Invariant Theory of Artin-Schelter Regular Algebras: A survey

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.