POISSON BRACKETS SYMMETRY FROM THE PENTAGON-WHEEL COCYCLE IN THE GRAPH COMPLEX

摘要

Kontsevich designed a scheme to generate infinitesimal symmetries P = Q(P) of Poisson brackets P on all affine manifolds M~r; every such deformation is encoded by oriented graphs onn + 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 flow P = Or(γ)(P) preserves the space of Poisson structures if γ is a cocycle with respect to the vertex-expanding differential d 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_5a consists of two graphs. We reconstruct the symmetry Q_5(P) = Or(γ_5)(P) and verify that Q_5 is a Poisson cocycle indeed: [P, Q_5(P)] = 0 via [P, P] = 0.