Semisimple Hopf actions on Weyl algebras

TL;DR

This paper proves that semisimple Hopf algebra actions on Weyl algebras in characteristic zero are essentially group actions, showing such actions must be cocommutative if inner faithful.

Contribution

It demonstrates that semisimple Hopf actions on Weyl algebras factor through group algebras, establishing a significant restriction on possible symmetries.

Findings

Hopf algebra actions on Weyl algebras are cocommutative

Actions factor through group algebras

Techniques include reduction modulo prime and division algebra analysis

Abstract

We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.