Decomposable Leavitt path algebras for arbitrary graphs.
type: Item
instance of: scholarly article
title: Decomposable Leavitt path algebras for arbitrary graphs.
zbMATH Open document ID: 1333.16008
author: Gonzalo Aranda Pino
author: Alireza Nasr-Isfahani
published in: Forum Mathematicum
publication date: 2015-11-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1603.04985
review text: Given any field \(K\) and a completely arbitrary graph \(E\), the paper studies those Leavitt path algebras \(L_K(E)\) that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. One of the notions involved is that of direct connectedness of a graph. This, takes into account the direction of the edges (that is, for every \(v,w\in E_0\) there exists a path \(\mu\) with \(s(\mu)=v\) and \(r(\mu)=w\)). However the concept of connectedness of a graph, does not (i.e., for every \(v,w\in E_0\) there exists a walk from \(v\) to \(w\)). It is proved that if \(E\) is a directly connected graph, then \(L_K(E)\) is a connected algebra, and that if \(L_K(E)\) is connected, then \(E\) is connected. Furthermore the implications can not be reversed. The paper uses a notion of \(X\)-compatible path, for a hereditary saturated set \(X\). By using this, a Leavitt path algebra \(L_K(E)\) is decomposable if and only if there exist disjoint hereditary and saturated subsets \(X\) and \(Y\) such that there are a finite and nonzero number of paths that are either \(X\)-compatible paths or \(Y\)-compatible paths. The work also studies the indecomposable ``atoms'' and proves that when the algebra decomposes, it actually does so as a direct sum of Leavitt path algebras for some suitable graphs. Under certain finiteness conditions, a unique indecomposable decomposition exists.
reviewed by: Candido Martín González
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: 6505810
zbMATH Keywords: indecomposable Leavitt path algebras
zbMATH Keywords: decomposable Leavitt path algebras
zbMATH Keywords: arbitrary graphs
zbMATH Keywords: direct connectedness of graphs
zbMATH Keywords: direct sums
zbMATH Keywords: indecomposable decompositions
MaRDI profile type: Publication
OpenAlex ID: W1979337118
arXiv ID: 1603.04985
DOI: 10.1515/FORUM-2013-0165
description: scientific article