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

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.

Some typos fixed. To appear in Canadian Mathematical Bulletin

References in corpus (5)

• Calabi-Yau algebras
• Actions of some pointed Hopf algebras on path algebras of quivers
• Auslander's Theorem for permutation actions on noncommutative algebras
• Growth of Graded Twisted Calabi-Yau Algebras
• Congenial Algebras: Extensions and Examples