On the Abelian Galois coverings of an algebra
type: Item
instance of: scholarly article
title: On the Abelian Galois coverings of an algebra
zbMATH Open document ID: 0589.16024
DOI: 10.1016/0021-8693(86)90131-6
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(\Lambda\) be a finite dimensional algebra over an algebraically closed field \(k\), given by a pair \((Q,I)\), where \(Q\) is a finite oriented graph without double arrows and \(I\) is an admissible ideal of the path algebra \(kQ\) of \(Q\), \(\pi\) be its fundamental group and \({\tilde \Lambda}\) its universal Galois covering. Then each group homomorphism \(\psi\colon \pi\to G\) produces a Galois covering \(p_{\psi}\colon \Lambda_{\psi}\to \Lambda\) with group \(G\), moreover \(\psi\) induces a certain action of the group \(X\) of characters of \(G\) on \(\Lambda\). The author shows that if \(G\) is a finite abelian group, whose order is prime with the characteristic of \(k\), then the Galois covering \(p_{\psi}\) is isomorphic to a certain covering \(p'\colon P\to \Lambda\), where \(P\) is a category of indecomposable projectives over a skew group algebra \(\Lambda X\) of \(X\) over \(\Lambda\), and consequently \(\Lambda_{\psi}\) and \(\Lambda X\) are Morita equivalent. Another proof of the last fact is given by \textit{I. Reiten} and \textit{C. Riedtmann} [J. Algebra 92, 224--282 (1985; Zbl 0549.16017)].
reviewed by: Piotr Dowbor
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 3946393
zbMATH Keywords: finite dimensional algebra
zbMATH Keywords: path algebra
zbMATH Keywords: fundamental group
zbMATH Keywords: universal Galois covering
zbMATH Keywords: characters
zbMATH Keywords: finite abelian group
zbMATH Keywords: category of indecomposable projectives
zbMATH Keywords: skew group algebra
author: José A. de la Peña
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90131-6
OpenAlex ID: W1994351585
cites work: The standard form of a representation-finite algebra
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cites work: The universal cover of a quiver with relations
cites work: Automorphisms of representation finite algebras
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cites work: Skew group algebras in the representation theory of Artin algebras
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description: scientific article; zbMATH DE number 3946393