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

TL;DR

This paper demonstrates how graph complexes induce universal Poisson cocycles on homogeneous Poisson structures, providing a uniform construction applicable across affine manifolds and illustrating with examples related to Lie algebra R-matrices.

Contribution

It introduces a universal method to construct Poisson cocycles from graph cocycles for homogeneous Poisson structures, extending the understanding of deformation cohomology.

Findings

Constructed universal Poisson cocycles from graph cocycles for homogeneous Poisson brackets.

Showed the cocycles are uniform across all finite-dimensional affine manifolds.

Provided examples involving cubic Poisson brackets related to Lie algebra R-matrices.

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.