The socle of a Leavitt path algebra.
type: Item
instance of: scholarly article
title: The socle of a Leavitt path algebra.
zbMATH Open document ID: 1134.16005
author: Gonzalo Aranda Pino
author: Dolores Martín Barquero
author: Mercedes Siles Molina
author: Candido Martín González
published in: Journal of Pure and Applied Algebra
publication date: 2007-12-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0701637
review text: Suppose that \(E\) is a directed countable graph such that for any vertex \(v\) there exist finitely many edges entering \(v\). For any edge \(e\) denote by \(s(e)\) its source and by \(r(e)\) its end. If \(k\) is a field then one can define the Leavitt path \(k\)-algebra \(L(E)\) as the associative algebra with a unit element generated by the set of orthogonal idempotents which are vertices \(v\) from \(E\) and by double copies \(\{e,e^*\}\) of the set of edges in \(E\). These generators are connected by defining relations \[ s(e)e=er(e)=e,\quad r(e)e^*=e^*s(e)= e^*,\quad e^*e'=\delta_{e,e'}r(e),\quad v=\sum_{\{e\mid s(e)=v\}}ee^*. \] Denote by \(P_1(E)\) the set of all vertices in \(E\) such that \(s^{-1}(v)\) contains at most one element and \(v\) is contained in a cycle at some vertex from a tree of \(v\). It is shown that if \(u\) is a vertex then \(L(E)u\) is a minimal left ideal in \(L(E)\) if and only if \(u\in P_1(E)\). Any minimal left ideal has this form. Hence the socle of \(L(E)\) is the ideal generated by \(P_1(E)\).
Mathematics Subject Classification ID: 16S88
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 5218381
zbMATH Keywords: quivers
zbMATH Keywords: ideals
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: orthogonal idempotents
zbMATH Keywords: defining relations
zbMATH Keywords: socle
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
OpenAlex ID: W2095894620
arXiv ID: math/0701637
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DOI: 10.1016/J.JPAA.2007.06.001
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description: scientific article