Representations of skew group algebras induced from isomorphically invariant modules over path algebras

TL;DR

This paper explores the relationship between indecomposable modules over path algebras and their skew group algebra counterparts, providing methods to induce and classify indecomposable modules in the presence of automorphisms.

Contribution

It establishes a precise criterion for indecomposability of modules over skew group algebras induced from automorphisms of path algebras, extending module classification techniques.

Findings

Characterizes indecomposable modules over skew group algebras

Provides a method to induce all indecomposable modules from invariant modules

Analyzes relationships for simple, projective, and injective modules

Abstract

Suppose that $Q$ is a connected quiver without oriented cycles and $\sigma$ is an automorphism of $Q$. Let $k$ be an algebraically closed field whose characteristic does not divide the order of the cyclic group $\langle\sigma\rangle$. The aim of this paper is to investigate the relationship between indecomposable $kQ$-modules and indecomposable $kQ\#k\langle\sigma\rangle$-modules. It has been shown by Hubery that any $kQ\#k\langle\sigma\rangle$-module is an isomorphically invariant $kQ$-module, i.e., ii-module (in this paper, we call it $\langle\sigma\rangle$-equivalent $kQ$-module), and conversely any $\langle\sigma\rangle$-equivalent $kQ$-module induces a $kQ\#k\langle\sigma\rangle$-module. In this paper, the authors prove that a $kQ\#k\langle\sigma\rangle$-module is indecomposable if and only if it is an indecomposable $\langle\sigma\rangle$-equivalent $kQ$-module. Namely, a method is given in order to induce all indecomposable $kQ\#k\langle\sigma\rangle$-modules from all indecomposable $\langle\sigma\rangle$-equivalent $kQ$-modules. The number of non-isomorphic indecomposable $kQ\#k\langle\sigma\rangle$-modules induced from the same indecomposable $\langle\sigma\rangle$-equivalent $kQ$-module is given. In particular, the authors give the relationship between indecomposable $kQ\#k\langle\sigma\rangle$-modules and indecomposable $kQ$-modules in the cases of indecomposable simple, projective and injective modules.