Hopf actions on Poisson algebras

TL;DR

This paper investigates finite-dimensional Hopf actions on Poisson algebras, revealing a rigidity phenomenon where such actions must factor through group algebras, and extends known results from associative to Poisson algebra contexts.

Contribution

It establishes quantum rigidity results for Poisson algebras, especially quadratic ones, and connects Hopf actions on Poisson algebras to their quantizations and universal enveloping algebras.

Findings

Hopf actions on quadratic Poisson algebras factor through group algebras

Lifting Hopf actions to Rees algebras enables classification of actions

Partial classification of Taft algebra actions on low-dimensional Poisson algebras

Abstract

We study finite-dimensional Hopf actions on Poisson algebras and explore the phenomenon of quantum rigidity in this context. Our main focus is on filtered (and especially quadratic) Poisson algebras, including the Weyl Poisson algebra in $2n$ variables and certain Poisson algebras in two variables. In particular, we show that any finite-dimensional Hopf algebra acting inner faithfully on these Poisson algebras must necessarily factor through a group algebra-mirroring well-known rigidity theorems for Weyl algebras in the associative setting. The proofs hinge on lifting the Hopf actions to associated Rees algebras, where we construct suitable noncommutative "quantizations" that allow us to leverage classification results for Hopf actions on quantum (or filtered) algebras. We also discuss how group actions on Poisson algebras extend to universal enveloping algebras, and we give partial classifications of Taft algebra actions on certain low-dimensional Poisson algebras.