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

Kontsevich designed a scheme to generate infinitesimal symmetries $\dot{P}=Q\left(P\right)$ 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 flow $\dot{P}=\text{O}\overrightarrow{\text{r}}\left(\gamma \right)\left(P\right)$ 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 ${\gamma }_{2\ell +1}$ is known to exist: marked by $\ell \in N,$ each of them contains a $(2\ell +1)$ -gon wheel with a nonzero coefficient. At $\ell =1$ the tetrahedron ${\gamma }_3$ itself is a cocycle; at $\ell =2$ the Kontsevich–Willwacher pentagon-wheel cocycle ${\gamma }_5$ consists of two graphs. We reconstruct the symmetry $Q_5\left(P\right)=\text{O}\overrightarrow{\text{r}}\left({\gamma }_5\right)\left(P\right)$ and verify that $Q_5$ is a Poisson cocycle indeed: $\left[\left[P,Q_5\left(P\right)\right]\right]\doteq 0$ via $\left[\left[P,P\right]\right]=0.$