Forms of algebras stably equivalent to self-injective algebras of polynomial growth
type: Item
instance of: scholarly article
title: Forms of algebras stably equivalent to self-injective algebras of polynomial growth
zbMATH Open document ID: 0924.16013
author: Zygmunt Pogorzały
published in: Journal of Algebra
publication date: 1999-09-21T00:00:00Z
review text: Let \(K\) be an algebraically closed field. By an algebra we mean a finite dimensional \(K\)-algebra. Two algebras \(A\) and \(B\) are said to be stably equivalent if there is an equivalence of the factor categories \(\underline{\text{mod}}(A)\) and \(\underline{\text{mod}}(B)\) of finite dimensional right modules \(\text{mod}(A)\) and \(\text{mod}(B)\) modulo the ideal consisting of all homomorphisms factorizing through projective modules. One of the main results of the paper is the following theorem. Let \(A\) be a self-injective standard representation-infinite algebra which is of tame representation type and of polynomial growth. If \(B\) is an algebra stably equivalent to \(A\) then: (a) \(B\) is self-injective standard representation-infinite of tame representation type and of polynomial growth, or (b) there is a degeneration algebra \(C\) of \(B\) which is of tame representation type and of polynomial growth, the Auslander-Reiten quivers of \(B\) and \(C\) coincide, and there exists an equivalence of categories \(\underline{\text{mod}}(B)\cong\underline{\text{mod}}(C)\) preserving simple modules.
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1218712
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: path algebras
zbMATH Keywords: tilted algebras
zbMATH Keywords: polynomial growth
zbMATH Keywords: components
zbMATH Keywords: degenerations
zbMATH Keywords: stably equivalent algebras
zbMATH Keywords: standard representation-infinite algebras
zbMATH Keywords: category equivalences
MaRDI profile type: Publication
Wikidata QID: Q122165996
OpenAlex ID: W2031848251
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DOI: 10.1006/JABR.1997.7387
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description: scientific article; zbMATH DE number 1218712