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

Suppose that Q is a connected quiver Without oriented cycles and or is all automorphism of Q. Let k, be ail algebraically closed field whose characteristic does not divide the order of the cyclic group (sigma). The aim of this paper is to investigate the relationship between indecomposable kQ modules and indecomposable kQ # k(sigma)-modules. It has been Shown by Hubery that any kQ # k(sigma)-module is ail isomorphically invariant kQ-module, i.e- ii-module (in this paper, we call it (sigma)-equivalent kQ-module), and conversely any (sigma)-equivalent kQ-module induces a kQ # k(sigma)-module. In this paper, the authors prove that a kQ # k(sigma)-module is indecomposable if and only if it is air indecomposable (sigma)-equivalent kQ-module. Namely, a method is given in order to induce all indecomposable kQ # k(sigma)-modules from all indecomposable (sigma)equivalent kQ-modules. The number of non-isomorphic indecomposable kQ # k(sigma)-modules induced from the same indecomposable (sigma)-equivalent kQ-module is given. In particular, the authors give the relationship between indecomposable kQ # k(sigma)-modules and indecomposable kQ-modules in the cases of indecomposable simple. projective and injective Modules. (C) 2008 Elsevier Inc. All rights reserved.