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

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 $k=8$.

International conference ISQS'25 on integrable systems and quantum symmetries (6-10 June 2017 in CVUT Prague, Czech Republic). Introductory paragraph I.1 follows p.3 in arXiv:1710.00658 [math.CO]; 13 pages, 3 figures, 2 tables

References in corpus (4)

• The Kontsevich tetrahedral flow revisited
• Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?
• Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex
• The heptagon-wheel cocycle in the Kontsevich graph complex

Cited by in corpus (4)

• Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex
• The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets
• Open problems in the Kontsevich graph construction of Poisson bracket symmetries
• The defining properties of the Kontsevich unoriented graph complex