Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex

TL;DR

This paper explores how the pentagon-wheel cocycle in the graph complex generates symmetries of Poisson brackets, reconstructing and verifying a specific Poisson cocycle related to the Kontsevich--Willwacher cocycle.

Contribution

It reconstructs the symmetry associated with the pentagon-wheel cocycle and proves it defines a Poisson cocycle, extending understanding of Poisson structure deformations.

Findings

Reconstructed the symmetry $ ext{Or}(oldsymbol{eta}_5)$ from the pentagon-wheel cocycle.

Verified that the reconstructed symmetry is a Poisson cocycle.

Confirmed the cocycle condition $[ ext{P}, ext{Q}_5( ext{P})]igg|_{[ ext{P}, ext{P}]=0}$ holds.

Abstract

Kontsevich designed a scheme to generate infinitesimal symmetries $\dot{\mathcal{P}} = \mathcal{Q}(\mathcal{P})$ of Poisson brackets $\mathcal{P}$ 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 $\dot{\mathcal{P}} = \text{Or}(\gamma)(\mathcal{P})$ preserves the space of Poisson structures if $\gamma$ is a cocycle with respect to the vertex-expanding differential in the graph complex. A class of such cocycles $\boldsymbol{\gamma}_{2\ell+1}$ is known to exist: marked by $\ell \in \mathbb{N}$, each of them contains a $(2\ell+1)$-gon wheel with a nonzero coefficient. At $\ell=1$ the tetrahedron $\boldsymbol{\gamma}_3$ itself is a cocycle; at $\ell=2$ the Kontsevich--Willwacher pentagon-wheel cocycle $\boldsymbol{\gamma}_5$ consists of two graphs. We reconstruct the symmetry $\mathcal{Q}_5(\mathcal{P}) = \text{Or}(\boldsymbol{\gamma}_5)(\mathcal{P})$ and verify that $\mathcal{Q}_5$ is a Poisson cocycle indeed: $[\![\mathcal{P},\mathcal{Q}_5(\mathcal{P})]\!]\doteq 0$ via $[\![\mathcal{P},\mathcal{P}]\!]=0$.