Invariant Theory of Artin-Schelter Regular Algebras: A survey

TL;DR

This survey explores how classical invariant theory concepts extend to actions of finite groups or Hopf algebras on Artin-Schelter regular algebras, focusing on properties like regularity and Gorenstein conditions.

Contribution

It summarizes recent results on invariants of Artin-Schelter regular algebras under group and Hopf algebra actions, highlighting conditions for regularity and Gorenstein properties.

Findings

Conditions for $A^H$ to be AS regular

Criteria for $A^H$ to be AS Gorenstein

Characterization of 'complete intersection' in this context

Abstract

This is a survey of results that extend notions of the classical invariant theory of linear actions by finite groups on $k[x_1, \dots, x_n]$ to the setting of finite group or Hopf algebra $H$ actions on an Artin-Schelter regular algebra $A$. We investigate when $A^H$ is AS regular, or AS Gorenstein, or a "complete intersection" in a sense that is defined. Directions of related research are explored briefly.