Unimodular graded Poisson Hopf algebras

TL;DR

This paper proves that certain finitely generated, connected graded Poisson Hopf algebras with homogeneous brackets are unimodular, extending a recent result from Hopf algebra theory to the Poisson setting.

Contribution

It establishes the unimodularity of a class of Poisson Hopf algebras under specific grading and homogeneity conditions, providing a Poisson analogue to known Hopf algebra results.

Findings

Poisson Hopf algebras with homogeneous brackets are unimodular

Unimodularity holds for finitely generated, connected graded Poisson Hopf algebras

Extension of Hopf algebra unimodularity results to Poisson Hopf algebras

Abstract

Let $A$ be a Poisson Hopf algebra over an algebraically closed field of characteristic zero. If $A$ is finitely generated and connected graded as an algebra and its Poisson bracket is homogeneous of degree $d \geq 0$, then $A$ is unimodular; that is, the modular derivation of $A$ is zero. This is a Poisson analogue of a recent result concerning Hopf algebras which are connected graded as algebras.