Refine search
Actions for selected content:
1 results
Auslander’s theorem for dihedral actions on preprojective algebras of type A
- Part of
-
- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 1 / March 2023
- Published online by Cambridge University Press:
- 09 June 2022, pp. 324-339
- Print publication:
- March 2023
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
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–Ueyama and Bao–He–Zhang has encouraged the study of this theorem in the context of Artin–Schelter regular algebras. We initiate a study of Auslander’s result in the setting of nonconnected 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.
