Finitely presented simple modules over Leavitt path algebras.
type: Item
instance of: scholarly article
title: Finitely presented simple modules over Leavitt path algebras.
zbMATH Open document ID: 1304.16002
author: Pere Ara
author: Kulumani M. Rangaswamy
published in: Journal of Algebra
publication date: 2014-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1311.4332
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 6339760
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: simple modules
zbMATH Keywords: primitive ideals
zbMATH Keywords: finitely presented modules
zbMATH Keywords: Chen modules
zbMATH Keywords: annihilators
reviewed by: Candido Martín González
MaRDI profile type: Publication
OpenAlex ID: W2964025813
arXiv ID: 1311.4332
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DOI: 10.1016/J.JALGEBRA.2014.06.032
review text: Recently, X. W. Chen has provided a method of constructing simple modules \(V_{[p]}\) over a Leavitt path algebra \(L_K(E)\) of an arbitrary graph \(E\) by using the equivalence class \([p]\) of infinite paths tail-equivalent to a fixed infinite path \(p\) in \(E\) (definitions included in the paper under review). He also constructed simple modules corresponding to various sinks \(w\) in \(E\).NEWLINENEWLINE In this paper, the authors introduce additional classes of simple modules using vertices which are infinite emitters and also exclusive cycles, and call all these simple modules over \(L_K(E)\) Chen modules. They give a description of the annihilators of the various Chen modules and, as a consequence, they show that every primitive ideal of \(L_K(E)\) can be realized as the annihilator of a Chen module. They also prove, that the Chen modules are pairwise non-isomorphic.NEWLINENEWLINE For an algebra \(A\), denote by \(\widetilde A\) the set of isoclasses of simple left \(A\)-modules, and by \(\text{Prim}(A)\) the set of primitive ideals of \(A\). There is a canonical map \(\widetilde A\to\text{Prim}(A)\) mapping \([N]\) to \(\text{Ann}_A(N)\), the annihilator of \(N\).NEWLINENEWLINE Then it is proved that: If \(E\) is finite graph, \(K\) an arbitrary field and \(L:=L_K(E)\), then the following conditions are equivalent:NEWLINENEWLINE (1) Every simple left \(L\)-module is finitely presented.NEWLINENEWLINE (2) Every simple Chen module is finitely presented.NEWLINENEWLINE (3) Every vertex \(v\) in \(E\) is the base of at most one cycle.NEWLINENEWLINE (4) The map \(\widetilde L\to\text{Prim}(L)\) is a bijection.NEWLINENEWLINE (5) All simple left \(L\)-modules are Chen modules.NEWLINENEWLINE (6) The Gelfand-Kirillov dimension of \(L_K(E)\) is finite.
description: scientific article; zbMATH DE number 6339760