6),(5,7),(6,1),(6,7),(7,0),(7,2)] -3600
[(2,4),(2,5),(3,5),(3,6),(4,3),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] -3600
[(2,4),(2,5),(3,5),(3,6),(4,3),(4,7),(5,6),(5,7),(6,1),(6,7),(7,0),(7,2)] 3600
[(2,6),(2,7),(3,5),(3,7),(4,3),(4,7),(5,2),(5,4),(6,1),(6,4),(7,0),(7,6)] 3600
[(2,5),(2,6),(3,2),(3,7),(4,3),(4,7),(5,4),(5,6),(6,1),(6,7),(7,0),(7,5)] -3600
[(2,3),(2,4),(3,5),(3,7),(4,5),(4,6),(5,6),(5,7),(6,1),(6,2),(7,0),(7,4)] -3600
[(2,5),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] 3600
[(2,5),(2,7),(3,5),(3,6),(4,3),(4,7),(5,4),(5,7),(6,1),(6,2),(7,0),(7,6)] -3600
[(2,5),(2,7),(3,6),(3,7),(4,3),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,5)] -3600
[(2,4),(2,7),(3,4),(3,5),(4,5),(4,6),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] -3600
[(2,5),(2,7),(3,5),(3,7),(4,2),(4,3),(5,4),(5,6),(6,1),(6,3),(7,0),(7,6)] 3600
[(2,5),(2,7),(3,2),(3,5),(4,3),(4,7),(5,4),(5,6),(6,1),(6,4),(7,0),(7,6)] 3600
[(2,4),(2,5),(3,2),(3,5),(4,3),(4,7),(5,6),(5,7),(6,1),(6,3),(7,0),(7,6)] 3600
[(2,6),(2,7),(3,5),(3,6),(4,3),(4,7),(5,2),(5,4),(6,1),(6,7),(7,0),(7,5)] -3600
[(2,5),(2,6),(3,6),(3,7),(4,3),(4,7),(5,4),(5,7),(6,1),(6,5),(7,0),(7,2)] 3600
[(2,3),(2,5),(3,5),(3,6),(4,6),(4,7),(5,4),(5,7),(6,1),(6,7),(7,0),(7,2)] -3600
[(2,5),(2,7),(3,5),(3,6),(4,3),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,3)] 3600
[(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,3),(5,7),(6,1),(6,5),(7,0),(7,2)] 3600
[(2,5),(2,6),(3,5),(3,6),(4,3),(4,7),(5,4),(5,7),(6,1),(6,7),(7,0),(7,2)] -3600
[(2,4),(2,5),(3,6),(3,7),(4,6),(4,7),(5,3),(5,7),(6,1),(6,2),(7,0),(7,2)] 3600
[(2,5),(2,7),(3,5),(3,7),(4,3),(4,6),(5,4),(5,6),(6,1),(6,2),(7,0),(7,4)] 3600
[(2,5),(2,7),(3,5),(3,6),(4,2),(4,3),(5,4),(5,6),(6,1),(6,7),(7,0),(7,3)] 3600
[(2,5),(2,7),(3,2),(3,5),(4,3),(4,6),(5,4),(5,6),(6,1),(6,7),(7,0),(7,4)] 3600
[(2,4),(2,7),(3,6),(3,7),(4,5),(4,7),(5,3),(5,6),(6,1),(6,2),(7,0),(7,5)] -3600
[(2,3),(2,5),(3,5),(3,7),(4,6),(4,7),(5,4),(5,7),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,4),(2,7),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,1),(6,2),(7,0),(7,3)] -3600
[(2,3),(2,5),(3,4),(3,5),(4,6),(4,7),(5,4),(5,7),(6,1),(6,2),(7,0),(7,6)] -3600
[(2,4),(2,7),(3,5),(3,6),(4,3),(4,5),(5,6),(5,7),(6,1),(6,2),(7,0),(7,4)] -3600
[(2,5),(2,7),(3,5),(3,6),(4,3),(4,6),(5,4),(5,7),(6,1),(6,2),(7,0),(7,4)] 3600
[(2,5),(2,7),(3,5),(3,6),(4,3),(4,6),(5,4),(5,7),(6,1),(6,2),(7,0),(7,3)] 3600
[(2,4),(2,5),(3,5),(3,7),(4,3),(4,7),(5,4),(5,6),(6,1),(6,2),(7,0),(7,6)] 3600
[(2,4),(2,5),(3,5),(3,7),(4,3),(4,6),(5,4),(5,7),(6,1),(6,2),(7,0),(7,6)] -3600
[(2,5),(2,7),(3,4),(3,6),(4,2),(4,5),(5,3),(5,7),(6,1),(6,4),(7,0),(7,6)] 3600
[(2,5),(2,6),(3,2),(3,5),(4,3),(4,7),(5,4),(5,7),(6,1),(6,3),(7,0),(7,6)] -3600
[(2,5),(2,6),(3,5),(3,7),(4,2),(4,3),(5,4),(5,6),(6,1),(6,7),(7,0),(7,4)] -3600
The Poisson differential [[P,Qγ5 ]] and its Leibniz graph factorization ♢γ5(P, [[P, P ]])
are available from https://rburing.nl/gcaops, namely as [P,Q_gamma_5].txt and
[P,Q_gamma_5]_leibniz.txt
Appendix E
Graph cocycles
Encodings of graph cocycles γ3, γ5, γ7, [γ3, γ5] in the Kontsevich unoriented graph complex
with the vertex-expanding differential.
E.1 The tetrahedron γ3
1*UndirectedGraph(4,[(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)])
E.2 The pentagon-wheel cocycle γ5
1*UndirectedGraph(6,[(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)]) + \
(5/2)*UndirectedGraph(6,[(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)])
E.3 The heptagon-wheel cocycle γ7
[(1,2),(1,4),(1,8),(2,3),(2,7),(3,5),(3,7),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)] -21/8
[(1,3),(1,4),(1,8),(2,3),(2,5),(2,8),(3,7),(4,6),(4,8),(5,6),(5,7),(6,7),(6,8),(7,8)] -77/4
[(1,2),(1,3),(1,5),(2,4),(2,7),(3,5),(3,6),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)] -35/8
[(1,2),(1,3),(1,8),(2,4),(2,6),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)] 49/8
[(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,5),(3,7),(4,6),(4,8),(5,6),(5,8),(6,7),(7,8)] 77/8
[(1,2),(1,3),(1,8),(2,6),(2,7),(3,5),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(6,8),(7,8)] -105/8
[(1,2),(1,4),(1,8),(2,3),(2,7),(3,6),(3,8),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(7,8)] 7/8
[(1,2),(1,4),(1,5),(2,3),(2,7),(3,5),(3,6),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)] 35/8
[(1,2),(1,3),(1,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)] -49/8
[(1,2),(1,3),(1,8),(2,5),(2,7),(3,4),(3,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)] 35/4
[(1,2),(1,3),(1,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,7),(5,8),(6,7),(6,8),(7,8)] -119/16
[(1,2),(1,3),(1,5),(2,4),(2,8),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(6,7),(6,8),(7,8)] 49/8
[(1,2),(1,3),(1,4),(2,3),(2,8),(3,7),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)] 77/4
[(1,2),(1,5),(1,7),(2,5),(2,6),(3,5),(3,6),(3,8),(4,5),(4,7),(4,8),(6,7),(6,8),(7,8)] -49/8
[(1,3),(1,5),(1,8),(2,4),(2,6),(2,8),(3,7),(3,8),(4,6),(4,7),(5,6),(5,7),(6,8),(7,8)] -49/4
[(1,3),(1,4),(1,8),(2,5),(2,6),(2,8),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(6,7),(7,8)] -49/4
[(1,2),(1,4),(1,8),(2,3),(2,8),(3,5),(3,7),(4,6),(4,8),(5,6),(5,7),(6,7),(6,8),(7,8)] -7
[(1,2),(1,4),(1,8),(2,3),(2,8),(3,6),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,7),(7,8)] -7
[(1,2),(1,5),(1,6),(2,5),(2,7),(3,5),(3,6),(3,8),(4,6),(4,7),(4,8),(5,8),(6,7),(7,8)] 49/8
[(1,2),(1,4),(1,8),(2,3),(2,8),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)] 49/8
[(1,2),(1,3),(1,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)] -7
[(1,6),(1,7),(1,8),(2,3),(2,5),(2,8),(3,4),(3,8),(4,6),(4,8),(5,7),(5,8),(6,8),(7,8)] 1
[(1,2),(1,3),(1,8),(2,4),(2,8),(3,5),(3,8),(4,6),(4,7),(5,7),(5,8),(6,7),(6,8),(7,8)] 7
[(1,2),(1,3),(1,8),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(6,8),(7,8)] -7
[(1,2),(1,4),(1,6),(2,3),(2,5),(3,6),(3,7),(4,5),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)] 77/8
[(1,3),(1,6),(1,7),(2,4),(2,5),(2,6),(3,5),(3,7),(4,5),(4,8),(5,8),(6,7),(6,8),(7,8)] -7
[(1,4),(1,5),(1,7),(2,3),(2,6),(2,8),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)] 49/4
[(1,2),(1,6),(1,8),(2,7),(2,8),(3,4),(3,6),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(7,8)] -147/8
[(1,2),(1,5),(1,6),(2,7),(2,8),(3,5),(3,6),(3,8),(4,5),(4,6),(4,7),(5,7),(6,8),(7,8)] -21/8
[(1,2),(1,4),(1,8),(2,3),(2,7),(3,5),(3,6),(4,5),(4,6),(5,7),(5,8),(6,7),(6,8),(7,8)] -35/8
[(1,4),(1,5),(1,6),(2,3),(2,6),(2,8),(3,7),(3,8),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)] -49/4
[(1,2),(1,5),(1,8),(2,3),(2,8),(3,4),(3,7),(4,6),(4,8),(5,6),(5,7),(6,7),(6,8),(7,8)] 105/8
569
570 APPENDIX E. GRAPH COCYCLES
[(1,2),(1,4),(1,7),(2,3),(2,6),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)] -49/8
[(1,2),(1,6),(1,8),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,8),(5,7),(6,8),(7,8)] 49/16
[(1,2),(1,3),(1,8),(2,5),(2,7),(3,5),(3,6),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)] 7
[(1,2),(1,4),(1,8),(2,5),(2,8),(3,4),(3,6),(3,8),(4,7),(5,7),(5,8),(6,7),(6,8),(7,8)] -7
[(1,2),(1,6),(1,8),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,8),(5,8),(6,7),(7,8)] -77/16
[(1,2),(1,4),(1,8),(2,3),(2,7),(3,5),(3,8),(4,6),(4,7),(5,7),(5,8),(6,7),(6,8),(7,8)] 77/4
[(1,2),(1,4),(1,5),(2,3),(2,7),(3,6),(3,8),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)] 35/2
[(1,2),(1,3),(1,8),(2,5),(2,7),(3,4),(3,6),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)] -105/8
[(1,2),(1,5),(1,6),(2,5),(2,7),(3,5),(3,6),(3,8),(4,6),(4,7),(4,8),(5,7),(6,8),(7,8)] -7
[(1,2),(1,3),(1,6),(2,5),(2,8),(3,4),(3,7),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)] -147/16
[(1,2),(1,3),(1,7),(2,5),(2,6),(3,5),(3,7),(4,5),(4,6),(4,8),(5,8),(6,7),(6,8),(7,8)] -77/4
[(1,2),(1,4),(1,7),(2,3),(2,7),(3,5),(3,8),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)] -49/8
[(1,2),(1,3),(1,5),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(5,8),(6,7),(6,8),(7,8)] -7/4
[(1,2),(1,4),(1,8),(2,3),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(7,8)] -7
E.4 The commutator [γ3, γ5] ∈ ker d on 9 vertices and
16 edges
(-60)*UndirectedGraph(9,[(0,3),(0,4),(0,8),(1,2),(1,4),(1,8),(2,3),(2,8),(3,7),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,2),(0,4),(0,8),(1,2),(1,3),(1,8),(2,8),(3,4),(3,7),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,8),(2,7),(3,4),(3,6),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,3),(0,8),(1,3),(1,8),(2,4),(2,5),(2,6),(3,7),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,4),(0,6),(1,5),(1,7),(2,3),(2,6),(2,8),(3,6),(3,8),(4,7),(4,8),(5,7),(5,8),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,5),(0,8),(1,4),(1,7),(2,3),(2,6),(2,8),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,4),(0,5),(0,8),(1,2),(1,3),(1,8),(2,3),(2,8),(3,7),(4,6),(4,8),(5,6),(5,7),(6,7),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,4),(0,8),(1,5),(1,7),(2,3),(2,6),(2,8),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,1),(0,3),(0,8),(1,2),(1,7),(2,4),(2,8),(3,4),(3,7),(4,6),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,1),(0,3),(0,8),(1,2),(1,7),(2,4),(2,7),(3,4),(3,8),(4,6),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,1),(0,3),(0,7),(1,2),(1,7),(2,4),(2,8),(3,4),(3,8),(4,6),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,4),(0,8),(1,3),(1,8),(2,3),(2,4),(2,8),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,7),(2,6),(3,4),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,3),(0,8),(1,3),(1,7),(2,4),(2,7),(2,8),(3,6),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,7),(2,7),(3,4),(3,6),(3,8),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,8),(2,6),(3,5),(3,6),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(6,7),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,6),(0,8),(1,6),(1,8),(2,3),(2,5),(2,8),(3,5),(3,7),(4,5),(4,7),(4,8),(5,7),(6,7),(6,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,3),(0,7),(1,3),(1,7),(2,4),(2,7),(2,8),(3,6),(4,6),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,5),(2,6),(3,5),(3,6),(3,8),(4,5),(4,7),(4,8),(5,7),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,5),(1,3),(1,6),(2,5),(2,8),(3,5),(3,8),(4,6),(4,7),(4,8),(5,7),(6,7),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,3),(0,5),(0,8),(1,2),(1,7),(1,8),(2,7),(2,8),(3,4),(3,6),(4,6),(4,7),(5,6),(5,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,4),(1,8),(2,3),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,2),(0,3),(0,8),(1,4),(1,5),(1,8),(2,3),(2,7),(3,6),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,5),(2,7),(3,5),(3,6),(3,8),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,3),(0,8),(1,2),(1,5),(2,5),(2,8),(3,5),(3,6),(4,6),(4,7),(4,8),(5,7),(6,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,8),(2,6),(3,5),(3,7),(3,8),(4,5),(4,6),(4,7),(5,7),(5,8),(6,7),(6,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,8),(2,7),(3,4),(3,5),(3,8),(4,5),(4,6),(5,6),(5,7),(6,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,5),(0,8),(1,5),(1,8),(2,3),(2,6),(2,7),(3,5),(3,6),(4,6),(4,7),(4,8),(5,8),(6,7),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,5),(0,8),(1,5),(1,8),(2,3),(2,6),(2,8),(3,6),(3,7),(4,5),(4,6),(4,7),(5,8),(6,7),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,3),(0,5),(0,8),(1,2),(1,4),(1,7),(2,4),(2,7),(3,7),(3,8),(4,6),(5,6),(5,8),(6,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,6),(2,5),(3,4),(3,7),(3,8),(4,5),(4,8),(5,6),(5,7),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,3),(1,5),(2,5),(2,8),(3,5),(3,6),(4,6),(4,7),(4,8),(5,7),(6,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,3),(0,5),(0,7),(1,2),(1,4),(1,7),(2,4),(2,8),(3,7),(3,8),(4,6),(5,6),(5,8),(6,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,7),(0,8),(1,7),(1,8),(2,3),(2,4),(2,7),(3,5),(3,6),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,1),(0,3),(0,8),(1,2),(1,8),(2,5),(2,8),(3,6),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,7),(1,5),(1,7),(2,7),(2,8),(3,4),(3,6),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,7),(0,8),(1,7),(1,8),(2,3),(2,5),(2,7),(3,4),(3,6),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,3),(0,8),(1,2),(1,8),(2,4),(2,8),(3,4),(3,8),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,4),(0,8),(1,4),(1,8),(2,3),(2,6),(2,8),(3,5),(3,8),(4,7),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,5),(0,8),(1,4),(1,6),(2,3),(2,7),(2,8),(3,7),(3,8),(4,6),(4,8),(5,6),(5,8),(6,7),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,8),(2,6),(3,4),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,3),(0,6),(0,8),(1,2),(1,4),(1,8),(2,4),(2,8),(3,5),(3,7),(4,7),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,3),(0,4),(0,8),(1,2),(1,6),(1,8),(2,5),(2,8),(3,4),(3,7),(4,7),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,3),(0,7),(1,2),(1,7),(2,7),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,7),(6,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,2),(0,7),(1,2),(1,7),(2,8),(3,4),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8)]) + \
(-60)*UndirectedGraph(9,[(0,3),(0,4),(0,6),(1,2),(1,5),(1,8),(2,5),(2,8),(3,7),(3,8),(4,6),(4,7),(5,7),(5,8),(6,7),(6,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,8),(2,5),(3,5),(3,6),(3,8),(4,5),(4,6),(4,7),(5,7),(6,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,2),(0,7),(1,2),(1,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(5,6),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,5),(2,6),(3,4),(3,5),(3,8),(4,6),(4,7),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,3),(0,8),(1,2),(1,5),(2,5),(2,6),(3,5),(3,7),(4,6),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,6),(2,5),(3,5),(3,7),(3,8),(4,5),(4,6),(4,7),(5,8),(6,7),(6,8),(7,8)]) + \
E.4. THE COMMUTATOR [γ3, γ5] ∈ ker d ON 9 VERTICES AND 16 EDGES 571
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,6),(1,3),(1,5),(2,5),(2,8),(3,5),(3,7),(4,6),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)]) + \
(-120)*UndirectedGraph(9,[(0,3),(0,4),(0,8),(1,2),(1,4),(1,8),(2,3),(2,5),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,7),(1,2),(1,6),(2,5),(3,5),(3,6),(3,8),(4,5),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,5),(1,3),(1,6),(2,5),(2,8),(3,5),(3,7),(4,6),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,5),(2,7),(3,4),(3,6),(3,8),(4,5),(4,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,4),(0,8),(1,3),(1,8),(2,3),(2,4),(2,7),(3,5),(4,6),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,1),(0,2),(0,6),(1,2),(1,5),(2,7),(3,4),(3,5),(3,8),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,8),(2,6),(3,4),(3,5),(3,6),(4,5),(4,7),(5,7),(5,8),(6,7),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,3),(0,4),(0,6),(1,2),(1,5),(1,8),(2,5),(2,8),(3,6),(3,7),(4,5),(4,7),(5,8),(6,7),(6,8),(7,8)]) + \
(120)*UndirectedGraph(9,[(0,4),(0,5),(0,8),(1,2),(1,3),(1,6),(2,3),(2,7),(3,8),(4,5),(4,6),(4,7),(5,7),(5,8),(6,7),(6,8)]) + \
(120)*UndirectedGraph(9,[(0,5),(0,7),(0,8),(1,4),(1,6),(1,8),(2,3),(2,5),(2,7),(3,6),(3,7),(4,5),(4,6),(4,8),(5,7),(6,8)]) + \
(-120)*UndirectedGraph(9,[(0,4),(0,5),(0,7),(1,3),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(4,5),(4,7),(4,8),(5,6),(5,7),(6,8)]) + \
(-120)*UndirectedGraph(9,[(0,2),(0,6),(0,8),(1,3),(1,7),(1,8),(2,3),(2,5),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(6,7)]) + \
(-120)*UndirectedGraph(9,[(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(2,3),(2,8),(3,6),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7)]) + \
(-60)*UndirectedGraph(9,[(0,3),(0,4),(0,7),(1,2),(1,5),(1,8),(2,5),(2,8),(3,6),(3,8),(4,6),(4,7),(5,7),(5,8),(6,7),(6,8)]) + \
(-60)*UndirectedGraph(9,[(0,1),(0,2),(0,8),(1,2),(1,8),(2,5),(3,4),(3,6),(3,8),(4,5),(4,7),(5,6),(5,7),(6,7),(6,8),(7,8)]) + \
(60)*UndirectedGraph(9,[(0,1),(0,2),(0,7),(1,2),(1,7),(2,8),(3,4),(3,5),(3,8),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)])

Appendix F
Reference documentation for the
gcaops software
The following appendix has been generated from the documentation strings included in
the source code of gcaops. In addition to an exhaustive listing of all available methods
on all objects, some basic usage examples are included.
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Release 1
Ricardo Buring
Jul 01, 2022

TABLE OF CONTENTS
1 Algebra 579
1.1 Superfunction algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
1.2 Superfunction algebra operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
1.3 Polydifferential operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
1.4 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
1.5 Differential polynomial ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
1.6 Homogeneous differential polynomial equation solver . . . . . . . . . . . . . . . . . . . . . . . . . 589
2 Abstract base classes 591
2.1 Graph basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
2.2 Graph vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
2.3 Graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
2.4 Graph vector (vector backend) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
2.5 Graph vector (dictionary backend) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
3 Undirected graphs 599
3.1 Undirected graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
3.2 Undirected graph basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
3.3 Undirected graph vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
3.4 Undirected graph operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
3.5 Undirected graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
4 Directed graphs 609
4.1 Directed graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
4.2 Directed graph basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
4.3 Directed graph vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612
4.4 Directed graph operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
4.5 Directed graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
5 Formality graphs 617
5.1 Formality graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
5.2 Formality graph basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
5.3 Formality graph vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
5.4 Formality graph operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
5.5 Formality graph complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
5.6 Formality graph operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
6 Graph cache 633
6.1 Graph file view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
6.2 Graph cache . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
i
Python Module Index 637
Index 639
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Import the package:
sage: import gcaops
578 TABLE OF CONTENTS
CHAPTER
ONE
ALGEBRA
1.1 Superfunction algebra
Superfunction algebra
class gcaops.algebra.superfunction_algebra.Superfunction(parent, monomial_coefficients)
Bases: object
Superfunction on a coordinate chart of a 𝑍2-graded space.
A polynomial in the odd coordinates, with coefficients in the base ring (of even degree 0 functions).
__add__(other)
Return this superfunction added to other.
__eq__(other)
Return True if this superfunction equals other and False otherwise.
Note: This takes the difference and calls is_zero() on it. For comparison with zero it is faster to call
is_zero() directly.
__getitem__(indices)
Return the coefficient of the monomial in the odd coordinates specified by indices.
__init__(parent, monomial_coefficients)
Initialize this superfunction.
INPUT:
• parent - a SuperfunctionAlgebra
• monomial_coefficients - a dictionary, taking a natural number m less than 2^parent.ngens()
to the coefficient of the monomial in the odd coordinates represented by m
__mul__(other)
Return this superfunction multiplied by other.
__neg__()
Return the negative of this superfunction.
__pos__()
Return a copy of this superfunction.
__pow__(exponent)
Return this superfunction raised to the power exponent.
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__radd__(other)
Return other added to this superfunction.
__repr__()
Return a string representation of this superfunction.
__rmul__(other)
Return other multiplied by this superfunction.
Note: This assumes that other commutes with this superfunction. It is justified because this function
only gets called when other is even.
__rsub__(other)
Return other minus this superfunction.
__setitem__(indices, new_value)
Set the coefficient of the monomial in the odd coordinates specified by indices to new_value.
__sub__(other)
Return this superfunction minus other.
__truediv__(other)
Return this superfunction divided by other.
bracket(other)
Return the Schouten bracket (odd Poisson bracket) of this superfunction with other.
copy()
Return a copy of this superfunction.
degree()
Return the degree of this superfunction as a polynomial in the odd coordinates.
degrees()
Return an iterator over the degrees of the monomials (in the odd coordinates) of this superfunction.
derivative(*args)
Return the derivative of this superfunction with respect to args.
INPUT:
• args – an odd coordinate or an even coordinate, or a list of such
diff(*args)
Return the derivative of this superfunction with respect to args.
INPUT:
• args – an odd coordinate or an even coordinate, or a list of such
homogeneous_part(degree)
Return the homogeneous part of this superfunction of total degree degree in the odd coordinates.
Note: Returns a Superfunction whose homogeneous component of degree degree is a reference to the
respective component of this superfunction.
indices(degree=None)
Return an iterator over indices of this superfunction, i.e. a tuple of exponents for each monomial in the
odd coordinates.
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INPUT:
• degree (default: None) – if not None, yield only indices of degree degree
is_zero()
Return True if this superfunction equals zero and False otherwise.
map_coefficients(f, new_parent=None)
Apply f to each of this superfunction’s coefficients and return the resulting superfunction.
parent()
Return the parent SuperfunctionAlgebra that this superfunction belongs to.
schouten_bracket(other)
Return the Schouten bracket (odd Poisson bracket) of this superfunction with other.
class gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra(base_ring,
even_coordinates=None,
names='xi', sim-
plify=None,
is_zero='is_zero')
Bases: object
Supercommutative algebra of superfunctions on a coordinate chart of a 𝑍2-graded space.
Consisting of polynomials in the odd (degree 1) coordinates, with coefficients in the base ring (of even degree
0 functions). It is a free module over the base ring with a basis consisting of sorted monomials in the odd
coordinates. The elements encode skew-symmetric multi-derivations of the base ring, or multi-vectors.
__call__(arg)
Return arg converted into an element of this superfunction algebra.
ASSUMPTIONS:
If arg is a PolyDifferentialOperator, it is assumed that its coefficients are skew-symmetric.
__init__(base_ring, even_coordinates=None, names='xi', simplify=None, is_zero='is_zero')
Initialize this superfunction algebra.
INPUT:
• base_ring – a commutative ring, considered as a ring of (even, degree 0) functions
• even_coordinates – (default: None) a list or tuple of elements of base_ring; if none is provided,
then it is set to base_ring.gens()
• names – (default: 'xi') a list or tuple of strings or a comma separated string, consisting of names
for the odd coordinates; or a single string consisting of a prefix that will be used to generate a list of
numbered names
• simplify – (default: None) a string, containing the name of a method of an element of the base ring;
that method should return a simplification of the element (will be used in each operation on elements
that affects coefficients), or None (which amounts to no simplification).
• is_zero – (default: 'is_zero') a string, containing the name of a method of an element of the base
ring; that method should return True when a simplified element of the base ring is equal to zero (will
be used to decide equality of elements, to calculate the degree of elements, and to skip terms in some
operations on elements)
__repr__()
Return a string representation of this superfunction algebra.
base_ring()
Return the base ring of this superfunction algebra, consisting of (even, degree 0) functions.
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dimension(degree)
Return the dimension of the graded component spanned by monomials of the given degree in the odd
coordinates (as a module over the base ring).
INPUT:
• degree – a natural number
even_coordinate(i)
Return the i-th even coordinate in the base ring of this superfunction algebra.
even_coordinates()
Return the even coordinates in the base ring of this superfunction algebra.
gen(i)
Return the i-th odd coordinate of this superfunction algebra.
gens()
Return the tuple of odd coordinates of this superfunction algebra.
graph_operation(graph_vector)
Return the operation (on this superfunction algebra) defined by graph_vector.
If the input is a graph cochain in a graph complex, then the operation that pre-symmetrizes the arguments
is returned.
ASSUMPTION:
Assumes each graph in graph_vector has the same number of vertices.
ngens()
Return the number of odd coordinates of this superfunction algebra.
odd_coordinate(i)
Return the i-th odd coordinate of this superfunction algebra.
odd_coordinates()
Return the tuple of odd coordinates of this superfunction algebra.
one()
Return the unit element of this superfunction algebra.
schouten_bracket()
Return the Schouten bracket (odd Poisson bracket) on this superfunction algebra.
tensor_power(n)
Return the n-th tensor power of this superfunction algebra.
zero()
Return the zero element of this superfunction algebra.
1.2 Superfunction algebra operation
Initialize self. See help(type(self)) for accurate signature.
class gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraDirectedGraphOperation(domain,
codomain,
graph_vector)
Bases: gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraOperation
A homogeneous n-ary multi-linear operation on a SuperfunctionAlgebra, defined by a
DirectedGraphVector.
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degree()
Return the degree of this operation.
class gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraOperation(domain,
codomain)
Bases: abc.ABC
A homogeneous n-ary multi-linear operation acting on a SuperfunctionAlgebra.
codomain()
Return the codomain of this operation.
degree()
Return the degree of this operation.
domain()
Return the domain of this operation.
class gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSchoutenBracket(domain,
codomain)
Bases: gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricOperation
Schouten bracket on a SuperfunctionAlgebra.
degree()
Return the degree of this operation.
class gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricBracketOperation(*args)
Bases: gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricOperation
A homogeneous symmetric n-ary multi-linear operation acting on a SuperfunctionAlgebra, given by the
Nijenhuis-Richardson bracket of two graded symmetric operations.
degree()
Return the degree of this operation.
class gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricDirectedGraphOperation(domain,
codomain,
graph_vector)
Bases: gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraDirectedGraphOperation,
gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricOperation
A homogeneous symmetric n-ary multi-linear operation acting on a SuperfunctionAlgebra, defined by a
DirectedGraphVector.
class gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricOperation(domain,
codomain)
Bases: gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraOperation
A homogeneous symmetric n-ary multi-linear operation acting on a SuperfunctionAlgebra.
bracket(other)
Return the Nijenhuis-Richardson bracket of this operation with the other operation.
class gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricUndirectedGraphOperation(domain,
codomain,
graph_vector)
Bases: gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraUndirectedGraphOperation,
gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricOperation
A homogeneous n-ary multi-linear symmetric operation acting on a SuperfunctionAlgebra, defined by a
UndirectedGraphVector.
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class gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraUndirectedGraphOperation(domain,
codomain,
graph_vector)
Bases: gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraOperation
A homogeneous n-ary multi-linear operation acting on a SuperfunctionAlgebra, defined by a
UndirectedGraphVector.
degree()
Return the degree of this operation.
1.3 Polydifferential operator
Polydifferential operator
class gcaops.algebra.polydifferential_operator.PolyDifferentialOperator(parent, coeffi-
cients)
Bases: object
Polydifferential operator on a coordinate chart.
A multi-linear polydifferential operator, with coefficients in the base ring (of functions).
__add__(other)
Return this polydifferential operator added to other.
__eq__(other)
Return True if this polydifferential operator equals other and False otherwise.
Note: This takes the difference and calls is_zero() on it. For comparison with zero it is faster to call
is_zero() directly.
__getitem__(multi_indices)
Return the coefficient of the differential monomial specified by multi_indices.
__init__(parent, coefficients)
Initialize this polydifferential operator.
INPUT:
• parent - a PolyDifferentialOperatorAlgebra (which has an ordered basis of monomials in the
odd coordinates)
• coefficients - a dictionary, mapping the arity m to a dictionary that maps m-tuples of multi-indices
to elements in the base ring of parent
__mul__(other)
Return this polydifferential operator multiplied by other.
Note: This is the pre-Lie product, a sum (with signs) of insertions of other into this polydifferential
operator. For unary operators, it is simply composition.
__neg__()
Return the negative of this polydifferential operator.
__pos__()
Return a copy of this polydifferential operator.
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__pow__(exponent)
Return this polydifferential operator raised to the power exponent.
__radd__(other)
Return other added to this polydifferential operator.
__repr__()
Return a string representation of this polydifferential operator.
__rmul__(other)
Return other multiplied by this polydifferential operator.
Note: This is only defined for elements of the base ring.
__rsub__(other)
Return other minus this polydifferential operator.
__setitem__(multi_indices, new_value)
Set the coefficient of the differential monomial specified by multi_indices to new_value.
__sub__(other)
Return this polydifferential operator minus other.
__truediv__(other)
Return this polydifferential operator divided by other.
arity()
Return the arity of this polydifferential operator.
ASSUMPTIONS:
Assumes this polydifferential operator is homogeneous.
bracket(other)
Return the Gerstenhaber bracket of this polydifferential operator with other.
coefficient(variable)
Return the coefficient of variable of this polydifferential operator.
copy()
Return a copy of this polydifferential operator.
gerstenhaber_bracket(other)
Return the Gerstenhaber bracket of this polydifferential operator with other.
hochschild_differential()
Return the Hochschild differential of this polydifferential operator, with respect to the multiplication oper-
ator of the parent.
homogeneous_part(arity)
Return the homogeneous part of this polydifferential operator of arity arity.
Note: Returns a polydifferential operator whose homogeneous component of arity arity is a reference
to the respective component of this polydifferential operator.
insertion(position, other)
Return the insertion of other into the position-th argument of this polydifferential operator.
is_zero()
Return True if this polydifferential operator equals zero and False otherwise.
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map_coefficients(f, new_parent=None)
Apply f to each of this polydifferential operator’s coefficients and return the resulting polydifferential
operator.
multi_indices()
Return an iterator over the multi-indices of the terms in this polydifferential operator.
parent()
Return the parent PolyDifferentialOperatorAlgebra that this polydifferential operator belongs to.
skew_symmetrization()
Return the polydifferential operator which is the skew-symmetrization of this polydifferential operator.
subs(*args, **kwargs)
Return this polydifferential operator with the subs method applied (with the given arguments) to each
coefficient.
symmetrization()
Return the polydifferential operator which is the symmetrization of this polydifferential operator.
class gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra(base_ring,
coordi-
nates=None,
names='ddx',
sim-
plify=None,
is_zero='is_zero')
Bases: object
Noncommutative algebra of polydifferential operators on a coordinate chart.
__call__(*args)
Return arg converted into an element of this polydifferential operator algebra.
__init__(base_ring, coordinates=None, names='ddx', simplify=None, is_zero='is_zero')
Initialize this polydifferential operator algebra.
INPUT:
• base_ring – a commutative ring, considered as a ring of functions
• coordinates – (default: None) a list or tuple of elements of base_ring; if none is provided, then it
is set to base_ring.gens()
• names – (default: 'ddx') a list or tuple of strings or a comma separated string, consisting of names
for the derivatives with respect to the coordinates; or a single string consisting of a prefix that will be
used to generate a list of numbered names
• simplify – (default: None) a string, containing the name of a method of an element of the base ring;
that method should return a simplification of the element (will be used in each operation on elements
that affects coefficients), or None (which amounts to no simplification).
• is_zero – (default: 'is_zero') a string, containing the name of a method of an element of the base
ring; that method should return True when a simplified element of the base ring is equal to zero (will
be used to decide equality of elements, to calculate the arity of elements, and to skip terms in some
operations on elements)
__repr__()
Return a string representation of this polydifferential operator algebra.
base_ring()
Return the base ring of this polydifferential operator algebra, consisting of functions.
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coordinate(i)
Return the i-th even coordinate in the base ring of this polydifferential operator algebra.
coordinates()
Return the coordinates in the base ring of this polydifferential operator algebra.
derivative(i)
Return the i-th derivative of this polydifferential operator algebra.
derivatives()
Return the tuple of derivatives of this polydifferential operator algebra.
gen(i)
Return the i-th derivative of this polydifferential operator algebra.
gens()
Return the tuple of derivatives of this polydifferential operator algebra.
identity_operator()
Return the (unary) identity operator of this polydifferential operator algebra.
multiplication_operator()
Return the (binary) multiplication operator of this polydifferential operator algebra.
ngens()
Return the number of derivatives of this polydifferential operator algebra.
tensor_product(*args)
Return the tensor product of args as an element of this polydifferential operator algebra.
zero()
Return the zero element of this polydifferential operator algebra.
1.4 Tensor product
Tensor product
class gcaops.algebra.tensor_product.TensorProduct(factors)
Bases: object
Tensor product of vector spaces.
factor(index)
Return the index-th factor of this tensor product.
factors()
Return the list of factors of this tensor product.
nfactors()
Return the number of factors of this tensor product.
class gcaops.algebra.tensor_product.TensorProductElement(parent, terms)
Bases: object
Element of a tensor product of vector spaces.
graded_symmetrization()
Return the graded symmetrization of this tensor product element.
ASSUMPTION:
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Assumes each factor in each term of this tensor product element has a degree method and is homogeneous
of that degree.
parent()
Return the parent TensorProduct that this tensor product element belongs to.
terms()
Return the list of terms of this tensor product element.
1.5 Differential polynomial ring
Differential polynomial ring
class gcaops.algebra.differential_polynomial_ring.DifferentialPolynomial(parent, polyno-
mial)
Bases: object
Differential polynomial.
derivative(*x)
Return the total derivative of this differential polynomial with respect to the base variables x.
diff(*x)
Return the total derivative of this differential polynomial with respect to the base variables x.
fibre_degrees()
Return the vector of degrees (with respect to each fibre variable) of this differential monomial.
parent()
Return the DifferentialPolynomialRing that this differential polynomial belongs to.
partial_derivative(*x)
Return the partial derivative of this differential polynomial with respect to the variables x.
pdiff(*x)
Return the partial derivative of this differential polynomial with respect to the variables x.
total_derivative(*x)
Return the total derivative of this differential polynomial with respect to the base variables x.
weights()
Return the vector of weights of this differential monomial.
class gcaops.algebra.differential_polynomial_ring.DifferentialPolynomialRing(base_ring,
fi-
bre_names,
base_names,
max_differential_orders)
Bases: object
Differential polynomial ring.
__init__(base_ring, fibre_names, base_names, max_differential_orders)
Initialize this differential polynomial ring.
INPUT:
• base_ring – a ring, the ring of coefficients
• fibre_names – a list of strings, the names of the fibre variables
• base_names – a list of strings, the names of the base variables
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• max_differential_orders – a list of natural numbers, the maximum differential order of each
fibre variable
base_variables()
Return the tuple of base variables of this differential polynomial ring.
element_class
alias of DifferentialPolynomial
fibre_variables()
Return the tuple of fibre variables of this differential polynomial ring.
homogeneous_monomials(fibre_degrees, weights, max_differential_orders=None)
Return the list of differential monomials with the given degrees and weights.
1.6 Homogeneous differential polynomial equation solver
Homogeneous differential polynomial equation solver
gcaops.algebra.differential_polynomial_solver.solve_homogeneous_diffpoly(target, source,
unknowns)
Return a solution of a homogeneous differential polynomial equation.
INPUT:
• target – a homogeneous differential polynomial, the right-hand side of the equation
• source – a homogeneous differential polynomial, the left-hand side of the equation
• unknowns – a list of fibre variables, such that the total derivatives of those variables appear in source
ALGORITHM:
Builds an ansatz based on the homogeneity, and solves the arising linear system.
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CHAPTER
TWO
ABSTRACT BASE CLASSES
2.1 Graph basis
Graph basis
class gcaops.graph.graph_basis.GraphBasis
Bases: abc.ABC
Basis of a module spanned by graphs.
A basis consists of tuples grading + (index, ...) where e.g. grading = (num_vertices,
num_edges) and grading + (index,) identifies the isomorphism class of the graph.
graph_properties()
Return a dictionary containing the properties of the graphs in this basis.
graph_to_key(graph)
Return a tuple consisting of the key in this basis and the sign factor such that graph equals the sign times
the graph identified by the key.
INPUT:
• graph – a graph
key_to_graph(key)
Return a tuple consisting of a graph and the sign factor such that the sign times the graph equals the graph
identified by the key.
INPUT:
• key – a key in this basis
2.2 Graph vector
Graph vector
class gcaops.graph.graph_vector.GraphModule
Bases: abc.ABC
Module spanned by graphs.
__call__(arg)
Convert arg into an element of this module.
__repr__()
Return a string representation of this module.
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base_ring()
Return the base ring of this module.
basis()
Return the basis of this module.
zero()
Return the zero vector in this module.
class gcaops.graph.graph_vector.GraphVector
Bases: abc.ABC
Vector representing a linear combination of graphs.
__add__(other)
Return this graph vector added to other.
__eq__(other)
Return True if this graph vector is equal to other and False otherwise.
__iter__()
Returns an iterator over this graph vector, yielding tuples of the form (coeff, graph).
__len__()
Return the number of graphs with nonzero coefficients in this graph vector.
__mul__(other)
Return this graph vector multiplied by other.
__neg__()
Return the negative of this graph vector.
__pos__()
Return a copy of this graph vector.
__radd__(other)
Return other added to this graph vector.
__repr__()
Return a string representation of this graph vector.
__rmul__(other)
Return other multiplied by this graph vector.
coefficient(monomial)
Return the coefficient of monomial in this graph vector.
copy()
Return a copy of this graph vector.
gradings()
Return the set of grading tuples such that this graph vector contains terms with those gradings.
homogeneous_part(*grading)
Return the homogeneous part of this graph vector consisting only of terms with the given grading.
insertion(position, other, **kwargs)
Return the insertion of other into this graph vector at the vertex position.
map_coefficients(f, new_parent=None)
Apply f to each of this graph vector’s coefficients and return the resulting graph vector.
map_graphs(f, new_parent=None)
Apply f to each of this graph vector’s graphs and return the resulting graph vector.
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nedges()
Return the number of edges in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of edges.
nvertices()
Return the number of vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of vertices.
parent()
Return the parent GraphModule that this graph vector belongs to.
plot(**options)
Return a plot of this graph vector.
show(**options)
Show this graph.
2.3 Graph complex
Graph complex
class gcaops.graph.graph_complex.GraphCochain
Bases: gcaops.graph.graph_vector.GraphVector
Cochain of a GraphComplex.
bracket(other)
Return the graph Lie bracket of this graph cochain with other.
differential()
Return the graph differential of this graph cochain.
class gcaops.graph.graph_complex.GraphComplex
Bases: gcaops.graph.graph_vector.GraphModule
Graph complex.
2.4 Graph vector (vector backend)
Graph vector (vector backend)
class gcaops.graph.graph_vector_vector.GraphModule_vector(base_ring, graph_basis,
vector_constructor, ma-
trix_constructor, sparse=True)
Bases: gcaops.graph.graph_vector.GraphModule
Module spanned by graphs (with elements stored as dictionaries of vectors).
__call__(arg)
Convert arg into an element of this module.
__init__(base_ring, graph_basis, vector_constructor, matrix_constructor, sparse=True)
Initialize this graph module.
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INPUT:
• base_ring – a ring, to be used as the ring of coefficients
• graph_basis – a GraphBasis
• vector_constructor – constructor of (sparse) vectors
• matrix_constructor – constructor of (sparse) matrices
• sparse – (default: True) a boolean, passed along to both constructors as a keyword argument
__repr__()
Return a string representation of this module.
base_ring()
Return the base ring of this module.
basis()
Return the basis of this module.
zero()
Return the zero vector in this module.
class gcaops.graph.graph_vector_vector.GraphVector_vector(parent, vectors)
Bases: gcaops.graph.graph_vector.GraphVector
Vector representing a linear combination of graphs (stored as a dictionary of vectors).
__add__(other)
Return this graph vector added to other.
__eq__(other)
Return True if this graph vector is equal to other and False otherwise.
__init__(parent, vectors)
Initialize this graph vector.
INPUT:
• parent – a GraphModule
• vectors – a dictionary, mapping gradings to (sparse) vectors of coefficients with respect to the basis
of parent
__iter__()
Facilitates iterating over this graph vector, yielding tuples of the form (coeff, graph).
__len__()
Return the number of graphs with nonzero coefficients in this graph vector.
__mul__(other)
Return this graph vector multiplied by other.
__neg__()
Return the negative of this graph vector.
__pos__()
Return a copy of this graph vector.
__radd__(other)
Return other added to this graph vector.
__repr__()
Return a string representation of this graph vector.
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__rmul__(other)
Return other multiplied by this graph vector.
__rsub__(other)
Return this graph vector subtracted from other.
__sub__(other)
Return other subtracted from this graph vector.
copy()
Return a copy of this graph vector.
gradings()
Return the set of grading tuples such that this graph vector contains terms with those gradings.
homogeneous_part(*grading)
Return the homogeneous part of this graph vector consisting only of terms with the given grading.
insertion(position, other, **kwargs)
Return the insertion of other into this graph vector at the vertex position.
map_coefficients(f, new_parent=None)
Apply f to each of this graph vector’s coefficients and return the resulting graph vector.
parent()
Return the parent GraphModule that this graph vector belongs to.
vector(*grading)
Return the vector of coefficients of graphs with the given grading.
2.5 Graph vector (dictionary backend)
Graph vector (dictionary backend)
class gcaops.graph.graph_vector_dict.GraphModule_dict(base_ring, graph_basis)
Bases: gcaops.graph.graph_vector.GraphModule
Module spanned by graphs (with elements stored as dictionaries).
__call__(arg)
Convert arg into an element of this module.
__init__(base_ring, graph_basis)
Initialize this graph module.
INPUT:
• base_ring – a ring, to be used as the ring of coefficients
• graph_basis – a GraphBasis
__repr__()
Return a string representation of this module.
base_ring()
Return the base ring of this module.
basis()
Return the basis of this module.
zero()
Return the zero vector in this module.
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class gcaops.graph.graph_vector_dict.GraphVector_dict(parent, vector)
Bases: gcaops.graph.graph_vector.GraphVector
Vector representing a linear combination of graphs (stored as a dictionary).
__add__(other)
Return this graph vector added to other.
__eq__(other)
Return True if this graph vector is equal to other and False otherwise.
__init__(parent, vector)
Initialize this graph vector.
INPUT:
• parent – a GraphModule
• vector – a dictionary, representing a sparse vector of coefficients with respect to the basis of parent
__iter__()
Facilitates iterating over this graph vector, yielding tuples of the form (coeff, graph).
__len__()
Return the number of graphs with nonzero coefficients in this graph vector.
__mul__(other)
Return this graph vector multiplied by other.
__neg__()
Return the negative of this graph vector.
__pos__()
Return a copy of this graph vector.
__radd__(other)
Return other added to this graph vector.
__repr__()
Return a string representation of this graph vector.
__rmul__(other)
Return other multiplied by this graph vector.
__rsub__(other)
Return this graph vector subtracted from other.
__sub__(other)
Return other subtracted from this graph vector.
copy()
Return a copy of this graph vector.
gradings()
Return the set of grading tuples such that this graph vector contains terms with those gradings.
homogeneous_part(*grading)
Return the homogeneous part of this graph vector consisting only of terms with the given grading.
insertion(position, other, **kwargs)
Return the insertion of other into this graph vector at the vertex position.
map_coefficients(f, new_parent=None)
Apply f to each of this graph vector’s coefficients and return the resulting graph vector.
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parent()
Return the parent GraphModule that this graph vector belongs to.
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CHAPTER
THREE
UNDIRECTED GRAPHS
3.1 Undirected graph
Undirected graph
class gcaops.graph.undirected_graph.UndirectedGraph(num_vertices, edges)
Bases: object
Undirected graph with vertices labeled by natural numbers and an ordered set of edges.
__eq__(other)
Return True if this graph equals other.
Note that this is not an isomorphism test, and the ordering of the list of edges is taken into account.
EXAMPLES:
sage: g = UndirectedGraph(3, [(0, 1), (1, 2), (2, 0)])
sage: h1 = UndirectedGraph(3, [(0, 1), (1, 2), (0, 2)])
sage: g == h1
True
sage: h2 = UndirectedGraph(3, [(0, 1), (0, 2), (1, 2)])
sage: g == h2
False
__init__(num_vertices, edges)
Initialize this undirected graph.
INPUT:
• num_vertices – a natural number, the number of vertices
• edges – a list of tuples of natural numbers
EXAMPLES:
1. Construct the graph consisting of a single edge:
sage: g = UndirectedGraph(2, [(0, 1)]); g
UndirectedGraph(2, [(0, 1)])
2. Construct the tetrahedron graph:
sage: g = UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]); g
UndirectedGraph(4, [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])
__len__()
Return the number of vertices of this graph.
EXAMPLES:
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sage: g = UndirectedGraph(3, [(0, 1), (1, 2), (2, 0)])
sage: len(g)
3
__repr__()
Return a string representation of this graph.
EXAMPLES:
sage: g = UndirectedGraph(3, [(0, 1), (1, 2), (2, 0)])
sage: repr(g)
'UndirectedGraph(3, [(0, 1), (1, 2), (0, 2)])'
canonicalize_edges()
Lexicographically order the edges of this graph and return the sign of that edge permutation.
EXAMPLES:
sage: g = UndirectedGraph(3, [(1, 2), (0, 1)])
sage: g.canonicalize_edges()
-1
sage: g
UndirectedGraph(3, [(0, 1), (1, 2)])
edges()
Return the list of edges of this graph.
EXAMPLES:
sage: g = UndirectedGraph(3, [(0, 1), (1, 2)])
sage: g.edges()
[(0, 1), (1, 2)]
get_pos()
Return the dictionary of positions of vertices in this graph (used for plotting).
EXAMPLES:
sage: g = UndirectedGraph(2, [(0, 1)])
sage: g.get_pos() is None
True
sage: g.set_pos({0: (0.0, 0.0), 1: (1.0, 0.0)})
sage: g.get_pos()
{0: (0.000000000000000, 0.000000000000000),
1: (1.00000000000000, 0.000000000000000)}
orientations()
Return a generator producing the DirectedGraphs which are obtained by orienting this graph in all possible
ways.
EXAMPLES:
sage: g = UndirectedGraph(2, [(0, 1)])
sage: list(g.orientations())
[DirectedGraph(2, [(0, 1)]), DirectedGraph(2, [(1, 0)])]
plot(**options)
Return a plot of this graph.
EXAMPLES:
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sage: g = UndirectedGraph(2, [(0, 1)])
sage: g.plot()
Graphics object consisting of 4 graphics primitives
relabeled(relabeling)
Return the graph obtained by relabeling this graph in the given way.
EXAMPLES:
sage: g = UndirectedGraph(3, [(0, 1), (1, 2)])
sage: g.relabeled({0: 1, 1: 0, 2: 2})
UndirectedGraph(3, [(0, 1), (0, 2)])
set_pos(new_pos)
Set the positions of vertices in this graph (used for plotting).
EXAMPLES:
sage: g = UndirectedGraph(2, [(0, 1)])
sage: g.set_pos({0: (0.0, 0.0), 1: (1.0, 0.0)})
sage: g.get_pos()
{0: (0.000000000000000, 0.000000000000000),
1: (1.00000000000000, 0.000000000000000)}
show(**options)
Show this graph.
EXAMPLES:
sage: g = UndirectedGraph(2, [(0, 1)])
sage: g.show()
3.2 Undirected graph basis
Undirected graph basis
class gcaops.graph.undirected_graph_basis.UndirectedGraphBasis
Bases: gcaops.graph.graph_basis.GraphBasis
Basis of a module spanned by undirected graphs.
A basis consists of keys (v,e,index,...) where (v,e,index) identifies the isomorphism class of the graph.
graph_class
alias of gcaops.graph.undirected_graph.UndirectedGraph
class gcaops.graph.undirected_graph_basis.UndirectedGraphComplexBasis(connected=None,
bicon-
nected=None,
min_degree=0)
Bases: gcaops.graph.undirected_graph_basis.UndirectedGraphBasis
Basis consisting of representatives of isomorphism classes of undirected graphs with no automorphisms that
induce an odd permutation on edges
cardinality(vertices, edges)
Return the number of graphs in this basis with the given amount of vertices and edges.
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graph_properties()
Return a dictionary containing the properties of the graphs in this basis.
graph_to_key(graph)
Return a tuple consisting of the key in this basis and the sign factor such that graph equals the sign times
the graph identified by the key.
INPUT:
• graph – an UndirectedGraph
OUTPUT:
Either (None, 1) if the input graph is not in the span of the basis, or a tuple consisting of a key and a
sign, where a key is a tuple consisting of the number of vertices, the number of edges, and the index of the
graph in the list.
graphs(vertices, edges)
Return the list of graphs in this basis with the given amount of vertices and edges.
key_to_graph(key)
Return a tuple consisting of an UndirectedGraph and the sign factor such that the sign times the graph
equals the graph identified by the key.
INPUT:
• key – a key in this basis
OUTPUT:
Either (None, 1) if the input key is not in the basis, or a tuple consisting of an UndirectedGraph and
a sign which is always +1.
class gcaops.graph.undirected_graph_basis.UndirectedGraphOperadBasis
Bases: gcaops.graph.undirected_graph_basis.UndirectedGraphBasis
Basis consisting of labeled undirected graphs with no automorphisms that induce an odd permutation on edges
graph_properties()
Return a dictionary containing the properties of the graphs in this basis.
graph_to_key(graph)
Return a tuple consisting of the key in this basis and the sign factor such that graph equals the sign times
the graph identified by the key.
INPUT:
• graph – an UndirectedGraph
OUTPUT:
Either (None, 1) if the input graph is not in the span of the basis, or a tuple consisting of a key and a
sign, where a key is a tuple consisting of the number of vertices, the number of edges, the index of the
graph in the list, followed by a permutation of vertices.
key_to_graph(key)
Return a tuple consisting of an UndirectedGraph and the sign factor such that the sign times the graph
equals the graph identified by the key.
INPUT:
• key – a key in this basis
OUTPUT:
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Either (None, 1) if the input key is not in the basis, or a tuple consisting of an UndirectedGraph and
a sign.
3.3 Undirected graph vector
Undirected graph vector
class gcaops.graph.undirected_graph_vector.UndirectedGraphModule
Bases: gcaops.graph.graph_vector.GraphModule
Module spanned by undirected graphs.
class gcaops.graph.undirected_graph_vector.UndirectedGraphModule_dict(base_ring,
graph_basis)
Bases: gcaops.graph.undirected_graph_vector.UndirectedGraphModule, gcaops.graph.
graph_vector_dict.GraphModule_dict
Module spanned by undirected graphs (with elements stored as dictionaries).
__init__(base_ring, graph_basis)
Initialize this undirected graph module.
INPUT:
• base_ring – a ring, to be used as the ring of coefficients
• graph_basis – an UndirectedGraphBasis
class gcaops.graph.undirected_graph_vector.UndirectedGraphModule_vector(base_ring,
graph_basis,
vec-
tor_constructor,
ma-
trix_constructor,
sparse=True)
Bases: gcaops.graph.undirected_graph_vector.UndirectedGraphModule, gcaops.graph.
graph_vector_vector.GraphModule_vector
Module spanned by undirected graphs (with elements stored as dictionaries of vectors).
__init__(base_ring, graph_basis, vector_constructor, matrix_constructor, sparse=True)
Initialize this undirected graph module.
INPUT:
• base_ring – a ring, to be used as the ring of coefficients
• graph_basis – an UndirectedGraphBasis
• vector_constructor – constructor of (sparse) vectors
• matrix_constructor – constructor of (sparse) matrices
• sparse – (default: True) a boolean, passed along to both constructors as a keyword argument
class gcaops.graph.undirected_graph_vector.UndirectedGraphVector
Bases: gcaops.graph.graph_vector.GraphVector
Vector representing a linear combination of undirected graphs.
class gcaops.graph.undirected_graph_vector.UndirectedGraphVector_dict(parent, vector)
Bases: gcaops.graph.undirected_graph_vector.UndirectedGraphVector, gcaops.graph.
graph_vector_dict.GraphVector_dict
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Vector representing a linear combination of undirected graphs (stored as a dictionary).
__init__(parent, vector)
Initialize this undirected graph vector.
INPUT:
• parent – an UndirectedGraphModule
• vector – a dictionary, representing a sparse vector of coefficients with respect to the basis of parent
nedges()
Return the number of edges in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of edges.
nvertices()
Return the number of vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of vertices.
class gcaops.graph.undirected_graph_vector.UndirectedGraphVector_vector(parent, vectors)
Bases: gcaops.graph.undirected_graph_vector.UndirectedGraphVector, gcaops.graph.
graph_vector_vector.GraphVector_vector
Vector representing a linear combination of undirected graphs (stored as a dictionary of vectors).
__init__(parent, vectors)
Initialize this graph vector.
INPUT:
• parent – an UndirectedGraphModule
• vectors – a dictionary, mapping bi-gradings to (sparse) vectors of coefficients with respect to the
basis of parent
nedges()
Return the number of edges in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of edges.
nvertices()
Return the number of vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of vertices.
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3.4 Undirected graph operad
Undirected graph operad
gcaops.graph.undirected_graph_operad.UndirectedGraphOperad(base_ring)
Return the operad of undirected graphs over the given base_ring.
class gcaops.graph.undirected_graph_operad.UndirectedGraphOperad_dict(base_ring)
Bases: gcaops.graph.undirected_graph_vector.UndirectedGraphModule_dict
Operad of undirected graphs (with elements stored as dictionaries).
__init__(base_ring)
Initialize this graph operad.
__repr__()
Return a string representation of this graph operad.
class gcaops.graph.undirected_graph_operad.UndirectedGraphOperation_dict(parent, vector)
Bases: gcaops.graph.undirected_graph_vector.UndirectedGraphVector_dict
Element of an UndirectedGraphOperad_dict (stored as a dictionary).
__init__(parent, vector)
Initialize this graph operation.
3.5 Undirected graph complex
Undirected graph complex
class gcaops.graph.undirected_graph_complex.UndirectedGraphCochain
Bases: gcaops.graph.graph_complex.GraphCochain, gcaops.graph.undirected_graph_vector.
UndirectedGraphVector
Cochain of an UndirectedGraphComplex_.
bracket(other)
Return the graph Lie bracket of this graph cochain with other.
class gcaops.graph.undirected_graph_complex.UndirectedGraphCochain_dict(parent, vector)
Bases: gcaops.graph.undirected_graph_complex.UndirectedGraphCochain, gcaops.graph.
undirected_graph_vector.UndirectedGraphVector_dict
Cochain of an UndirectedGraphComplex_dict (stored as a dictionary).
__init__(parent, vector)
Initialize this graph cochain.
differential()
Return the graph differential of this graph cochain.
class gcaops.graph.undirected_graph_complex.UndirectedGraphCochain_vector(parent, vec-
tor)
Bases: gcaops.graph.undirected_graph_complex.UndirectedGraphCochain, gcaops.graph.
undirected_graph_vector.UndirectedGraphVector_vector
Cochain of an UndirectedGraphComplex_vector (stored as a dictionary of vectors).
__init__(parent, vector)
Initialize this graph cochain.
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differential(use_cache=True)
Return the graph differential of this graph cochain.
is_coboundary(certificate=False)
Return True if this graph cochain is a coboundary.
INPUT:
• certificate - if True, return a tuple where the first element is the truth value, and the second
element is a graph cochain such that its differential is this graph cochain (or None).
gcaops.graph.undirected_graph_complex.UndirectedGraphComplex(base_ring, connected=None,
biconnected=None,
min_degree=0, imple-
mentation='dict', vec-
tor_constructor=None,
matrix_constructor=None,
sparse=True)
Return the undirected graph complex over base_ring with the given properties.
class gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_
Bases: gcaops.graph.graph_complex.GraphComplex, gcaops.graph.undirected_graph_vector.
UndirectedGraphModule
Undirected graph complex.
class gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_dict(base_ring, con-
nected=None,
bicon-
nected=None,
min_degree=0)
Bases: gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_, gcaops.graph.
undirected_graph_vector.UndirectedGraphModule_dict
Undirected graph complex (with elements stored as dictionaries).
__init__(base_ring, connected=None, biconnected=None, min_degree=0)
Initialize this graph complex.
__repr__()
Return a string representation of this graph complex.
class gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_vector(base_ring,
vec-
tor_constructor,
ma-
trix_constructor,
sparse=True,
con-
nected=None,
bicon-
nected=None,
min_degree=0)
Bases: gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_, gcaops.graph.
undirected_graph_vector.UndirectedGraphModule_vector
Undirected graph complex (with elements stored as dictionaries of vectors).
__init__(base_ring, vector_constructor, matrix_constructor, sparse=True, connected=None, bicon-
nected=None, min_degree=0)
Initialize this graph complex.
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INPUT:
• base_ring – a ring, to be used as the ring of coefficients
• vector_constructor – constructor of (sparse) vectors
• matrix_constructor – constructor of (sparse) matrices
• sparse – (default: True) a boolean, passed along to both constructors as a keyword argument
__repr__()
Return a string representation of this graph complex.
cohomology_basis(vertices, edges)
Return a basis of the cohomology in the given bi-grading (vertices, edges).
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FOUR
DIRECTED GRAPHS
4.1 Directed graph
Directed graph
class gcaops.graph.directed_graph.DirectedGraph(num_vertices, edges)
Bases: object
Directed graph with vertices labeled by natural numbers and an ordered set of edges.
canonicalize_edges()
Lexicographically order the edges of this graph and return the sign of that edge permutation.
EXAMPLES:
sage: g = DirectedGraph(3, [(2, 1), (2, 0)])
sage: g.canonicalize_edges()
-1
sage: g
DirectedGraph(3, [(2, 0), (2, 1)])
edges()
Return the list of edges of this graph.
EXAMPLES:
sage: g = DirectedGraph(3, [(0, 1), (1, 2)])
sage: g.edges()
[(0, 1), (1, 2)]
get_pos()
Return the dictionary of positions of vertices in this graph (used for plotting).
EXAMPLES:
sage: g = DirectedGraph(2, [(0, 1)])
sage: g.get_pos() is None
True
sage: g.set_pos({0: (0.0, 0.0), 1: (1.0, 0.0)})
sage: g.get_pos()
{0: (0.000000000000000, 0.000000000000000),
1: (1.00000000000000, 0.000000000000000)}
in_degrees()
Return the tuple of in-degrees of vertices of this graph.
EXAMPLES:
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sage: g = DirectedGraph(4, [(1, 0), (2, 0), (3, 0), (1, 2), (2, 3), (3, 1)])
sage: g.in_degrees()
(3, 1, 1, 1)
out_degrees()
Return the tuple of out-degrees of vertices of this graph.
EXAMPLES:
sage: g = DirectedGraph(4, [(1, 0), (2, 0), (3, 0), (1, 2), (2, 3), (3, 1)])
sage: g.out_degrees()
(0, 2, 2, 2)
plot(**options)
Return a plot of this graph.
EXAMPLES:
sage: g = DirectedGraph(2, [(0, 1)])
sage: g.plot()
Graphics object consisting of 4 graphics primitives
relabeled(relabeling)
Return the graph obtained by relabeling this graph in the given way.
EXAMPLES:
sage: g = DirectedGraph(3, [(2, 0), (2, 1)])
sage: g.relabeled({0: 1, 1: 0, 2: 2})
DirectedGraph(3, [(2, 1), (2, 0)])
set_pos(new_pos)
Set the positions of vertices in this graph (used for plotting).
EXAMPLES:
sage: g = DirectedGraph(2, [(0, 1)])
sage: g.set_pos({0: (0.0, 0.0), 1: (1.0, 0.0)})
sage: g.get_pos()
{0: (0.000000000000000, 0.000000000000000),
1: (1.00000000000000, 0.000000000000000)}
show(**options)
Show this graph.
EXAMPLES:
sage: g = DirectedGraph(2, [(0, 1)])
sage: g.show()
4.2 Directed graph basis
Directed graph basis
class gcaops.graph.directed_graph_basis.DirectedGraphBasis
Bases: gcaops.graph.graph_basis.GraphBasis
Basis of a module spanned by directed graphs.
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A basis consists of keys (v,e,index,...) where (v,e,index) identifies the isomorphism class of the graph.
graph_class
alias of gcaops.graph.directed_graph.DirectedGraph
class gcaops.graph.directed_graph_basis.DirectedGraphComplexBasis(connected=None,
biconnected=None,
min_degree=0,
loops=True)
Bases: gcaops.graph.directed_graph_basis.DirectedGraphBasis
Basis consisting of representatives of isomorphism classes of directed graphs with no automorphisms that induce
an odd permutation on edges
cardinality(vertices, edges)
Return the number of graphs in this basis with the given amount of vertices and edges.
graph_properties()
Return a dictionary containing the properties of the graphs in this basis.
graph_to_key(graph)
Return a tuple consisting of the key in this basis and the sign factor such that graph equals the sign times
the graph identified by the key.
INPUT:
• graph – a DirectedGraph
OUTPUT:
Either (None, 1) if the input graph is not in the span of the basis, or a tuple consisting of a key and a
sign, where a key is a tuple consisting of the number of vertices, the number of edges, and the index of the
graph in the list.
graphs(vertices, edges)
Return the list of graphs in this basis with the given amount of vertices and edges.
key_to_graph(key)
Return a tuple consisting of a DirectedGraph and the sign factor such that the sign times the graph equals
the graph identified by the key.
INPUT:
• key – a key in this basis
OUTPUT:
Either (None, 1) if the input key is not in the basis, or a tuple consisting of a DirectedGraph and a
sign which is always +1.
class gcaops.graph.directed_graph_basis.DirectedGraphOperadBasis
Bases: gcaops.graph.directed_graph_basis.DirectedGraphBasis
Basis consisting of labeled directed graphs with no automorphisms that induce an odd permutation on edges
graph_properties()
Return a dictionary containing the properties of the graphs in this basis.
graph_to_key(graph)
Return a tuple consisting of the key in this basis and the sign factor such that graph equals the sign times
the graph identified by the key.
INPUT:
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• graph – a DirectedGraph
OUTPUT:
Either (None, 1) if the input graph is not in the span of the basis, or a tuple consisting of a key and a
sign, where a key is a tuple consisting of the number of vertices, the number of edges, the index of the
graph in the list, followed by a permutation of vertices.
key_to_graph(key)
Return a tuple consisting of a DirectedGraph and the sign factor such that the sign times the graph equals
the graph identified by the key.
INPUT:
• key – a key in this basis
OUTPUT:
Either (None, 1) if the input key is not in the basis, or a tuple consisting of a DirectedGraph and a
sign.
4.3 Directed graph vector
Directed graph vector
class gcaops.graph.directed_graph_vector.DirectedGraphModule
Bases: gcaops.graph.graph_vector.GraphModule
Module spanned by directed graphs.
__call__(arg)
Convert arg into an element of this module.
class gcaops.graph.directed_graph_vector.DirectedGraphModule_dict(base_ring,
graph_basis)
Bases: gcaops.graph.directed_graph_vector.DirectedGraphModule, gcaops.graph.
graph_vector_dict.GraphModule_dict
Module spanned by directed graphs (with elements stored as dictionaries).
__init__(base_ring, graph_basis)
Initialize this directed graph module.
INPUT:
• base_ring – a ring, to be used as the ring of coefficients
• graph_basis – a DirectedGraphBasis
class gcaops.graph.directed_graph_vector.DirectedGraphModule_vector(base_ring,
graph_basis, vec-
tor_constructor,
matrix_constructor,
sparse=True)
Bases: gcaops.graph.directed_graph_vector.DirectedGraphModule, gcaops.graph.
graph_vector_vector.GraphModule_vector
Module spanned by directed graphs (with elements stored as dictionaries of vectors).
__init__(base_ring, graph_basis, vector_constructor, matrix_constructor, sparse=True)
Initialize this directed graph module.
INPUT:
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• base_ring – a ring, to be used as the ring of coefficients
• graph_basis – a DirectedGraphBasis
• vector_constructor – constructor of (sparse) vectors
• matrix_constructor – constructor of (sparse) matrices
• sparse – (default: True) a boolean, passed along to both constructors as a keyword argument
class gcaops.graph.directed_graph_vector.DirectedGraphVector
Bases: gcaops.graph.graph_vector.GraphVector
Vector representing a linear combination of directed graphs.
filter(max_out_degree=None)
Return the graph vector which is the summand of this graph vector containing exactly those graphs that
pass the filter.
class gcaops.graph.directed_graph_vector.DirectedGraphVector_dict(parent, vector)
Bases: gcaops.graph.directed_graph_vector.DirectedGraphVector, gcaops.graph.
graph_vector_dict.GraphVector_dict
Vector representing a linear combination of directed graphs (stored as a dictionary).
__init__(parent, vector)
Initialize this directed graph vector.
INPUT:
• parent – a DirectedGraphModule
• vector – a dictionary, representing a sparse vector of coefficients with respect to the basis of parent
filter(max_out_degree=None)
Return the graph vector which is the summand of this graph vector containing exactly those graphs that
pass the filter.
nedges()
Return the number of edges in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of edges.
nvertices()
Return the number of vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of vertices.
class gcaops.graph.directed_graph_vector.DirectedGraphVector_vector(parent, vectors)
Bases: gcaops.graph.directed_graph_vector.DirectedGraphVector, gcaops.graph.
graph_vector_vector.GraphVector_vector
Vector representing a linear combination of directed graphs (stored as a dictionary of vectors).
__init__(parent, vectors)
Initialize this graph vector.
INPUT:
• parent – a DirectedGraphModule
• vectors – a dictionary, mapping bi-gradings to (sparse) vectors of coefficients with respect to the
basis of parent
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filter(max_out_degree=None)
Return the graph vector which is the summand of this graph vector containing exactly those graphs that
pass the filter.
nedges()
Return the number of edges in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of edges.
nvertices()
Return the number of vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of vertices.
4.4 Directed graph operad
Directed graph operad
gcaops.graph.directed_graph_operad.DirectedGraphOperad(base_ring)
Return the operad of directed graphs over the given base_ring.
class gcaops.graph.directed_graph_operad.DirectedGraphOperad_dict(base_ring)
Bases: gcaops.graph.directed_graph_vector.DirectedGraphModule_dict
Operad of directed graphs (with elements stored as dictionaries).
__init__(base_ring)
Initialize this graph operad.
__repr__()
Return a string representation of this graph operad.
class gcaops.graph.directed_graph_operad.DirectedGraphOperation_dict(parent, vector)
Bases: gcaops.graph.directed_graph_vector.DirectedGraphVector_dict
Element of a DirectedGraphOperad_dict (stored as a dictionary).
__init__(parent, vector)
Initialize this graph operation.
4.5 Directed graph complex
Directed graph complex
class gcaops.graph.directed_graph_complex.DirectedGraphCochain
Bases: gcaops.graph.graph_complex.GraphCochain, gcaops.graph.directed_graph_vector.
DirectedGraphVector
Cochain of a DirectedGraphComplex_.
bracket(other)
Return the graph Lie bracket of this graph cochain with other.
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class gcaops.graph.directed_graph_complex.DirectedGraphCochain_dict(parent, vector)
Bases: gcaops.graph.directed_graph_complex.DirectedGraphCochain, gcaops.graph.
directed_graph_vector.DirectedGraphVector_dict
Cochain of a DirectedGraphComplex_dict (stored as a dictionary).
__init__(parent, vector)
Initialize this graph cochain.
differential()
Return the graph differential of this graph cochain.
class gcaops.graph.directed_graph_complex.DirectedGraphCochain_vector(parent, vector)
Bases: gcaops.graph.directed_graph_complex.DirectedGraphCochain, gcaops.graph.
directed_graph_vector.DirectedGraphVector_vector
Cochain of a DirectedGraphComplex_vector (stored as a dictionary of vectors).
__init__(parent, vector)
Initialize this graph cochain.
differential(use_cache=True)
Return the graph differential of this graph cochain.
is_coboundary(certificate=False)
Return True if this graph cochain is a coboundary.
INPUT:
• certificate - if True, return a tuple where the first element is the truth value, and the second
element is a graph cochain such that its differential is this graph cochain (or None).
gcaops.graph.directed_graph_complex.DirectedGraphComplex(base_ring, connected=None, bi-
connected=None, min_degree=0,
loops=True, implementation='dict',
vector_constructor=None,
matrix_constructor=None,
sparse=True)
Return the directed graph complex over base_ring with the given properties.
class gcaops.graph.directed_graph_complex.DirectedGraphComplex_
Bases: gcaops.graph.graph_complex.GraphComplex, gcaops.graph.directed_graph_vector.
DirectedGraphModule
Directed graph complex.
class gcaops.graph.directed_graph_complex.DirectedGraphComplex_dict(base_ring, con-
nected=None, bi-
connected=None,
min_degree=0,
loops=True)
Bases: gcaops.graph.directed_graph_complex.DirectedGraphComplex_, gcaops.graph.
directed_graph_vector.DirectedGraphModule_dict
Directed graph complex (with elements stored as dictionaries).
__init__(base_ring, connected=None, biconnected=None, min_degree=0, loops=True)
Initialize this graph complex.
__repr__()
Return a string representation of this graph complex.
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class gcaops.graph.directed_graph_complex.DirectedGraphComplex_vector(base_ring, vec-
tor_constructor,
ma-
trix_constructor,
sparse=True, con-
nected=None, bi-
connected=None,
min_degree=0,
loops=True)
Bases: gcaops.graph.directed_graph_complex.DirectedGraphComplex_, gcaops.graph.
directed_graph_vector.DirectedGraphModule_vector
Directed graph complex (with elements stored as dictionaries of vectors).
__init__(base_ring, vector_constructor, matrix_constructor, sparse=True, connected=None, bicon-
nected=None, min_degree=0, loops=True)
Initialize this graph complex.
__repr__()
Return a string representation of this graph complex.
cohomology_basis(vertices, edges)
Return a basis of the cohomology in the given bi-grading (vertices, edges).
616 Chapter 4. Directed graphs
CHAPTER
FIVE
FORMALITY GRAPHS
5.1 Formality graph
Formality graph
class gcaops.graph.formality_graph.FormalityGraph(num_ground_vertices, num_aerial_vertices,
edges)
Bases: object
Directed graph with an ordered set of edges, and vertices labeled by natural numbers, the first of which are
ordered ground vertices without outgoing edges.
aerial_product(other)
Return the product of this graph with the other graph (i.e. the disjoint union followed by the identification
of the ground vertices).
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.aerial_product(g)
FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 0), (3, 1)])
automorphism_group()
Return the automorphism group of this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 0), (3, 1)])
sage: g.automorphism_group()
Permutation Group with generators [(2,3)]
canonicalize_edges()
Lexicographically order the edges of this graph and return the sign of that edge permutation.
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 1), (2, 0)])
sage: g.canonicalize_edges()
-1
sage: g
FormalityGraph(2, 1, [(2, 0), (2, 1)])
differential_orders()
Return the tuple of in-degrees of the ground vertices of this graph.
EXAMPLES:
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sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.differential_orders()
(1, 1)
sage: h = FormalityGraph(3, 2, [(3, 0), (3, 1), (3, 2), (4, 3), (4, 2)])
sage: h.differential_orders()
(1, 1, 2)
edge_contraction_graph(edge)
Return the FormalityGraph which is obtained by contracting the edge edge between aerial vertices in
this graph.
EXAMPLES:
sage: g = FormalityGraph(3, 2, [(3, 0), (3, 1), (4, 3), (4, 2)])
sage: g.edge_contraction_graph((4, 3))
FormalityGraph(3, 1, [(3, 0), (3, 1), (3, 2)])
edges()
Return the list of edges of this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.edges()
[(2, 0), (2, 1)]
edges_in_air()
Return the list of edges between aerial vertices of this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 2), (3, 1)])
sage: g.edges_in_air()
[(3, 2)]
sage: h = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: h.edges_in_air()
[]
static from_kgs_encoding(kgs_encoding)
Return a tuple consisting of a sign and a FormalityGraph built of wedges, as specified by the given encod-
ing.
INPUT:
• kgs_encoding – a string, containing a graph encoding as used in Buring’s
kontsevich_graph_series-cpp programs
See also:
kgs_encoding()
EXAMPLES:
sage: FormalityGraph.from_kgs_encoding('2 1 1 0 1')
(1, FormalityGraph(2, 1, [(2, 0), (2, 1)]))
sage: FormalityGraph.from_kgs_encoding('2 2 1 0 1 2 1')
(1, FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 2), (3, 1)]))
static from_kontsevint_encoding(kontsevint_encoding)
Return the Formalitygraph specified by the given encoding.
INPUT:
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• kontsevint_encoding – a string, containing a graph encoding as used in Panzer’s kontsevint
program
See also:
kontsevint_encoding()
EXAMPLES:
sage: FormalityGraph.from_kontsevint_encoding('[[L, R]]')
FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: FormalityGraph.from_kontsevint_encoding('[[p1, p2, p3]]')
FormalityGraph(3, 1, [(3, 0), (3, 1), (3, 2)])
sage: FormalityGraph.from_kontsevint_encoding('[[L, R], [1, R]]')
FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 2), (3, 1)])
get_pos()
Return the dictionary of positions of vertices in this graph (used for plotting).
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.get_pos() is None
True
sage: g.set_pos({0: (0.0, 0.0), 1: (1.0, 0.0), 2: (0.5, 1.0)})
sage: g.get_pos()
{0: (0.000000000000000, 0.000000000000000),
1: (1.00000000000000, 0.000000000000000),
2: (0.500000000000000, 1.00000000000000)}
ground_relabeled(relabeling)
Return a ground vertex relabeling of this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.ground_relabeled({0: 1, 1: 0})
FormalityGraph(2, 1, [(2, 1), (2, 0)])
has_eye_on_ground()
Return True if this graph contains a 2-cycle between two aerial vertices which are connected to the same
ground vertex, and False otherwise.
EXAMPLES:
sage: g = FormalityGraph(1, 2, [(1, 2), (2, 1), (1, 0), (2, 0)])
sage: g.has_eye_on_ground()
True
sage: h = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: h.has_eye_on_ground()
False
has_loops()
Return True if this graph contains an edge which is a loop, and False otherwise.
EXAMPLES:
sage: g = FormalityGraph(1, 1, [(1, 1)])
sage: g.has_loops()
True
sage: h = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: h.has_loops()
False
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has_multiple_edges()
Return True if this graph contains multiple edges, and False otherwise.
EXAMPLES:
sage: g = FormalityGraph(1, 1, [(1, 0), (1, 0)])
sage: g.has_multiple_edges()
True
sage: h = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: h.has_multiple_edges()
False
has_odd_automorphism()
Return True if this graph has an automorphism that induces an odd permutation on its ordered set of edges.
EXAMPLES:
sage: g = FormalityGraph(2, 3, [(2, 0), (2, 1), (3, 0), (3, 1), (4, 2), (4, 3)])
sage: g.has_odd_automorphism()
True
sage: h = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: h.has_odd_automorphism()
False
in_degrees()
Return the tuple of in-degrees of vertices of this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.in_degrees()
(1, 1, 0)
sage: h = FormalityGraph(3, 2, [(3, 0), (3, 1), (3, 2), (4, 3), (4, 2)])
sage: h.in_degrees()
(1, 1, 2, 1, 0)
kgs_encoding()
Return the encoding of this graph for use in Buring’s kontsevich_graph_series-cpp programs.
ASSUMPTIONS:
Assumes that this graph is built of wedges (i.e. each aerial vertex has out-degree two).
See also:
from_kgs_encoding()
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.kgs_encoding()
'2 1 1 0 1'
sage: h = FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 2), (3, 1)])
sage: h.kgs_encoding()
'2 2 1 0 1 2 1'
kontsevint_encoding()
Return the encoding of this graph for use in Panzer’s kontsevint program.
ASSUMPTIONS:
Assumes len(self.edges()) == 2*self.num_aerial_vertices() - 2 + self.
num_ground_vertices().
See also:
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from_kontsevint_encoding()
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.kontsevint_encoding()
'[[p1,p2]]'
sage: h = FormalityGraph(2, 2, [(2, 0), (2, 3), (3, 1), (3, 2)])
sage: h.kontsevint_encoding()
'[[p1,2],[p2,1]]'
multiplicity()
Return the number of formality graphs isomorphic to this one, under isomorphisms that preserve the
ground vertices pointwise.
EXAMPLES:
sage: g1 = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g1.multiplicity()
2
sage: g2 = FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 2), (3, 1)])
sage: g2.multiplicity()
8
sage: g3 = FormalityGraph(2, 2, [(2, 0), (2, 1), (3, 0), (3, 1)])
sage: g3.multiplicity()
4
num_aerial_vertices()
Return the number of aerial vertices of this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.num_aerial_vertices()
1
num_ground_vertices()
Return the number of ground vertices of this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.num_ground_vertices()
2
out_degrees()
Return the tuple of out-degrees of vertices of this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.out_degrees()
(0, 0, 2)
sage: h = FormalityGraph(3, 2, [(3, 0), (3, 1), (3, 2), (4, 3), (4, 2)])
sage: h.out_degrees()
(0, 0, 0, 3, 2)
plot(**options)
Return a plot of this graph.
EXAMPLES:
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sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.plot()
Graphics object consisting of 6 graphics primitives
relabeled(relabeling)
Return a vertex relabeling of this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.relabeled({0: 1, 1: 0, 2: 2})
FormalityGraph(2, 1, [(2, 1), (2, 0)])
set_pos(new_pos)
Set the positions of vertices in this graph (used for plotting).
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.set_pos({0: (0.0, 0.0), 1: (1.0, 0.0), 2: (0.5, 1.0)})
sage: g.get_pos()
{0: (0.000000000000000, 0.000000000000000),
1: (1.00000000000000, 0.000000000000000),
2: (0.500000000000000, 1.00000000000000)}
show(**options)
Show this graph.
EXAMPLES:
sage: g = FormalityGraph(2, 1, [(2, 0), (2, 1)])
sage: g.show()
5.2 Formality graph basis
Formality graph basis
class gcaops.graph.formality_graph_basis.FormalityGraphBasis
Bases: gcaops.graph.graph_basis.GraphBasis
Basis of a module spanned by formality graphs.
A basis consists of keys (gv,av,e,index,...) where (gv,av,e,index) identifies the isomorphism class of
the graph.
graph_class
alias of gcaops.graph.formality_graph.FormalityGraph
class gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis(positive_differential_order=None,
connected=None,
loops=None)
Bases: gcaops.graph.formality_graph_basis.FormalityGraphBasis
Basis consisting of representatives of isomorphism classes of formality graphs with no automorphisms that
induce an odd permutation on edges.
cardinality(num_ground_vertices, num_aerial_vertices, num_edges)
Return the number of graphs in this basis with the given num_ground_vertices,
num_aerial_vertices and num_edges.
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graph_properties()
Return a dictionary containing the properties of the graphs in this basis.
graph_to_key(graph)
Return a tuple consisting of the key in this basis and the sign factor such that graph equals the sign times
the graph identified by the key.
INPUT:
• graph – a FormalityGraph
OUTPUT:
Either (None, 1) if the input graph is not in the span of the basis, or a tuple consisting of a key and a
sign, where a key is a tuple consisting of the number of ground vertices, the number of aerial vertices, the
number of edges, and the index of the graph in the list.
graphs(num_ground_vertices, num_aerial_vertices, num_edges)
Return the list of graphs in this basis with the given num_ground_vertices, num_aerial_vertices
and num_edges.
key_to_graph(key)
Return a tuple consisting of a FormalityGraph and the sign factor such that the sign times the graph
equals the graph identified by the key.
INPUT:
• key – a key in this basis
OUTPUT:
Either (None, 1) if the input key is not in the basis, or a tuple consisting of a FormalityGraph and a
sign which is always +1.
class gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis_lazy(positive_differential_order=None,
con-
nected=None,
loops=None)
Bases: gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis
Basis consisting of representatives of isomorphism classes of formality graphs with no automorphisms that
induce an odd permutation on edges.
graph_to_key(graph)
Return a tuple consisting of the key in this basis and the sign factor such that graph equals the sign times
the graph identified by the key.
INPUT:
• graph – a FormalityGraph
OUTPUT:
Either (None, 1) if the input graph is not in the span of the basis, or a tuple consisting of a key and a
sign, where a key is a tuple containing the number of ground vertices, the number of aerial vertices, and
the number of edges, followed by all the edges in the graph.
key_to_graph(key)
Return a tuple consisting of a FormalityGraph and the sign factor such that the sign times the graph
equals the graph identified by the key.
INPUT:
• key – a key in this basis
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OUTPUT:
Either (None, 1) if the input key is not in the basis, or a tuple consisting of a FormalityGraph and a
sign which is always +1.
class gcaops.graph.formality_graph_basis.FormalityGraphOperadBasis(positive_differential_order=None,
connected=None,
loops=None)
Bases: gcaops.graph.formality_graph_basis.FormalityGraphBasis
Basis consisting of labeled formality graphs with no automorphisms that induce an odd permutation on edges
graph_properties()
Return a dictionary containing the properties of the graphs in this basis.
graph_to_key(graph)
Return a tuple consisting of the key in this basis and the sign factor such that graph equals the sign times
the graph identified by the key.
INPUT:
• graph – a FormalityGraph
OUTPUT:
Either (None, 1) if the input graph is not in the span of the basis, or a tuple consisting of a key and a
sign, where a key is a tuple consisting of the number of ground vertices, the number of aerial vertices, the
number of edges, the index of the graph in the list, followed by a permutation of vertices.
key_to_graph(key)
Return a tuple consisting of a FormalityGraph and the sign factor such that the sign times the graph
equals the graph identified by the key.
INPUT:
• key – a key in this basis
OUTPUT:
Either (None, 1) if the input key is not in the basis, or a tuple consisting of a FormalityGraph and a
sign.
class gcaops.graph.formality_graph_basis.KontsevichGraphBasis(positive_differential_order=None,
connected=None,
loops=None,
mod_ground_permutations=False,
max_aerial_in_degree=None)
Bases: gcaops.graph.formality_graph_basis.QuantizationGraphBasis
Basis consisting of representatives of isomorphism classes of Kontsevich graphs (built of wedges) with no
automorphisms that induce an odd permutation on edges.
flipping_weight_relations(num_ground_vertices, num_aerial_vertices)
Return a matrix in which each row represents a linear relation between the weights of the graphs in the
basis at the given bi-grading.
The relations are those induced by a single orientation-reversing coordinate change on the upper half-plane,
applied to each factor of the configuration space.
ASSUMPTION:
Assumes num_ground_vertices == 2, and assumes that the weights are real-valued (e.g. defined using
the harmonic propagators).
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class gcaops.graph.formality_graph_basis.LeibnizGraphBasis(positive_differential_order=None,
connected=None, loops=None,
mod_ground_permutations=False,
max_aerial_in_degree=None)
Bases: gcaops.graph.formality_graph_basis.QuantizationGraphBasis
Basis consisting of representatives of isomorphism classes of Leibniz graphs (built of one tripod wedges) with
no automorphisms that induce an odd permutation on edges.
5.3 Formality graph vector
Formality graph vector
class gcaops.graph.formality_graph_vector.FormalityGraphModule
Bases: gcaops.graph.graph_vector.GraphModule
Module spanned by formality graphs.
__call__(arg)
Return the result of converting arg into an element of this module.
element_from_kgs_encoding(kgs_encoding, hbar=1)
Return the linear combination of Kontsevich graphs specified by an encoding, as an element of this module.
INPUT:
• kgs_encoding – a string, containing an encoding of a graph series expansion as used in Buring’s
kontsevich_graph_series-cpp program
• hbar (default: 1) – an element of the base ring, to be used as the graph series expansion parameter
class gcaops.graph.formality_graph_vector.FormalityGraphModule_dict(base_ring,
graph_basis)
Bases: gcaops.graph.formality_graph_vector.FormalityGraphModule, gcaops.graph.
graph_vector_dict.GraphModule_dict
Module spanned by formality graphs (with elements stored as dictionaries).
__init__(base_ring, graph_basis)
Initialize this formality graph module.
INPUT:
• base_ring – a ring, to be used as the ring of coefficients
• graph_basis – a FormalityGraphBasis
class gcaops.graph.formality_graph_vector.FormalityGraphModule_vector(base_ring,
graph_basis, vec-
tor_constructor,
ma-
trix_constructor,
sparse=True)
Bases: gcaops.graph.formality_graph_vector.FormalityGraphModule, gcaops.graph.
graph_vector_vector.GraphModule_vector
Module spanned by formality graphs (with elements stored as dictionaries of vectors).
__init__(base_ring, graph_basis, vector_constructor, matrix_constructor, sparse=True)
Initialize this formality graph module.
INPUT:
5.3. Formality graph vector 625
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• base_ring – a ring, to be used as the ring of coefficients
• graph_basis – a FormalityGraphBasis
• vector_constructor – constructor of (sparse) vectors
• matrix_constructor – constructor of (sparse) matrices
• sparse – (default: True) a boolean, passed along to both constructors as a keyword argument
class gcaops.graph.formality_graph_vector.FormalityGraphVector
Bases: gcaops.graph.graph_vector.GraphVector
Vector representing a linear combination of formality graphs.
attach_to_ground(degrees)
Return the non-aerial graph vector that represents the polydifferential operator which results from evaluat-
ing this graph vector at multi-vectors of the given degrees.
ASSUMPTIONS:
Assumes that this graph vector is aerial.
differential_orders()
Return an iterator over the tuples of in-degrees of ground vertices of graphs in this graph vector.
filter(max_aerial_in_degree=None)
Return the graph vector which is the summand of this graph vector containing exactly those graphs that
pass the filter.
ground_skew_symmetrization()
Return the skew-symmetrization (or anti-symmetrization) of this graph vector with respect to the ground
vertices.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of ground vertices.
ground_symmetrization()
Return the symmetrization of this graph vector with respect to the ground vertices.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of ground vertices.
insertion(position, other, **kwargs)
Return the insertion of other into this graph vector at the vertex position.
is_aerial()
Return True if this graph vector is aerial, and False otherwise.
kgs_encoding()
Return an encoding of this graph vector for use in Buring’s kontsevich_graph_series-cpp program.
ASSUMPTIONS:
Assumes all graphs in this graph vector are built of wedges (i.e. with each aerial vertex having out-degree
two).
nground()
Return the number of ground vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of ground vertices.
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part_of_differential_order(diff_order)
Return the graph vector which is the summand of this graph vector containing only graphs such that the
in-degrees of the ground vertices are diff_order.
set_aerial(is_aerial=True)
Set this graph vector to be aerial if is_aerial is True, respectively not aerial if is_aerial is False.
class gcaops.graph.formality_graph_vector.FormalityGraphVector_dict(parent, vector,
is_aerial=False)
Bases: gcaops.graph.formality_graph_vector.FormalityGraphVector, gcaops.graph.
graph_vector_dict.GraphVector_dict
Vector representing a linear combination of formality graphs (stored as a dictionary).
__init__(parent, vector, is_aerial=False)
Initialize this formality graph vector.
INPUT:
• parent – a FormalityGraphModule
• vector – a dictionary, representing a sparse vector of coefficients with respect to the basis of parent
• is_aerial – (default: False) a boolean, if True then this graph vector will be aerial
filter(max_aerial_in_degree=None)
Return the graph vector which is the summand of this graph vector containing exactly those graphs that
pass the filter.
is_aerial()
Return True if this graph vector is aerial, and False otherwise.
nedges()
Return the number of edges in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of edges.
nground()
Return the number of ground vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of ground vertices.
nvertices()
Return the number of vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of vertices.
set_aerial(is_aerial=True)
Set this graph vector to be aerial if is_aerial is True, respectively not aerial if is_aerial is False.
class gcaops.graph.formality_graph_vector.FormalityGraphVector_vector(parent, vectors,
is_aerial=False)
Bases: gcaops.graph.formality_graph_vector.FormalityGraphVector, gcaops.graph.
graph_vector_vector.GraphVector_vector
Vector representing a linear combination of formality graphs (stored as a dictionary of vectors).
__init__(parent, vectors, is_aerial=False)
Initialize this graph vector.
5.3. Formality graph vector 627
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INPUT:
• parent – a FormalityGraphModule
• vectors – a dictionary, mapping tri-gradings to (sparse) vectors of coefficients with respect to the
basis of parent
• is_aerial – (default: False) a boolean, if True then this graph vector will be aerial
filter(max_aerial_in_degree=None)
Return the graph vector which is the summand of this graph vector containing exactly those graphs that
pass the filter.
is_aerial()
Return True if this graph vector is aerial, and False otherwise.
nedges()
Return the number of edges in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of edges.
nground()
Return the number of ground vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of ground vertices.
nvertices()
Return the number of vertices in each graph in this graph vector.
ASSUMPTIONS:
Assumes all graphs in this graph vector have the same number of vertices.
set_aerial(is_aerial=True)
Set this graph vector to be aerial if is_aerial is True, respectively not aerial if is_aerial is False.
5.4 Formality graph operad
Formality graph operad
gcaops.graph.formality_graph_operad.FormalityGraphOperad(base_ring)
Return the operad of formality graphs over the given base_ring.
class gcaops.graph.formality_graph_operad.FormalityGraphOperad_dict(base_ring)
Bases: gcaops.graph.formality_graph_vector.FormalityGraphModule_dict
Operad of formality graphs (with elements stored as dictionaries).
__init__(base_ring)
Initialize this graph operad.
__repr__()
Return a string representation of this graph operad.
class gcaops.graph.formality_graph_operad.FormalityGraphOperation_dict(parent, vector)
Bases: gcaops.graph.formality_graph_vector.FormalityGraphVector_dict
Element of a FormalityGraphOperad_dict (stored as a dictionary).
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__init__(parent, vector)
Initialize this graph operation.
5.5 Formality graph complex
Formality graph complex
class gcaops.graph.formality_graph_complex.FormalityGraphCochain
Bases: gcaops.graph.graph_complex.GraphCochain, gcaops.graph.formality_graph_vector.
FormalityGraphVector
Cochain of a FormalityGraphComplex_.
bracket(other, **kwargs)
Return the graph Gerstenhaber bracket of this graph cochain with other.
gerstenhaber_bracket(other, **kwargs)
Return the graph Gerstenhaber bracket of this graph cochain with other.
schouten_bracket(other, **kwargs)
Return the graph analogue of the Schouten bracket (or Schouten-Nijenhuis bracket) of this graph cochain
with other.
ASSUMPTIONS:
Assumes that this graph vector and other both are skew-symmetric and have differential order equal to
one on each ground vertex.
class gcaops.graph.formality_graph_complex.FormalityGraphCochain_dict(parent, vector)
Bases: gcaops.graph.formality_graph_complex.FormalityGraphCochain, gcaops.graph.
formality_graph_vector.FormalityGraphVector_dict
Cochain of a FormalityGraphComplex_dict (stored as a dictionary).
__init__(parent, vector)
Initialize this graph cochain.
differential()
Return the Hochschild differential of this graph cochain.
hochschild_differential()
Return the Hochschild differential of this graph cochain.
class gcaops.graph.formality_graph_complex.FormalityGraphCochain_vector(parent, vector)
Bases: gcaops.graph.formality_graph_complex.FormalityGraphCochain, gcaops.graph.
formality_graph_vector.FormalityGraphVector_vector
Cochain of a FormalityGraphComplex_vector (stored as a dictionary of vectors).
__init__(parent, vector)
Initialize this graph cochain.
differential(use_cache=False)
Return the graph differential of this graph cochain.
hochschild_differential(use_cache=False)
Return the graph differential of this graph cochain.
5.5. Formality graph complex 629
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gcaops.graph.formality_graph_complex.FormalityGraphComplex(base_ring, connected=None,
loops=None, implemen-
tation='dict', lazy=False,
vector_constructor=None,
matrix_constructor=None,
sparse=True)
Return the Formality graph complex over base_ring with the given properties.
class gcaops.graph.formality_graph_complex.FormalityGraphComplex_
Bases: gcaops.graph.graph_complex.GraphComplex, gcaops.graph.formality_graph_vector.
FormalityGraphModule
Formality graph complex.
class gcaops.graph.formality_graph_complex.FormalityGraphComplex_dict(base_ring, con-
nected=None,
loops=None,
lazy=False)
Bases: gcaops.graph.formality_graph_complex.FormalityGraphComplex_, gcaops.graph.
formality_graph_vector.FormalityGraphModule_dict
Formality graph complex (with elements stored as dictionaries).
__init__(base_ring, connected=None, loops=None, lazy=False)
Initialize this graph complex.
__repr__()
Return a string representation of this graph complex.
class gcaops.graph.formality_graph_complex.FormalityGraphComplex_vector(base_ring, vec-
tor_constructor,
ma-
trix_constructor,
sparse=True,
con-
nected=None,
loops=None)
Bases: gcaops.graph.formality_graph_complex.FormalityGraphComplex_, gcaops.graph.
formality_graph_vector.FormalityGraphModule_vector
Formality graph complex (with elements stored as dictionaries of vectors).
__init__(base_ring, vector_constructor, matrix_constructor, sparse=True, connected=None,
loops=None)
Initialize this graph complex.
__repr__()
Return a string representation of this graph complex.
cohomology_basis(ground_vertices, aerial_vertices, edges)
Return a basis of the cohomology in the given tri-grading (ground_vertices, aerial_vertices,
edges).
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5.6 Formality graph operator
Formality graph operator
class gcaops.graph.formality_graph_operator.FormalityGraphOperator(domain, codomain,
graph_vector)
Bases: object
A homogeneous n-ary multi-linear operator on a SuperfunctionAlgebra with values in a
PolydifferentialOperatorAlgebra, defined by a FormalityGraphVector.
__call__(*args)
Return the evaluation of this operator at args.
__init__(domain, codomain, graph_vector)
Initialize this operator.
__repr__()
Return a string representation of this operator.
codomain()
Return the codomain of this operator.
domain()
Return the domain of this operator.
value_at_copies_of(arg)
Return the evaluation of this operator at copies of arg.
class gcaops.graph.formality_graph_operator.FormalityGraphSymmetricOperator
Bases: object
A homogeneous n-ary multi-linear symmetric operator on a SuperfunctionAlgebra with values in a
PolydifferentialOperatorAlgebra, defined by a FormalityGraphVector.
__call__(*args)
Return the evaluation of this operator at args.
gcaops.graph.formality_graph_operator.formality_graph_operator(graph_vector, domain,
codomain)
Factory.
5.6. Formality graph operator 631
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632 Chapter 5. Formality graphs
CHAPTER
SIX
GRAPH CACHE
6.1 Graph file view
Graph file view
class gcaops.graph.graph_file.DirectedGraphFileView(filename, num_vertices, num_edges)
Bases: gcaops.graph.graph_file.GraphFileView
Directed graph database file view
class gcaops.graph.graph_file.FormalityGraphFileView(filename, num_ground_vertices,
num_aerial_vertices, num_edges)
Bases: gcaops.graph.graph_file.GraphFileView
Formality graph database file view
__init__(filename, num_ground_vertices, num_aerial_vertices, num_edges)
Initialize this formality graph database file view.
INPUT:
• filename – a string, the path to an SQLite database file (the file will be created if it does not yet exist)
• num_ground_vertices – a natural number, the number of ground vertices in each graph
• num_aerial_vertices – a natural number, the number of aerial vertices in each graph
• num_edges – a natural number, the number of edges in each graph
class gcaops.graph.graph_file.GraphFileView(filename, num_vertices, num_edges)
Bases: abc.ABC
Graph database file view
__getitem__(index)
Return the graph at the given index in this database file.
__getstate__()
Return the state of this object as a dictionary (used for pickling).
__init__(filename, num_vertices, num_edges)
Initialize this graph database file view.
INPUT:
• filename – a string, the path to an SQLite database file (the file will be created if it does not yet exist)
• num_vertices – a natural number, the number of vertices in each graph
• num_edges – a natural number, the number of edges in each graph
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__iter__()
Return an iterator over the graphs in this database file.
__len__()
Return the number of graphs in this database file.
__setstate__(state_dict)
Set the state of this object from a dictionary (used for pickling).
append(g)
Insert the given graph into this database file.
commit()
Commit any changes to this database file.
index(g)
Return the index of the given graph in this database file.
class gcaops.graph.graph_file.UndirectedGraphFileView(filename, num_vertices, num_edges)
Bases: gcaops.graph.graph_file.GraphFileView
Undirected graph database file view
class gcaops.graph.graph_file.UndirectedToDirectedGraphFileView(filename)
Bases: object
Undirected to directed graph database file view
__getstate__()
Return the state of this object as a dictionary (used for pickling).
__init__(filename)
Initialize this “undirected to directed graph” database file view.
INPUT:
• filename – a string, the path to an SQLite database file (the file will be created if it does not yet exist)
__iter__()
Return an iterator over the rows in the “undirected to directed graph” database file.
__len__()
Return the number of rows in the “undirected to directed graph” database file.
__setstate__(state_dict)
Set the state of this object from a dictionary (used for pickling).
append(row)
Insert a row into the “undirected to directed graph” database file.
commit()
Commit any changes to this database file.
undirected_to_directed_coeffs(undirected_graph_idx)
Return an iterator over the (directed_graph_idx, coefficient) tuples related to the undirected
graph with the given index.
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6.2 Graph cache
Graph cache
class gcaops.graph.graph_cache.DirectedGraphCache(undirected_graph_cache)
Bases: gcaops.graph.graph_cache.GraphCache
Directed graph cache
__init__(undirected_graph_cache)
Initialize this directed graph cache.
INPUT:
• undirected_graph_cache – an UndirectedGraphCache
canonicalize_graph(graph)
Return a tuple consisting the normal form of the graph and the sign factor relating the input graph to the
normal form.
file_view
alias of gcaops.graph.graph_file.DirectedGraphFileView
graphs(bi_grading, connected=False, biconnected=False, min_degree=0, loops=True,
has_odd_automorphism=True)
Return a view (a list or a GraphFileView) of the graphs in the cache with the given options.
class gcaops.graph.graph_cache.FormalityGraphCache
Bases: gcaops.graph.graph_cache.GraphCache
Formality graph cache
canonicalize_graph(graph)
Return a tuple consisting the normal form of the graph, an isomorphism from the normal form to the input
graph, and the sign of the induced permutation on edges.
file_view
alias of gcaops.graph.graph_file.FormalityGraphFileView
graphs(tri_grading, connected=None, max_out_degree=None, num_verts_of_max_out_degree=None,
sorted_out_degrees=None, max_aerial_in_degree=None, loops=None,
prime=None, has_odd_automorphism=None, positive_differential_order=None,
mod_ground_permutations=False)
Return a view (a list or a GraphFileView) of the graphs in the cache with the given options.
class gcaops.graph.graph_cache.GraphCache
Bases: abc.ABC
Graph cache
canonicalize_graph(graph)
Return a tuple consisting the normal form of the graph, followed by data relating the input graph to the
normal form (e.g. a sign factor).
graphs(grading, **options)
Return a view (e.g. a list or a GraphFileView) of the graphs in the cache with the given options.
class gcaops.graph.graph_cache.UndirectedGraphCache
Bases: gcaops.graph.graph_cache.GraphCache
Undirected graph cache
6.2. Graph cache 635
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canonicalize_graph(graph)
Return a tuple consisting the normal form of the graph and the sign factor relating the input graph to the
normal form.
file_view
alias of gcaops.graph.graph_file.UndirectedGraphFileView
graphs(bi_grading, connected=False, biconnected=False, min_degree=0,
has_odd_automorphism=True)
Return a view (a list or a GraphFileView) of the graphs in the cache with the given options.
636 Chapter 6. Graph cache
PYTHON MODULE INDEX
g
gcaops.algebra.differential_polynomial_ring, 588
gcaops.algebra.differential_polynomial_solver, 589
gcaops.algebra.polydifferential_operator, 584
gcaops.algebra.superfunction_algebra, 579
gcaops.algebra.superfunction_algebra_operation, 582
gcaops.algebra.tensor_product, 587
gcaops.graph.directed_graph, 609
gcaops.graph.directed_graph_basis, 610
gcaops.graph.directed_graph_complex, 614
gcaops.graph.directed_graph_operad, 614
gcaops.graph.directed_graph_vector, 612
gcaops.graph.formality_graph, 617
gcaops.graph.formality_graph_basis, 622
gcaops.graph.formality_graph_complex, 629
gcaops.graph.formality_graph_operad, 628
gcaops.graph.formality_graph_operator, 631
gcaops.graph.formality_graph_vector, 625
gcaops.graph.graph_basis, 591
gcaops.graph.graph_cache, 635
gcaops.graph.graph_complex, 593
gcaops.graph.graph_file, 633
gcaops.graph.graph_vector, 591
gcaops.graph.graph_vector_dict, 595
gcaops.graph.graph_vector_vector, 593
gcaops.graph.undirected_graph, 599
gcaops.graph.undirected_graph_basis, 601
gcaops.graph.undirected_graph_complex, 605
gcaops.graph.undirected_graph_operad, 605
gcaops.graph.undirected_graph_vector, 603
637
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638 Python Module Index
INDEX
Symbols
__add__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 584
__add__() (gcaops.algebra.superfunction_algebra.Superfunction method), 579
__add__() (gcaops.graph.graph_vector.GraphVector method), 592
__add__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__add__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__call__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 586
__call__() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 581
__call__() (gcaops.graph.directed_graph_vector.DirectedGraphModule method), 612
__call__() (gcaops.graph.formality_graph_operator.FormalityGraphOperator method), 631
__call__() (gcaops.graph.formality_graph_operator.FormalityGraphSymmetricOperator method), 631
__call__() (gcaops.graph.formality_graph_vector.FormalityGraphModule method), 625
__call__() (gcaops.graph.graph_vector.GraphModule method), 591
__call__() (gcaops.graph.graph_vector_dict.GraphModule_dict method), 595
__call__() (gcaops.graph.graph_vector_vector.GraphModule_vector method), 593
__eq__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 584
__eq__() (gcaops.algebra.superfunction_algebra.Superfunction method), 579
__eq__() (gcaops.graph.graph_vector.GraphVector method), 592
__eq__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__eq__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__eq__() (gcaops.graph.undirected_graph.UndirectedGraph method), 599
__getitem__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 584
__getitem__() (gcaops.algebra.superfunction_algebra.Superfunction method), 579
__getitem__() (gcaops.graph.graph_file.GraphFileView method), 633
__getstate__() (gcaops.graph.graph_file.GraphFileView method), 633
__getstate__() (gcaops.graph.graph_file.UndirectedToDirectedGraphFileView method), 634
__init__() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomialRing method), 588
__init__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 584
__init__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 586
__init__() (gcaops.algebra.superfunction_algebra.Superfunction method), 579
__init__() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 581
__init__() (gcaops.graph.directed_graph_complex.DirectedGraphCochain_dict method), 615
__init__() (gcaops.graph.directed_graph_complex.DirectedGraphCochain_vector method), 615
__init__() (gcaops.graph.directed_graph_complex.DirectedGraphComplex_dict method), 615
__init__() (gcaops.graph.directed_graph_complex.DirectedGraphComplex_vector method), 616
__init__() (gcaops.graph.directed_graph_operad.DirectedGraphOperad_dict method), 614
__init__() (gcaops.graph.directed_graph_operad.DirectedGraphOperation_dict method), 614
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__init__() (gcaops.graph.directed_graph_vector.DirectedGraphModule_dict method), 612
__init__() (gcaops.graph.directed_graph_vector.DirectedGraphModule_vector method), 612
__init__() (gcaops.graph.directed_graph_vector.DirectedGraphVector_dict method), 613
__init__() (gcaops.graph.directed_graph_vector.DirectedGraphVector_vector method), 613
__init__() (gcaops.graph.formality_graph_complex.FormalityGraphCochain_dict method), 629
__init__() (gcaops.graph.formality_graph_complex.FormalityGraphCochain_vector method), 629
__init__() (gcaops.graph.formality_graph_complex.FormalityGraphComplex_dict method), 630
__init__() (gcaops.graph.formality_graph_complex.FormalityGraphComplex_vector method), 630
__init__() (gcaops.graph.formality_graph_operad.FormalityGraphOperad_dict method), 628
__init__() (gcaops.graph.formality_graph_operad.FormalityGraphOperation_dict method), 628
__init__() (gcaops.graph.formality_graph_operator.FormalityGraphOperator method), 631
__init__() (gcaops.graph.formality_graph_vector.FormalityGraphModule_dict method), 625
__init__() (gcaops.graph.formality_graph_vector.FormalityGraphModule_vector method), 625
__init__() (gcaops.graph.formality_graph_vector.FormalityGraphVector_dict method), 627
__init__() (gcaops.graph.formality_graph_vector.FormalityGraphVector_vector method), 627
__init__() (gcaops.graph.graph_cache.DirectedGraphCache method), 635
__init__() (gcaops.graph.graph_file.FormalityGraphFileView method), 633
__init__() (gcaops.graph.graph_file.GraphFileView method), 633
__init__() (gcaops.graph.graph_file.UndirectedToDirectedGraphFileView method), 634
__init__() (gcaops.graph.graph_vector_dict.GraphModule_dict method), 595
__init__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__init__() (gcaops.graph.graph_vector_vector.GraphModule_vector method), 593
__init__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__init__() (gcaops.graph.undirected_graph.UndirectedGraph method), 599
__init__() (gcaops.graph.undirected_graph_complex.UndirectedGraphCochain_dict method), 605
__init__() (gcaops.graph.undirected_graph_complex.UndirectedGraphCochain_vector method), 605
__init__() (gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_dict method), 606
__init__() (gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_vector method), 606
__init__() (gcaops.graph.undirected_graph_operad.UndirectedGraphOperad_dict method), 605
__init__() (gcaops.graph.undirected_graph_operad.UndirectedGraphOperation_dict method), 605
__init__() (gcaops.graph.undirected_graph_vector.UndirectedGraphModule_dict method), 603
__init__() (gcaops.graph.undirected_graph_vector.UndirectedGraphModule_vector method), 603
__init__() (gcaops.graph.undirected_graph_vector.UndirectedGraphVector_dict method), 604
__init__() (gcaops.graph.undirected_graph_vector.UndirectedGraphVector_vector method), 604
__iter__() (gcaops.graph.graph_file.GraphFileView method), 633
__iter__() (gcaops.graph.graph_file.UndirectedToDirectedGraphFileView method), 634
__iter__() (gcaops.graph.graph_vector.GraphVector method), 592
__iter__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__iter__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__len__() (gcaops.graph.graph_file.GraphFileView method), 634
__len__() (gcaops.graph.graph_file.UndirectedToDirectedGraphFileView method), 634
__len__() (gcaops.graph.graph_vector.GraphVector method), 592
__len__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__len__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__len__() (gcaops.graph.undirected_graph.UndirectedGraph method), 599
__mul__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 584
__mul__() (gcaops.algebra.superfunction_algebra.Superfunction method), 579
__mul__() (gcaops.graph.graph_vector.GraphVector method), 592
__mul__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__mul__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
640 Index
Documentation of gcaops, Release 1
__neg__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 584
__neg__() (gcaops.algebra.superfunction_algebra.Superfunction method), 579
__neg__() (gcaops.graph.graph_vector.GraphVector method), 592
__neg__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__neg__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__pos__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 584
__pos__() (gcaops.algebra.superfunction_algebra.Superfunction method), 579
__pos__() (gcaops.graph.graph_vector.GraphVector method), 592
__pos__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__pos__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__pow__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 584
__pow__() (gcaops.algebra.superfunction_algebra.Superfunction method), 579
__radd__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
__radd__() (gcaops.algebra.superfunction_algebra.Superfunction method), 579
__radd__() (gcaops.graph.graph_vector.GraphVector method), 592
__radd__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__radd__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__repr__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
__repr__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 586
__repr__() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
__repr__() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 581
__repr__() (gcaops.graph.directed_graph_complex.DirectedGraphComplex_dict method), 615
__repr__() (gcaops.graph.directed_graph_complex.DirectedGraphComplex_vector method), 616
__repr__() (gcaops.graph.directed_graph_operad.DirectedGraphOperad_dict method), 614
__repr__() (gcaops.graph.formality_graph_complex.FormalityGraphComplex_dict method), 630
__repr__() (gcaops.graph.formality_graph_complex.FormalityGraphComplex_vector method), 630
__repr__() (gcaops.graph.formality_graph_operad.FormalityGraphOperad_dict method), 628
__repr__() (gcaops.graph.formality_graph_operator.FormalityGraphOperator method), 631
__repr__() (gcaops.graph.graph_vector.GraphModule method), 591
__repr__() (gcaops.graph.graph_vector.GraphVector method), 592
__repr__() (gcaops.graph.graph_vector_dict.GraphModule_dict method), 595
__repr__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__repr__() (gcaops.graph.graph_vector_vector.GraphModule_vector method), 594
__repr__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__repr__() (gcaops.graph.undirected_graph.UndirectedGraph method), 600
__repr__() (gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_dict method), 606
__repr__() (gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_vector method), 607
__repr__() (gcaops.graph.undirected_graph_operad.UndirectedGraphOperad_dict method), 605
__rmul__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
__rmul__() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
__rmul__() (gcaops.graph.graph_vector.GraphVector method), 592
__rmul__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__rmul__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 594
__rsub__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
__rsub__() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
__rsub__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__rsub__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 595
__setitem__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
__setitem__() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
__setstate__() (gcaops.graph.graph_file.GraphFileView method), 634
Index 641
Documentation of gcaops, Release 1
__setstate__() (gcaops.graph.graph_file.UndirectedToDirectedGraphFileView method), 634
__sub__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
__sub__() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
__sub__() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
__sub__() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 595
__truediv__() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
__truediv__() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
A
aerial_product() (gcaops.graph.formality_graph.FormalityGraph method), 617
append() (gcaops.graph.graph_file.GraphFileView method), 634
append() (gcaops.graph.graph_file.UndirectedToDirectedGraphFileView method), 634
arity() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
attach_to_ground() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
automorphism_group() (gcaops.graph.formality_graph.FormalityGraph method), 617
B
base_ring() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 586
base_ring() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 581
base_ring() (gcaops.graph.graph_vector.GraphModule method), 591
base_ring() (gcaops.graph.graph_vector_dict.GraphModule_dict method), 595
base_ring() (gcaops.graph.graph_vector_vector.GraphModule_vector method), 594
base_variables() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomialRing method), 589
basis() (gcaops.graph.graph_vector.GraphModule method), 592
basis() (gcaops.graph.graph_vector_dict.GraphModule_dict method), 595
basis() (gcaops.graph.graph_vector_vector.GraphModule_vector method), 594
bracket() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
bracket() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
bracket() (gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricOperation method),
583
bracket() (gcaops.graph.directed_graph_complex.DirectedGraphCochain method), 614
bracket() (gcaops.graph.formality_graph_complex.FormalityGraphCochain method), 629
bracket() (gcaops.graph.graph_complex.GraphCochain method), 593
bracket() (gcaops.graph.undirected_graph_complex.UndirectedGraphCochain method), 605
C
canonicalize_edges() (gcaops.graph.directed_graph.DirectedGraph method), 609
canonicalize_edges() (gcaops.graph.formality_graph.FormalityGraph method), 617
canonicalize_edges() (gcaops.graph.undirected_graph.UndirectedGraph method), 600
canonicalize_graph() (gcaops.graph.graph_cache.DirectedGraphCache method), 635
canonicalize_graph() (gcaops.graph.graph_cache.FormalityGraphCache method), 635
canonicalize_graph() (gcaops.graph.graph_cache.GraphCache method), 635
canonicalize_graph() (gcaops.graph.graph_cache.UndirectedGraphCache method), 635
cardinality() (gcaops.graph.directed_graph_basis.DirectedGraphComplexBasis method), 611
cardinality() (gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis method), 622
cardinality() (gcaops.graph.undirected_graph_basis.UndirectedGraphComplexBasis method), 601
codomain() (gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraOperation method), 583
codomain() (gcaops.graph.formality_graph_operator.FormalityGraphOperator method), 631
coefficient() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
coefficient() (gcaops.graph.graph_vector.GraphVector method), 592
642 Index
Documentation of gcaops, Release 1
cohomology_basis() (gcaops.graph.directed_graph_complex.DirectedGraphComplex_vector method), 616
cohomology_basis() (gcaops.graph.formality_graph_complex.FormalityGraphComplex_vector method), 630
cohomology_basis() (gcaops.graph.undirected_graph_complex.UndirectedGraphComplex_vector method), 607
commit() (gcaops.graph.graph_file.GraphFileView method), 634
commit() (gcaops.graph.graph_file.UndirectedToDirectedGraphFileView method), 634
coordinate() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 586
coordinates() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 587
copy() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
copy() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
copy() (gcaops.graph.graph_vector.GraphVector method), 592
copy() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
copy() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 595
D
degree() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
degree() (gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraDirectedGraphOperation
method), 582
degree() (gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraOperation method), 583
degree() (gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSchoutenBracket method), 583
degree() (gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraSymmetricBracketOperation
method), 583
degree() (gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraUndirectedGraphOperation
method), 584
degrees() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
derivative() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomial method), 588
derivative() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 587
derivative() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
derivatives() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 587
diff() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomial method), 588
diff() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
differential() (gcaops.graph.directed_graph_complex.DirectedGraphCochain_dict method), 615
differential() (gcaops.graph.directed_graph_complex.DirectedGraphCochain_vector method), 615
differential() (gcaops.graph.formality_graph_complex.FormalityGraphCochain_dict method), 629
differential() (gcaops.graph.formality_graph_complex.FormalityGraphCochain_vector method), 629
differential() (gcaops.graph.graph_complex.GraphCochain method), 593
differential() (gcaops.graph.undirected_graph_complex.UndirectedGraphCochain_dict method), 605
differential() (gcaops.graph.undirected_graph_complex.UndirectedGraphCochain_vector method), 605
differential_orders() (gcaops.graph.formality_graph.FormalityGraph method), 617
differential_orders() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
DifferentialPolynomial (class in gcaops.algebra.differential_polynomial_ring), 588
DifferentialPolynomialRing (class in gcaops.algebra.differential_polynomial_ring), 588
dimension() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 581
DirectedGraph (class in gcaops.graph.directed_graph), 609
DirectedGraphBasis (class in gcaops.graph.directed_graph_basis), 610
DirectedGraphCache (class in gcaops.graph.graph_cache), 635
DirectedGraphCochain (class in gcaops.graph.directed_graph_complex), 614
DirectedGraphCochain_dict (class in gcaops.graph.directed_graph_complex), 614
DirectedGraphCochain_vector (class in gcaops.graph.directed_graph_complex), 615
DirectedGraphComplex() (in module gcaops.graph.directed_graph_complex), 615
DirectedGraphComplex_ (class in gcaops.graph.directed_graph_complex), 615
Index 643
Documentation of gcaops, Release 1
DirectedGraphComplex_dict (class in gcaops.graph.directed_graph_complex), 615
DirectedGraphComplex_vector (class in gcaops.graph.directed_graph_complex), 615
DirectedGraphComplexBasis (class in gcaops.graph.directed_graph_basis), 611
DirectedGraphFileView (class in gcaops.graph.graph_file), 633
DirectedGraphModule (class in gcaops.graph.directed_graph_vector), 612
DirectedGraphModule_dict (class in gcaops.graph.directed_graph_vector), 612
DirectedGraphModule_vector (class in gcaops.graph.directed_graph_vector), 612
DirectedGraphOperad() (in module gcaops.graph.directed_graph_operad), 614
DirectedGraphOperad_dict (class in gcaops.graph.directed_graph_operad), 614
DirectedGraphOperadBasis (class in gcaops.graph.directed_graph_basis), 611
DirectedGraphOperation_dict (class in gcaops.graph.directed_graph_operad), 614
DirectedGraphVector (class in gcaops.graph.directed_graph_vector), 613
DirectedGraphVector_dict (class in gcaops.graph.directed_graph_vector), 613
DirectedGraphVector_vector (class in gcaops.graph.directed_graph_vector), 613
domain() (gcaops.algebra.superfunction_algebra_operation.SuperfunctionAlgebraOperation method), 583
domain() (gcaops.graph.formality_graph_operator.FormalityGraphOperator method), 631
E
edge_contraction_graph() (gcaops.graph.formality_graph.FormalityGraph method), 618
edges() (gcaops.graph.directed_graph.DirectedGraph method), 609
edges() (gcaops.graph.formality_graph.FormalityGraph method), 618
edges() (gcaops.graph.undirected_graph.UndirectedGraph method), 600
edges_in_air() (gcaops.graph.formality_graph.FormalityGraph method), 618
element_class (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomialRing attribute), 589
element_from_kgs_encoding() (gcaops.graph.formality_graph_vector.FormalityGraphModule method), 625
even_coordinate() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
even_coordinates() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
F
factor() (gcaops.algebra.tensor_product.TensorProduct method), 587
factors() (gcaops.algebra.tensor_product.TensorProduct method), 587
fibre_degrees() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomial method), 588
fibre_variables() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomialRing method), 589
file_view (gcaops.graph.graph_cache.DirectedGraphCache attribute), 635
file_view (gcaops.graph.graph_cache.FormalityGraphCache attribute), 635
file_view (gcaops.graph.graph_cache.UndirectedGraphCache attribute), 636
filter() (gcaops.graph.directed_graph_vector.DirectedGraphVector method), 613
filter() (gcaops.graph.directed_graph_vector.DirectedGraphVector_dict method), 613
filter() (gcaops.graph.directed_graph_vector.DirectedGraphVector_vector method), 613
filter() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
filter() (gcaops.graph.formality_graph_vector.FormalityGraphVector_dict method), 627
filter() (gcaops.graph.formality_graph_vector.FormalityGraphVector_vector method), 628
flipping_weight_relations() (gcaops.graph.formality_graph_basis.KontsevichGraphBasis method), 624
formality_graph_operator() (in module gcaops.graph.formality_graph_operator), 631
FormalityGraph (class in gcaops.graph.formality_graph), 617
FormalityGraphBasis (class in gcaops.graph.formality_graph_basis), 622
FormalityGraphCache (class in gcaops.graph.graph_cache), 635
FormalityGraphCochain (class in gcaops.graph.formality_graph_complex), 629
FormalityGraphCochain_dict (class in gcaops.graph.formality_graph_complex), 629
FormalityGraphCochain_vector (class in gcaops.graph.formality_graph_complex), 629
644 Index
Documentation of gcaops, Release 1
FormalityGraphComplex() (in module gcaops.graph.formality_graph_complex), 629
FormalityGraphComplex_ (class in gcaops.graph.formality_graph_complex), 630
FormalityGraphComplex_dict (class in gcaops.graph.formality_graph_complex), 630
FormalityGraphComplex_vector (class in gcaops.graph.formality_graph_complex), 630
FormalityGraphComplexBasis (class in gcaops.graph.formality_graph_basis), 622
FormalityGraphComplexBasis_lazy (class in gcaops.graph.formality_graph_basis), 623
FormalityGraphFileView (class in gcaops.graph.graph_file), 633
FormalityGraphModule (class in gcaops.graph.formality_graph_vector), 625
FormalityGraphModule_dict (class in gcaops.graph.formality_graph_vector), 625
FormalityGraphModule_vector (class in gcaops.graph.formality_graph_vector), 625
FormalityGraphOperad() (in module gcaops.graph.formality_graph_operad), 628
FormalityGraphOperad_dict (class in gcaops.graph.formality_graph_operad), 628
FormalityGraphOperadBasis (class in gcaops.graph.formality_graph_basis), 624
FormalityGraphOperation_dict (class in gcaops.graph.formality_graph_operad), 628
FormalityGraphOperator (class in gcaops.graph.formality_graph_operator), 631
FormalityGraphSymmetricOperator (class in gcaops.graph.formality_graph_operator), 631
FormalityGraphVector (class in gcaops.graph.formality_graph_vector), 626
FormalityGraphVector_dict (class in gcaops.graph.formality_graph_vector), 627
FormalityGraphVector_vector (class in gcaops.graph.formality_graph_vector), 627
from_kgs_encoding() (gcaops.graph.formality_graph.FormalityGraph static method), 618
from_kontsevint_encoding() (gcaops.graph.formality_graph.FormalityGraph static method), 618
G
gcaops.algebra.differential_polynomial_ring
module, 588
gcaops.algebra.differential_polynomial_solver
module, 589
gcaops.algebra.polydifferential_operator
module, 584
gcaops.algebra.superfunction_algebra
module, 579
gcaops.algebra.superfunction_algebra_operation
module, 582
gcaops.algebra.tensor_product
module, 587
gcaops.graph.directed_graph
module, 609
gcaops.graph.directed_graph_basis
module, 610
gcaops.graph.directed_graph_complex
module, 614
gcaops.graph.directed_graph_operad
module, 614
gcaops.graph.directed_graph_vector
module, 612
gcaops.graph.formality_graph
module, 617
gcaops.graph.formality_graph_basis
module, 622
gcaops.graph.formality_graph_complex
Index 645
Documentation of gcaops, Release 1
module, 629
gcaops.graph.formality_graph_operad
module, 628
gcaops.graph.formality_graph_operator
module, 631
gcaops.graph.formality_graph_vector
module, 625
gcaops.graph.graph_basis
module, 591
gcaops.graph.graph_cache
module, 635
gcaops.graph.graph_complex
module, 593
gcaops.graph.graph_file
module, 633
gcaops.graph.graph_vector
module, 591
gcaops.graph.graph_vector_dict
module, 595
gcaops.graph.graph_vector_vector
module, 593
gcaops.graph.undirected_graph
module, 599
gcaops.graph.undirected_graph_basis
module, 601
gcaops.graph.undirected_graph_complex
module, 605
gcaops.graph.undirected_graph_operad
module, 605
gcaops.graph.undirected_graph_vector
module, 603
gen() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 587
gen() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
gens() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 587
gens() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
gerstenhaber_bracket() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
gerstenhaber_bracket() (gcaops.graph.formality_graph_complex.FormalityGraphCochain method), 629
get_pos() (gcaops.graph.directed_graph.DirectedGraph method), 609
get_pos() (gcaops.graph.formality_graph.FormalityGraph method), 619
get_pos() (gcaops.graph.undirected_graph.UndirectedGraph method), 600
graded_symmetrization() (gcaops.algebra.tensor_product.TensorProductElement method), 587
gradings() (gcaops.graph.graph_vector.GraphVector method), 592
gradings() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
gradings() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 595
graph_class (gcaops.graph.directed_graph_basis.DirectedGraphBasis attribute), 611
graph_class (gcaops.graph.formality_graph_basis.FormalityGraphBasis attribute), 622
graph_class (gcaops.graph.undirected_graph_basis.UndirectedGraphBasis attribute), 601
graph_operation() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
graph_properties() (gcaops.graph.directed_graph_basis.DirectedGraphComplexBasis method), 611
graph_properties() (gcaops.graph.directed_graph_basis.DirectedGraphOperadBasis method), 611
646 Index
Documentation of gcaops, Release 1
graph_properties() (gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis method), 622
graph_properties() (gcaops.graph.formality_graph_basis.FormalityGraphOperadBasis method), 624
graph_properties() (gcaops.graph.graph_basis.GraphBasis method), 591
graph_properties() (gcaops.graph.undirected_graph_basis.UndirectedGraphComplexBasis method), 601
graph_properties() (gcaops.graph.undirected_graph_basis.UndirectedGraphOperadBasis method), 602
graph_to_key() (gcaops.graph.directed_graph_basis.DirectedGraphComplexBasis method), 611
graph_to_key() (gcaops.graph.directed_graph_basis.DirectedGraphOperadBasis method), 611
graph_to_key() (gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis method), 623
graph_to_key() (gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis_lazy method), 623
graph_to_key() (gcaops.graph.formality_graph_basis.FormalityGraphOperadBasis method), 624
graph_to_key() (gcaops.graph.graph_basis.GraphBasis method), 591
graph_to_key() (gcaops.graph.undirected_graph_basis.UndirectedGraphComplexBasis method), 602
graph_to_key() (gcaops.graph.undirected_graph_basis.UndirectedGraphOperadBasis method), 602
GraphBasis (class in gcaops.graph.graph_basis), 591
GraphCache (class in gcaops.graph.graph_cache), 635
GraphCochain (class in gcaops.graph.graph_complex), 593
GraphComplex (class in gcaops.graph.graph_complex), 593
GraphFileView (class in gcaops.graph.graph_file), 633
GraphModule (class in gcaops.graph.graph_vector), 591
GraphModule_dict (class in gcaops.graph.graph_vector_dict), 595
GraphModule_vector (class in gcaops.graph.graph_vector_vector), 593
graphs() (gcaops.graph.directed_graph_basis.DirectedGraphComplexBasis method), 611
graphs() (gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis method), 623
graphs() (gcaops.graph.graph_cache.DirectedGraphCache method), 635
graphs() (gcaops.graph.graph_cache.FormalityGraphCache method), 635
graphs() (gcaops.graph.graph_cache.GraphCache method), 635
graphs() (gcaops.graph.graph_cache.UndirectedGraphCache method), 636
graphs() (gcaops.graph.undirected_graph_basis.UndirectedGraphComplexBasis method), 602
GraphVector (class in gcaops.graph.graph_vector), 592
GraphVector_dict (class in gcaops.graph.graph_vector_dict), 595
GraphVector_vector (class in gcaops.graph.graph_vector_vector), 594
ground_relabeled() (gcaops.graph.formality_graph.FormalityGraph method), 619
ground_skew_symmetrization() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
ground_symmetrization() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
H
has_eye_on_ground() (gcaops.graph.formality_graph.FormalityGraph method), 619
has_loops() (gcaops.graph.formality_graph.FormalityGraph method), 619
has_multiple_edges() (gcaops.graph.formality_graph.FormalityGraph method), 619
has_odd_automorphism() (gcaops.graph.formality_graph.FormalityGraph method), 620
hochschild_differential() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
hochschild_differential() (gcaops.graph.formality_graph_complex.FormalityGraphCochain_dict method),
629
hochschild_differential() (gcaops.graph.formality_graph_complex.FormalityGraphCochain_vector method),
629
homogeneous_monomials() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomialRing method),
589
homogeneous_part() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
homogeneous_part() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
homogeneous_part() (gcaops.graph.graph_vector.GraphVector method), 592
Index 647
Documentation of gcaops, Release 1
homogeneous_part() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
homogeneous_part() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 595
I
identity_operator() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 587
in_degrees() (gcaops.graph.directed_graph.DirectedGraph method), 609
in_degrees() (gcaops.graph.formality_graph.FormalityGraph method), 620
index() (gcaops.graph.graph_file.GraphFileView method), 634
indices() (gcaops.algebra.superfunction_algebra.Superfunction method), 580
insertion() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
insertion() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
insertion() (gcaops.graph.graph_vector.GraphVector method), 592
insertion() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
insertion() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 595
is_aerial() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
is_aerial() (gcaops.graph.formality_graph_vector.FormalityGraphVector_dict method), 627
is_aerial() (gcaops.graph.formality_graph_vector.FormalityGraphVector_vector method), 628
is_coboundary() (gcaops.graph.directed_graph_complex.DirectedGraphCochain_vector method), 615
is_coboundary() (gcaops.graph.undirected_graph_complex.UndirectedGraphCochain_vector method), 606
is_zero() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
is_zero() (gcaops.algebra.superfunction_algebra.Superfunction method), 581
K
key_to_graph() (gcaops.graph.directed_graph_basis.DirectedGraphComplexBasis method), 611
key_to_graph() (gcaops.graph.directed_graph_basis.DirectedGraphOperadBasis method), 612
key_to_graph() (gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis method), 623
key_to_graph() (gcaops.graph.formality_graph_basis.FormalityGraphComplexBasis_lazy method), 623
key_to_graph() (gcaops.graph.formality_graph_basis.FormalityGraphOperadBasis method), 624
key_to_graph() (gcaops.graph.graph_basis.GraphBasis method), 591
key_to_graph() (gcaops.graph.undirected_graph_basis.UndirectedGraphComplexBasis method), 602
key_to_graph() (gcaops.graph.undirected_graph_basis.UndirectedGraphOperadBasis method), 602
kgs_encoding() (gcaops.graph.formality_graph.FormalityGraph method), 620
kgs_encoding() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
KontsevichGraphBasis (class in gcaops.graph.formality_graph_basis), 624
kontsevint_encoding() (gcaops.graph.formality_graph.FormalityGraph method), 620
L
LeibnizGraphBasis (class in gcaops.graph.formality_graph_basis), 624
M
map_coefficients() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 585
map_coefficients() (gcaops.algebra.superfunction_algebra.Superfunction method), 581
map_coefficients() (gcaops.graph.graph_vector.GraphVector method), 592
map_coefficients() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
map_coefficients() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 595
map_graphs() (gcaops.graph.graph_vector.GraphVector method), 592
module
gcaops.algebra.differential_polynomial_ring, 588
gcaops.algebra.differential_polynomial_solver, 589
gcaops.algebra.polydifferential_operator, 584
648 Index
Documentation of gcaops, Release 1
gcaops.algebra.superfunction_algebra, 579
gcaops.algebra.superfunction_algebra_operation, 582
gcaops.algebra.tensor_product, 587
gcaops.graph.directed_graph, 609
gcaops.graph.directed_graph_basis, 610
gcaops.graph.directed_graph_complex, 614
gcaops.graph.directed_graph_operad, 614
gcaops.graph.directed_graph_vector, 612
gcaops.graph.formality_graph, 617
gcaops.graph.formality_graph_basis, 622
gcaops.graph.formality_graph_complex, 629
gcaops.graph.formality_graph_operad, 628
gcaops.graph.formality_graph_operator, 631
gcaops.graph.formality_graph_vector, 625
gcaops.graph.graph_basis, 591
gcaops.graph.graph_cache, 635
gcaops.graph.graph_complex, 593
gcaops.graph.graph_file, 633
gcaops.graph.graph_vector, 591
gcaops.graph.graph_vector_dict, 595
gcaops.graph.graph_vector_vector, 593
gcaops.graph.undirected_graph, 599
gcaops.graph.undirected_graph_basis, 601
gcaops.graph.undirected_graph_complex, 605
gcaops.graph.undirected_graph_operad, 605
gcaops.graph.undirected_graph_vector, 603
multi_indices() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 586
multiplication_operator() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra
method), 587
multiplicity() (gcaops.graph.formality_graph.FormalityGraph method), 621
N
nedges() (gcaops.graph.directed_graph_vector.DirectedGraphVector_dict method), 613
nedges() (gcaops.graph.directed_graph_vector.DirectedGraphVector_vector method), 614
nedges() (gcaops.graph.formality_graph_vector.FormalityGraphVector_dict method), 627
nedges() (gcaops.graph.formality_graph_vector.FormalityGraphVector_vector method), 628
nedges() (gcaops.graph.graph_vector.GraphVector method), 592
nedges() (gcaops.graph.undirected_graph_vector.UndirectedGraphVector_dict method), 604
nedges() (gcaops.graph.undirected_graph_vector.UndirectedGraphVector_vector method), 604
nfactors() (gcaops.algebra.tensor_product.TensorProduct method), 587
ngens() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 587
ngens() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
nground() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
nground() (gcaops.graph.formality_graph_vector.FormalityGraphVector_dict method), 627
nground() (gcaops.graph.formality_graph_vector.FormalityGraphVector_vector method), 628
num_aerial_vertices() (gcaops.graph.formality_graph.FormalityGraph method), 621
num_ground_vertices() (gcaops.graph.formality_graph.FormalityGraph method), 621
nvertices() (gcaops.graph.directed_graph_vector.DirectedGraphVector_dict method), 613
nvertices() (gcaops.graph.directed_graph_vector.DirectedGraphVector_vector method), 614
nvertices() (gcaops.graph.formality_graph_vector.FormalityGraphVector_dict method), 627
Index 649
Documentation of gcaops, Release 1
nvertices() (gcaops.graph.formality_graph_vector.FormalityGraphVector_vector method), 628
nvertices() (gcaops.graph.graph_vector.GraphVector method), 593
nvertices() (gcaops.graph.undirected_graph_vector.UndirectedGraphVector_dict method), 604
nvertices() (gcaops.graph.undirected_graph_vector.UndirectedGraphVector_vector method), 604
O
odd_coordinate() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
odd_coordinates() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
one() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
orientations() (gcaops.graph.undirected_graph.UndirectedGraph method), 600
out_degrees() (gcaops.graph.directed_graph.DirectedGraph method), 610
out_degrees() (gcaops.graph.formality_graph.FormalityGraph method), 621
P
parent() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomial method), 588
parent() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 586
parent() (gcaops.algebra.superfunction_algebra.Superfunction method), 581
parent() (gcaops.algebra.tensor_product.TensorProductElement method), 588
parent() (gcaops.graph.graph_vector.GraphVector method), 593
parent() (gcaops.graph.graph_vector_dict.GraphVector_dict method), 596
parent() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 595
part_of_differential_order() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 626
partial_derivative() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomial method), 588
pdiff() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomial method), 588
plot() (gcaops.graph.directed_graph.DirectedGraph method), 610
plot() (gcaops.graph.formality_graph.FormalityGraph method), 621
plot() (gcaops.graph.graph_vector.GraphVector method), 593
plot() (gcaops.graph.undirected_graph.UndirectedGraph method), 600
PolyDifferentialOperator (class in gcaops.algebra.polydifferential_operator), 584
PolyDifferentialOperatorAlgebra (class in gcaops.algebra.polydifferential_operator), 586
R
relabeled() (gcaops.graph.directed_graph.DirectedGraph method), 610
relabeled() (gcaops.graph.formality_graph.FormalityGraph method), 622
relabeled() (gcaops.graph.undirected_graph.UndirectedGraph method), 601
S
schouten_bracket() (gcaops.algebra.superfunction_algebra.Superfunction method), 581
schouten_bracket() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
schouten_bracket() (gcaops.graph.formality_graph_complex.FormalityGraphCochain method), 629
set_aerial() (gcaops.graph.formality_graph_vector.FormalityGraphVector method), 627
set_aerial() (gcaops.graph.formality_graph_vector.FormalityGraphVector_dict method), 627
set_aerial() (gcaops.graph.formality_graph_vector.FormalityGraphVector_vector method), 628
set_pos() (gcaops.graph.directed_graph.DirectedGraph method), 610
set_pos() (gcaops.graph.formality_graph.FormalityGraph method), 622
set_pos() (gcaops.graph.undirected_graph.UndirectedGraph method), 601
show() (gcaops.graph.directed_graph.DirectedGraph method), 610
show() (gcaops.graph.formality_graph.FormalityGraph method), 622
show() (gcaops.graph.graph_vector.GraphVector method), 593
show() (gcaops.graph.undirected_graph.UndirectedGraph method), 601
650 Index
Documentation of gcaops, Release 1
skew_symmetrization() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 586
solve_homogeneous_diffpoly() (in module gcaops.algebra.differential_polynomial_solver), 589
subs() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 586
Superfunction (class in gcaops.algebra.superfunction_algebra), 579
SuperfunctionAlgebra (class in gcaops.algebra.superfunction_algebra), 581
SuperfunctionAlgebraDirectedGraphOperation (class in gcaops.algebra.superfunction_algebra_operation),
582
SuperfunctionAlgebraOperation (class in gcaops.algebra.superfunction_algebra_operation), 583
SuperfunctionAlgebraSchoutenBracket (class in gcaops.algebra.superfunction_algebra_operation), 583
SuperfunctionAlgebraSymmetricBracketOperation (class in gcaops.algebra.superfunction_algebra_operation),
583
SuperfunctionAlgebraSymmetricDirectedGraphOperation (class in gcaops.algebra.superfunction_algebra_operation),
583
SuperfunctionAlgebraSymmetricOperation (class in gcaops.algebra.superfunction_algebra_operation), 583
SuperfunctionAlgebraSymmetricUndirectedGraphOperation (class in gcaops.algebra.superfunction_algebra_operation),
583
SuperfunctionAlgebraUndirectedGraphOperation (class in gcaops.algebra.superfunction_algebra_operation),
583
symmetrization() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperator method), 586
T
tensor_power() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
tensor_product() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 587
TensorProduct (class in gcaops.algebra.tensor_product), 587
TensorProductElement (class in gcaops.algebra.tensor_product), 587
terms() (gcaops.algebra.tensor_product.TensorProductElement method), 588
total_derivative() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomial method), 588
U
undirected_to_directed_coeffs() (gcaops.graph.graph_file.UndirectedToDirectedGraphFileView method),
634
UndirectedGraph (class in gcaops.graph.undirected_graph), 599
UndirectedGraphBasis (class in gcaops.graph.undirected_graph_basis), 601
UndirectedGraphCache (class in gcaops.graph.graph_cache), 635
UndirectedGraphCochain (class in gcaops.graph.undirected_graph_complex), 605
UndirectedGraphCochain_dict (class in gcaops.graph.undirected_graph_complex), 605
UndirectedGraphCochain_vector (class in gcaops.graph.undirected_graph_complex), 605
UndirectedGraphComplex() (in module gcaops.graph.undirected_graph_complex), 606
UndirectedGraphComplex_ (class in gcaops.graph.undirected_graph_complex), 606
UndirectedGraphComplex_dict (class in gcaops.graph.undirected_graph_complex), 606
UndirectedGraphComplex_vector (class in gcaops.graph.undirected_graph_complex), 606
UndirectedGraphComplexBasis (class in gcaops.graph.undirected_graph_basis), 601
UndirectedGraphFileView (class in gcaops.graph.graph_file), 634
UndirectedGraphModule (class in gcaops.graph.undirected_graph_vector), 603
UndirectedGraphModule_dict (class in gcaops.graph.undirected_graph_vector), 603
UndirectedGraphModule_vector (class in gcaops.graph.undirected_graph_vector), 603
UndirectedGraphOperad() (in module gcaops.graph.undirected_graph_operad), 605
UndirectedGraphOperad_dict (class in gcaops.graph.undirected_graph_operad), 605
UndirectedGraphOperadBasis (class in gcaops.graph.undirected_graph_basis), 602
UndirectedGraphOperation_dict (class in gcaops.graph.undirected_graph_operad), 605
Index 651
Documentation of gcaops, Release 1
UndirectedGraphVector (class in gcaops.graph.undirected_graph_vector), 603
UndirectedGraphVector_dict (class in gcaops.graph.undirected_graph_vector), 603
UndirectedGraphVector_vector (class in gcaops.graph.undirected_graph_vector), 604
UndirectedToDirectedGraphFileView (class in gcaops.graph.graph_file), 634
V
value_at_copies_of() (gcaops.graph.formality_graph_operator.FormalityGraphOperator method), 631
vector() (gcaops.graph.graph_vector_vector.GraphVector_vector method), 595
W
weights() (gcaops.algebra.differential_polynomial_ring.DifferentialPolynomial method), 588
Z
zero() (gcaops.algebra.polydifferential_operator.PolyDifferentialOperatorAlgebra method), 587
zero() (gcaops.algebra.superfunction_algebra.SuperfunctionAlgebra method), 582
zero() (gcaops.graph.graph_vector.GraphModule method), 592
zero() (gcaops.graph.graph_vector_dict.GraphModule_dict method), 595
zero() (gcaops.graph.graph_vector_vector.GraphModule_vector method), 594
652 Index