Semisimple Hopf actions on commutative domains

TL;DR

The paper proves that semisimple Hopf algebra actions on commutative domains are essentially group actions, answering longstanding questions and extending results to positive characteristic fields.

Contribution

It establishes that semisimple Hopf actions on commutative domains must come from group algebras, resolving key open questions in the field.

Findings

Semisimple Hopf actions on commutative domains are group algebra actions.

Results extend to positive characteristic fields.

Applications to Hopf actions on Weyl algebras.

Abstract

Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if A arises as an H-module algebra via an inner faithful H-action, then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen. The main results of this article extend to working over k of positive characteristic. On the other hand, we obtain results on Hopf actions on Weyl algebras as a consequence of the main theorem.