Auslander's Theorem for dihedral actions on preprojective algebras of type A

TL;DR

This paper investigates Auslander's theorem in the context of preprojective algebras of type A under dihedral group actions, establishing conditions for the Auslander map to be an isomorphism.

Contribution

It extends Auslander's theorem to non-connected graded Calabi-Yau algebras, specifically for dihedral group actions on preprojective algebras of type A, identifying precise conditions for isomorphism.

Findings

The Auslander map is an isomorphism if and only if the group does not contain all reflections through a vertex.

The study applies to non-connected graded Calabi-Yau algebras.

Provides new insights into automorphism group actions on preprojective algebras.

Abstract

Given an algebra $R$ and $G$ a finite group of automorphisms of $R$, there is a natural map $\eta_{R,G}:R\#G \to \mathrm{End}_{R^G} R$, called the Auslander map. A theorem of Auslander shows that $\eta_{R,G}$ is an isomorphism when $R=\mathbb{C}[V]$ and $G$ is a finite group acting linearly and without reflections on the finite-dimensional vector space $V$. The work of Mori and Bao-He-Zhang has encouraged study of this theorem in the context of Artin-Schelter regular algebras. We initiate a study of Auslander's result in the setting of non-connected graded Calabi-Yau algebras. When $R$ is a preprojective algebra of type $A$ and $G$ is a finite subgroup of $D_n$ acting on $R$ by automorphism, our main result shows that $\eta_{R,G}$ is an isomorphism if and only if $G$ does not contain all of the reflections through a vertex.