Actions of Taft Algebras on Noetherian Down-Up Algebras

TL;DR

This paper classifies actions of Taft algebras on noetherian graded down-up algebras, analyzing the properties of the invariant rings and revealing limitations of previous conjectures when semisimplicity is not assumed.

Contribution

It provides a complete classification of Taft algebra actions on these algebras and characterizes the invariant rings' properties, challenging existing assumptions in the literature.

Findings

Invariant rings can be non-commutative or non-Gorenstein.

Many conjectures fail without semisimplicity.

Conditions for Artin-Schelter regularity are explicitly identified.

Abstract

We consider actions of Taft algebras on noetherian graded down-up algebras. We classify all such actions and determine properties of the corresponding invariant rings $A^T$. We identify precisely when $A^T$ is commutative, when it is Artin-Schelter regular, and give sufficient conditions for it to be Artin-Schelter Gorenstein. Our results show that many results and conjectures in the literature concerning actions of semisimple Hopf algebras on Artin-Schelter regular algebras can fail when the semisimple hypothesis is omitted.