Maps between representations of zero-relation algebras
type: Item
instance of: scholarly article
title: Maps between representations of zero-relation algebras
zbMATH Open document ID: 0685.16018
DOI: 10.1016/0021-8693(89)90304-9
author: William Crawley-Boevey
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \(A=kQ/I\) be a zero-relation \(k\)-algebra over a field \(k\), \(R=k\widetilde Q/\widetilde I\) the universal Galois covering of \(A\) and \(F_\lambda:\text{mod\,}R\to\text{mod\,}A\) the associated push-down functor between the categories of finite-dimensional right modules. Then \(\widetilde Q\) is a locally finite tree and \(\widetilde I\) is generated by paths in \(\widetilde Q\). For a finite subtree \(S\) of \(\widetilde Q\) which does not contain subpaths from \(\widetilde I\), denote by \(V_ S\) the unique \(R\)-module (\(k\)-representation of \((\widetilde Q,\widetilde I)\)) which is one dimensional at each vertex of \(S\) and in which the linear maps corresponding to each arrow of \(S\) are the identity. The modules of the form \(F_\lambda V_S\) are called tree modules over \(A\). The author describes a basis for the space of homomorphisms between two tree modules.
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16D80
zbMATH DE Number: 4122164
zbMATH Keywords: quivers
zbMATH Keywords: zero-relation algebras
zbMATH Keywords: universal Galois coverings
zbMATH Keywords: push-down functor
zbMATH Keywords: categories of finite-dimensional right modules
zbMATH Keywords: locally finite trees
zbMATH Keywords: representations
zbMATH Keywords: tree modules
reviewed by: Andrzej Skowroński
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90304-9
OpenAlex ID: W2091617216
cites work: Covering spaces in representation-theory
cites work: Auslander-reiten sequences with few middle terms and applications to string algebrass
cites work: Galois coverings of representation-infinite algebras
cites work: Q3937542
cites work: Q4744409
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description: scientific article; zbMATH DE number 4122164