On the Morita Reduced Versions of Skew Group Algebras of Path Algebras

Abstract

Let R be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita reduced algebra associated to R.Reiten and Riedtmann proved that there exists an idempotent e of R such that the algebra eRe is both Morita equivalent to R and isomorphic to the path algebra of some quiver which was described by Demonet. This article gives explicit formulas for the decomposition of any element of eRe as a linear combination of paths in the quiver described by Demonet. This is done by expressing appropriate compositions and pairings in a suitable monoidal category which takes into account the representation theory of the finite group.

Quivers with potentials and actions of finite abelian groups

“…More recently, W G was computed in [GP19] for G any cyclic group under some assumptions on the action. Finally, an algorithm to compute W G for any finite group was given in [LM18b], but it relies on inverting a possibly large matrix and is thus not practical for computing examples.…”