Poisson Brackets Symmetry from the Pentagon-Wheel Cocycle in the Graph Complex

Abstract

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 → [·, ·].

Open problems in the Kontsevich graph construction of Poisson bracket symmetries

“…[22] and [4]). We observed in [2,7] [22] and [4,19]). The right-hand sides of evolutionary PDE systems d dε P = Q(P) are nonlinear of polynomial degree n and, generally speaking, proper, i.e.…”

“…[12,15] Problems 11-12 are typical in the geometry of differential equations (see [26] and [16,Lect. 3,7]): a symmetry is induced by the projection from a differential covering, and a PDE system is consistently constrained by its own symmetry. 7.…”

“…(See [4] and references therein for a detailed analysis why the graph orientation morphism O r actually works.) 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].…”

“…Find the smooth-atlas globalisation(s) of the Kontsevich symmetries Q 1: 6 P) = O r(γ 3 )(P), Q 5 (P) = O r(γ 5 )(P), and Q 7 (P) = O r(γ 7 )(P), found in[2],[7], and[4] respectively.…”

“…In other words, do the power series converge (uniformly over M r aff ) ? The PDE systemsṖ = Q(P) which we know from[3,2,4,7] consist of r(r − 1)/2 evolution equations with differential-polynomial right-hand sides.…”