A note on a paper by Cuadra, Etingof and Walton

TL;DR

This paper examines and extends a proof showing that actions of semisimple Hopf algebras on certain algebraic structures in characteristic zero are essentially group actions, broadening the scope to iterated Ore extensions.

Contribution

It generalizes previous results by demonstrating that semisimple Hopf algebra actions on iterated Ore extensions also factor through group algebras, not just Weyl algebras.

Findings

Actions of semisimple Hopf algebras on Weyl algebras factor through group algebras.

The methods extend to iterated Ore extensions of derivation type in characteristic zero.

Provides a unified approach to understanding symmetries of these algebraic structures.

Abstract

We analyse the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebra $H$ on the $n$th Weyl algebra $A=A_n(K)$ over a field $K$ of characteristic $0$ factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebra $H$ on an iterated Ore extension of derivation type $A=K[x_1;d_1][x_2;d_2][\cdots][x_n;d_n]$ in characteristic zero factors through a group algebra.