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 σ is an automorphism of Q . Let k be an algebraically closed field whose characteristic does not divide the order of the cyclic group 〈 σ 〉 . The aim of this paper is to investigate the relationship between indecomposable kQ -modules and indecomposable k Q # k 〈 σ 〉 -modules. It has been shown by Hubery that any k Q # k 〈 σ 〉 -module is an isomorphically invariant kQ -module, i.e., ii-module (in this paper, we call it 〈 σ 〉 -equivalent kQ -module), and conversely any 〈 σ 〉 -equivalent kQ -module induces a k Q # k 〈 σ 〉 -module. In this paper, the authors prove that a k Q # k 〈 σ 〉 -module is indecomposable if and only if it is an indecomposable 〈 σ 〉 -equivalent kQ -module. Namely, a method is given in order to induce all indecomposable k Q # k 〈 σ 〉 -modules from all indecomposable 〈 σ 〉 -equivalent kQ -modules. The number of non-isomorphic indecomposable k Q # k 〈 σ 〉 -modules induced from the same indecomposable 〈 σ 〉 -equivalent kQ -module is given. In particular, the authors give the relationship between indecomposable k Q # k 〈 σ 〉 -modules and indecomposable kQ -modules in the cases of indecomposable simple, projective and injective modules.