Crossed-products of Calabi-Yau algebras by finite groups

TL;DR

This paper investigates conditions under which the skew group algebra formed from a Calabi-Yau algebra and a finite group action retains the Calabi-Yau property, with applications to cluster categories and Auslander-Reiten theories.

Contribution

It provides necessary and sufficient conditions for the skew group algebra to be Calabi-Yau, especially for Ginzburg dg algebras, and explores their implications in representation theory.

Findings

A*G is Calabi-Yau when A is Ginzburg dg algebra with invariant potential.

A*G is Morita equivalent to a Ginzburg dg algebra under certain conditions.

Applications include comparing cluster categories and higher Auslander-Reiten theories.

Abstract

Let a finite group G act on a differential graded algebra A. This article presents necessary conditions and sufficient conditions for the skew group algebra A*G to be Calabi-Yau. In particular, when A is the Ginzburg dg algebra of a quiver with an invariant potential, then A*G is Calabi-Yau and Morita equivalent to a Ginzburg dg algebra. Some applications of these results are derived to compare the generalised cluster categories of A and A*G when they are defined and to compare the higher Auslander-Reiten theories of A and A*G when A is a finite dimensional algebra.