Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms

Abstract

The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$, but not quadratic (the coefficients of $P$ are not degree-two homogeneous polynomials), and whenever its velocity bi-vector $\dot{P}=Q(P)$, also homogeneous w.r.t. $\vec{V}$ by $L_{\vec{V}}(Q)=n\cdot Q$ whenever $Q(P)= Or(\gamma)(P^{\otimes^n})$ is obtained using the orientation morphism $Or$ from a graph cocycle $\gamma$ on $n$ vertices and $2n-2$ edges in each term, then the $1$-vector $\vec{X}=Or(\gamma)(\vec{V}\otimes P^{\otimes^{n-1}})$ is a Poisson cocycle. Its construction is uniform for all Poisson bi-vectors $P$ satisfying the above assumptions, on all finite-dimensional affine manifolds $M$. Still, if the bi-vector $Q\not\equiv 0$ is exact in the respective Poisson cohomology, so there exists a vector field $\vec{Y}$ such that $Q(P)=[\![\vec{Y},P]\!]$, then the universal cocycle $\vec{X}$ does not belong to the coset of $\vec{Y}$ mod $\ker[\![P,\cdot]\!]$. We illustrate the construction using two examples of cubic-coefficient Poisson brackets associated with the $R$-matrices for the Lie algebra $\mathfrak{gl}(2)$. Keywords: Graph complex, Poisson bracket, deformation cohomology, affine manifold, tetrahedral flow, diffeomorphism.