Cycles in Leavitt path algebras by means of idempotents.
type: Item
instance of: scholarly article
title: Cycles in Leavitt path algebras by means of idempotents.
zbMATH Open document ID: 1311.16004
author: Gonzalo Aranda Pino
author: Jose Brox
author: Mercedes Siles Molina
published in: Forum Mathematicum
publication date: 2015-01-19T00:00:00Z
review text: This paper studies two graph properties known as Condition (NE) and Condition (L). The first of them asks for all the cycles of the graph to have no exits, while the second one demands that every cycle has an exit. The authors present ring theoretic characterizations of both conditions for arbitrary graphs, in terms of idempotents. Concretely, they establish that \(E\) satisfies Condition (NE) if and only if \(L_K(E)\) does not present infinite idempotents (and identify the algebraic structure of \(L_K(E)\)). Then, they prove that \(E\) satisfies Condition (L) if and only if \(L_K(E)\) has no non-minimal primitive idempotents. A refining of the so-called Reduction Theorem is achieved and is used to prove that \(I(P_c(E))\), the ideal of the vertices which are base of cycles without exits of the graph \(E\), is a ring isomorphism invariant for arbitrary Leavitt path algebras. The authors also determine its structure in any case. To achieve the main results, two different tools are developed: the study of primitive idempotents and the notion of dual graph. For the result related to \(I(P_c(E))\), a combination of both tools is required.
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16S88
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 6387824
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: graph \(C^*\)-algebras
zbMATH Keywords: condition (L)
zbMATH Keywords: condition (NE)
zbMATH Keywords: primitive idempotents
zbMATH Keywords: infinite idempotents
zbMATH Keywords: dual graphs
reviewed by: Candido Martín González
MaRDI profile type: Publication
full work available at URL: forum-2011-0134
OpenAlex ID: W2319565084
DOI: 10.1515/FORUM-2011-0134
description: scientific article; zbMATH DE number 6387824