On the Irreducible Morphisms for Skew group algebras

TL;DR

This paper investigates the module categories of skew group algebras arising from finite abelian group actions on path algebras, introducing Galois semi-coverings and describing irreducible morphisms and almost split sequences.

Contribution

It introduces Galois semi-coverings for skew group algebras and provides a complete description of irreducible morphisms and almost split sequences in this context.

Findings

Complete description of irreducible morphisms for skew group algebras

Determination of the stable rank of skew-gentle algebras

Preservation of stable rank under skewness

Abstract

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional path algebra KQ of a quiver Q. Under certain assumptions on the action of G, we show the existence of a certain kind of covering that we call a Galois semi-covering functor, which becomes a Galois covering when the group action is free. We study the module category of its skew group algebra under this functor. As an application, we obtain a complete description of the irreducible morphisms and almost split sequences of skew group algebras and show that the (stable) rank is preserved under skewness. In particular, we determine the stable rank of skew-gentle algebras.