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

摘要

Kontsevich designed a scheme to generate infinitesimal symmetries (P)over dot = L(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 gamma on n vertices and 2n - 2 edges. The bi-vector flow (P)over dot = O (r) over right arrow(gamma)(P) preserves the space of Poisson structures if gamma is a cocycle with respect to the vertex-expanding differential d in the graph complex. A class of such cocycles gamma(2l+1) is known to exist: marked by l is an element of N each of them contains (2l + 1)-gon wheel with a nonzero coefficient. At l = 1 the tetrahedron gamma(3) itself is a cocycle; at l = 2 the Kontsevich-Willwacher pentagon-wheel cocycle gamma(5) consists of two graphs. We reconstruct the symmetry L-5(P) = O (r) over right arrow(gamma(5))(P) and verify that L-5 is a Poisson cocycle indeed: [P,L-5(P) (=)over dot 0 via [P,P] = 0.