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.

The defining properties of the Kontsevich unoriented graph complex

“…For example, the tetrahedron γ 3 ∈ ker d from [5], as well as the pentagon-and heptagon-wheel cocycles γ 5 , γ 7 ∈ ker d are presented in [2] (see also references therein). The Poisson structure symmetry which corresponds to γ 5 is found in [3,4]. Likewise, the symmetry built from γ 7 is reported in [1].…”

“…From [4] and references therein we recall that the bi-linear operation of insertion • i of a graph γ 1 into a graph γ 2 yields the sum of graphs γ 1 • i γ 2 = v∈Vert(γ 2 ) (γ 1 −→ v in γ 2 ) such that the edges in a graph γ 2 incident to a vertex v are redirected to vertices in γ 1 , each edge running over the vertices of γ 1 by its own Leibniz rule. By definition, for each term g in γ 1 • i γ 2 , the edge ordering is E(g) := E(γ 1 ) ∧ E(γ 2 ).…”

“…The knowledge of cocycles in the Kontsevich unoriented graph complex (Gra, d) is important because under the orientation morphism O r, the d-cocycles on n vertices and 2n−2 edges are mapped to the universal deformations of Poisson brackets on finite-dimensional affine manifolds (see [5] or [2,3,4] and [1]). For example, the tetrahedron γ 3 ∈ ker d from [5], as well as the pentagon-and heptagon-wheel cocycles γ 5 , γ 7 ∈ ker d are presented in [2] (see also references therein).…”

Open problems in the Kontsevich graph construction of Poisson bracket symmetries

“…Examples of this construction for known graph cocycles γ 2ℓ+1 on n vertices and 2n − 2 edges (namely, n = 2ℓ + 2 = 4, 6, and 8) have been given in [2,3], [7], and [4], respectively. Practical calculation of graph cocycles is addressed in [8,30]; the algorithms to verify the Poisson cocycle factorisation through the Jacobi identity are available from [9]. In the fundamental work [29], see also [27], Willwacher related the unoriented graph complex to generators of the Grothendieck-Teichmüller Lie algebra grt.…”

“…In the fundamental work [29], see also [27], Willwacher related the unoriented graph complex to generators of the Grothendieck-Teichmüller Lie algebra grt. In what follows, we expect some familiarity with the subject, e.g., on the basis of [22,24] and [2,4,9,19,28] or the lectures [18] available on-line from the Steklov MI RAS; the notation is standard.…”