Kontsevich designed a scheme to generate infinitesimal symmetriesṖ = Q(P) of Poisson 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 flowṖ = O r(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 γ 2ℓ+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 γ 3 itself is a cocycle; at ℓ = 2 the Kontsevich-Willwacher pentagon-wheel cocycle γ 5 consists of two graphs. We reconstruct the symmetry Q 5 (P) = O r(γ 5 )(P) and verify that Q 5 is a Poisson cocycle indeed: [[P, Q 5 (P)]] . = 0 via [[P, P]] = 0. . 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. 1 The dilationṖ = P is an example of symmetry for Jacobi identity; we study nonlinear flowsṖ = Q(P) which are universal w.r.t. all affine manifolds and should persist under the quantization i {·, ·} P → [·, ·].
The real vector space of non-oriented graphs is known to carry a differential 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 $(2\ell+1)$-wheel graph with a nonzero coefficient at every $\ell\in\mathbb{N}$. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at $\ell = 1$ and $\ell = 2$ of one and two graphs respectively, the cocycle condition $d(\gamma) = 0$ is verified by hand. For the next, heptagon-wheel cocycle (known to exist at $\ell = 3$), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.Comment: Special Issue JNMP 2017 `Local and nonlocal symmetries in Mathematical Physics'; 17 journal-style pages, 54 figures, 3 tables; v2 accepte
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.
Consider the real vector space of formal sums of non-empty, finite unoriented graphs without multiple edges and loops. Let the vertices of graphs be unlabelled but let every graph γ be endowed with an ordered set of edges E(γ). Denote by Gra the vector space of formal sums of graphs modulo the relation (γ 1 , E(γ 1 )) − sign(σ)(γ 2 , E(γ 2 )) = 0 for topologically equal graphs γ 1 and γ 2 whose edge orderings differ by a permutation σ. The zero class in Gra is represented by sums of graphs that cancel via the above relation. The Lie bracket of graphs with ordered edge sets is defined using the insertion of a graph into vertices of the other one. We give an explicit proof of the theorems which state that the space Gra is a well-defined differential graded Lie algebra: both the Lie bracket [·, ·] and the vertex-expanding differential d = [•−•, ·] respect the calculus modulo zero graphs.
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