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

TL;DR

This paper develops methods for explicit computation in Morita reduced skew group algebras of path algebras, linking algebraic structures with quiver representations and group actions.

Contribution

It provides explicit formulas for decomposing elements in the Morita reduced algebra, enhancing computational tools for skew group algebras of path algebras.

Findings

Explicit formulas for element decomposition in eRe

Connection between algebraic elements and quiver paths

Methods incorporate group representation theory

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.