Finite dimensional Hopf actions on deformation quantizations

TL;DR

This paper investigates conditions under which finite dimensional Hopf actions on deformation quantizations of commutative domains are essentially group actions, especially when the Poisson center of the fraction field is trivial.

Contribution

It establishes that Hopf actions on deformation quantizations factor through group algebras when the Poisson center of the fraction field is trivial.

Findings

Hopf actions factor through group algebras under certain conditions

Trivial Poisson center of the fraction field implies factorization

Provides criteria for Hopf actions on deformation quantizations

Abstract

We study when a finite dimensional Hopf action on a quantum formal deformation A of a commutative domain A_0 (i.e., a deformation quantization) must factor through a group algebra. In particular, we show that this occurs when the Poisson center of the fraction field of A_0 is trivial.