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

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 kQ#k〈σ〉-modules. It has been shown by Hubery that any kQ#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 kQ#k〈σ〉-module. In this paper, the authors prove that a kQ#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 kQ#k〈σ〉-modules from all indecomposable 〈σ〉-equivalent kQ-modules. The number of non-isomorphic indecomposable kQ#k〈σ〉-modules induced from the same indecomposable 〈σ〉-equivalent kQ-module is given. In particular, the authors give the relationship between indecomposable kQ#k〈σ〉-modules and indecomposable kQ-modules in the cases of indecomposable simple, projective and injective modules.