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=k[x1,…,xn] and G is a finite subgroup of Autgr(R)≅GL(n,k) which contains no reflections, then there is a natural graded isomorphism R#G≅EndRG(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 Autgr(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 Autgr(A). In all but one case, we also write down explicit presentations for the invariant rings AG, and show that they are isomorphic to factors of AS regular algebras.