Actions of Small Groups on Two-Dimensional Artin-Schelter Regular Algebras

Abstract

In commutative invariant theory, a classical result due to Auslander says that if $R=\mathbb{k}[x_1,\dots ,x_n]$ and $G$ is a finite subgroup of ${\text{Aut}}_{\text{gr}}(R)\cong \text{GL}(n,\mathbb{k})$ which contains no reflections, then there is a natural graded isomorphism $R\#G\cong {\text{End}}_{R^G}(R)$. In this paper, we show that a version of Auslander's Theorem holds if we replace $R$ by an Artin-Schelter regular algebra $A$ of global dimension 2, and $G$ by a finite subgroup of ${\text{Aut}}_{\text{gr}}(A)$ which contains no quasi-reflections. This extends work of Chan-Kirkman-Walton-Zhang. As part of the proof, we classify all such pairs $(A,G)$, up to conjugation of $G$ by an element of ${\text{Aut}}_{\text{gr}}(A)$. In all but one case, we also write down explicit presentations for the invariant rings $A^G$, and show that they are isomorphic to factors of AS regular algebras.

Minor changes to the introduction, a few typos have been corrected, and the results of section 5 have been slightly strengthened