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

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.