Tilted string algebras
type: Item
instance of: scholarly article
title: Tilted string algebras
zbMATH Open document ID: 0962.16009
DOI: 10.1016/S0022-4049(99)00101-2
author: Francois Huard
author: Shiping Liu
published in: Journal of Pure and Applied Algebra
publication date: 2001-06-10T00:00:00Z
review text: In the paper under review the authors give combinatorial criteria for string algebras to be tilted or quasitilted. Recall that a string algebra is an algebra \(A=kQ/I\) where \(k\) is a field and the ideal \(I\) is generated by paths and the quiver \(Q\) has the property that each vertex is the start-point of at most two arrows and the end-point of at most two arrows and, moreover, for each arrow \(\alpha\) there is at most one arrow \(\beta\) such that \(\alpha\beta\not\in I\) and at most one arrow \(\gamma\) such that \(\gamma\alpha\not\in I\) [\textit{M. C. R. Butler} and \textit{C. M. Ringel}, Commun. Algebra 15, 145-179 (1987; Zbl 0612.16013)]. It is shown that a string algebra \(A=kQ/I\) is quasitilted if and only if there is no sectional pair of zero-relations in \((Q,I)\). Moreover, \(A\) is tilted if and only if the following two conditions are satisfied: (1) there is no sectional pair of zero-relations in \((Q,I)\). (2) If \(\Theta\) is a simple cycle of \(Q\) containing no zero-relation, then the arrows attached to \(\Theta\) either all enter \(\Theta\) or all leave \(\Theta\).
reviewed by: Hagen Meltzer
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G60
zbMATH DE Number: 1538663
zbMATH Keywords: tilted algebras
zbMATH Keywords: quasitilted algebras
zbMATH Keywords: string algebras
zbMATH Keywords: quivers
zbMATH Keywords: zero-relations
MaRDI profile type: Publication
cites work: Tilted algebras of type An
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cites work: Tilted gentle algebras
cites work: Maps between tree and band modules
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cites work: Tame algebras and integral quadratic forms
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description: scientific article; zbMATH DE number 1538663