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Representations of skew group algebras induced from isomorphically invariant modules over path algebras. (English) Zbl 1207.16015

Summary: 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 A. Hubery [J. Lond. Math. Soc., II. Ser. 69, No. 1, 79-96 (2004; Zbl 1062.16021)] 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.

MSC:

16G20 Representations of quivers and partially ordered sets
16S35 Twisted and skew group rings, crossed products

Citations:

Zbl 1062.16021
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References:

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