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 $R = \Bbbk[x_1, \dots, x_n]$ and $G$ is a finite subgroup of $\text{Aut}_{\text{gr}}(R) \cong \text{GL}(n,\Bbbk)$ which contains no reflections, then there is a natural graded isomorphism $R \hspace{1pt} \# \hspace{1pt} G \cong \text{End}_{R^G}(R)$. 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.