Auslander-Reiten components for clans
type: Item
instance of: scholarly article
title: Auslander-Reiten components for clans
zbMATH Open document ID: 0959.16013
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2001-01-31T00:00:00Z
review text: It is proved that if the repetitive algebra \(\widehat\Lambda\) of a gentle algebra \(\Lambda\) is of non-polynomial growth, then the stable Auslander-Reiten quiver of \(\widehat\Lambda\) contains a finite number of components of type \(\mathbb{Z}\mathbb{A}_\infty\). Then, if the gentle algebra \(\Lambda\) has finite global dimension, \(\Lambda\) is derived equivalent to a hereditary algebra of type \(\mathbb{A}\) or \(\widetilde\mathbb{A}\), if and only if the stable Auslander-Reiten quiver of \(\widehat\Lambda\) does not contain components of type \(\mathbb{Z}\mathbb{A}_\infty\) or \(\mathbb{Z}\mathbb{A}_\infty^\infty\). These results extend to a more general situation, that is, when \(\widehat\Lambda\) is quasi-clanish. The notion of a quasi-clanish algebra generalizes the notion of clanish algebras and of special biserial algebras. Also, they study certain skew group algebras. All these results are independent of the base field \(k\).
reviewed by: Gladys Chalom
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 1444141
zbMATH Keywords: quasi-clanish algebras
zbMATH Keywords: stable Auslander-Reiten quivers
zbMATH Keywords: repetitive algebras
zbMATH Keywords: gentle algebras
zbMATH Keywords: hereditary algebras
author: Christof Geiss
author: José A. de la Peña
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1444141