Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus

TL;DR

This paper explores infinitesimal deformations of Poisson structures using Kontsevich graph calculus, providing algorithms for graph generation and classifying solutions up to four internal vertices, including new cocycles at higher levels.

Contribution

It introduces algorithms for generating Kontsevich graphs related to Poisson deformations and classifies solutions for graphs with up to four internal vertices, including new cocycles at higher levels.

Findings

Classified all solutions for graphs with up to 4 internal vertices.

Constructed the heptagon-wheel cocycle at k=8.

Reproduced the pentagon-wheel structure at k=6.

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 $k \leqslant 4$ we present all solutions of the deformation problem. For $k \geqslant 5$, 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 $k=8$.