The action of Kontsevich's graph complex on Poisson structures and star products: an implementation
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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 star-product up to o(h^4);
- we assemble ★ mod o(h^6) from external data by Banks-Panzer-Pym and we obtain the star product mod o(h^7) for affine Poisson brackets;
- we verify that graph weights found by Banks-Panzer-Pym up to o(h^6) satisfy many known relations;
- we illustrate the explicit proof of the associativity for the full star product modulo o(h^6) and for the affine star product modulo o(h^7);
- 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.