Examples of non-semisimple Hopf algebra actions on Artin-Schelter regular algebras

TL;DR

This paper classifies all inner-faithful actions of a specific non-semisimple Hopf algebra on certain Artin-Schelter regular algebras of low dimension, revealing new examples of such algebraic actions in positive characteristic.

Contribution

It provides a complete classification of inner-faithful actions of the restricted enveloping algebra of a 2D nonabelian Lie algebra on noetherian Koszul Artin-Schelter regular algebras of dimension up to three, a novel result in the field.

Findings

Classified all inner-faithful actions of the algebra U on the specified regular algebras.

Identified new examples of non-semisimple Hopf algebra actions in positive characteristic.

Extended understanding of algebra actions in noncommutative algebraic geometry.

Abstract

Let $\Bbbk$ be a base field of characteristic $p>0$ and let $U$ be the restricted enveloping algebra of a 2-dimensional nonabelian restricted Lie algebra. We classify all inner-faithful $U$-actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three.