Lie algebras and coverings
type: Item
instance of: scholarly article
title: Lie algebras and coverings
zbMATH Open document ID: 0913.16007
DOI: 10.1007/BF02564488
author: Ch. Riedtmann
published in: Commentarii Mathematici Helvetici
publication date: 1999-06-07T00:00:00Z
full work available at URL: https://eudml.org/doc/140340
review text: Let \(\Lambda\) be an associative unitary finite-dimensional \(\mathbb{C}\)-algebra which is representation finite. This means that the number of isomorphism classes of indecomposable finite-dimensional \(\Lambda\)-left modules is finite. Let us fix a set \(\mathcal F\) of representatives for these isomorphism classes. We showed [\textit{C. Riedtmann}, J. Algebra 170, No. 2, 526-546 (1994; Zbl 0841.16018)] that the free \(\mathbb{Z}\)-module \(L(\Lambda)=\bigoplus_{A\in{\mathcal F}}\mathbb{Z} v_A\) generated by the symbols \(\{v_A:A\in{\mathcal F}\}\) can be made into a \(\mathbb{Z}\)-Lie algebra. This is the complex version of \textit{C. M. Ringel}'s construction of Lie algebras via Hall algebras over finite fields [Banach Center Publ. 26, 433-447 (1990; Zbl 0778.16004)]. The construction of \(L(\Lambda)\) carries over easily to the case where \(\Lambda\) is a locally representation finite \(\mathbb{C}\)-category. Theorem. Let \(\Lambda\) be a locally representation finite \(\mathbb{C}\)-category with universal cover \(\widetilde\Lambda\) and fundamental group \(G\). Then \(L(\Lambda)\) is isomorphic to \(L(\widetilde\Lambda)/G\).
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 720098
zbMATH Keywords: finite-dimensional algebras
zbMATH Keywords: representation finite algebras
zbMATH Keywords: indecomposable finite-dimensional left modules
zbMATH Keywords: Lie algebras
zbMATH Keywords: Hall algebras over finite fields
zbMATH Keywords: locally representation finite categories
zbMATH Keywords: universal covers
zbMATH Keywords: fundamental groups
MaRDI profile type: Publication
OpenAlex ID: W2016188656
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description: scientific article; zbMATH DE number 720098