Title: Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex,
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
$[\![\mathcal{P},\mathcal{P}]\!]=0$.