Tame biserial algebras
type: Item
instance of: scholarly article
title: Tame biserial algebras
zbMATH Open document ID: 0567.16017
DOI: 10.1016/0021-8693(85)90119-X
author: Burkhard Wald
author: Josef Waschbüsch
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: A basic connected finite dimensional algebra \(A\) over an algebraically closed field \(K\) is called special biserial if \(A\) is isomorphic to a bounden quiver algebra \(KQ/I\) where \((Q,I)\) satisfies the following two conditions: (1) the number of arrows starting, respectively ending, at a fixed vertex of \(Q\) is bounded by 2; (2) for any arrow \(\alpha\) of \(Q\) there is at most one arrow \(\gamma\) and at most one arrow \(\sigma\) such that \(\alpha\gamma\) and \(\sigma\alpha\) are not in \(I\). It was shown by \textit{A. Skowroński} and \textit{J. Waschbüsch} [J. Reine Angew. Math. 345, 172--181 (1983; Zbl 0511.16021)] that the class of representation-finite special biserial algebras coincides with the class of representation-finite algebras \(A\) with \(\beta(A)\leq 2\), where \(\beta (A)\) records the largest possible number of non-projective-injective indecomposable direct summands in the middle terms of Auslander-Reiten sequences over \(A\). Here, using combinatorial Gelfand-Ponomarev methods [\textit{I. M. Gel'fand} and \textit{V. A. Ponomarev}, Usp. Mat. Nauk 23, No. 2(140), 3--60 (1968; Zbl 0236.22012)], one proves that any special biserial algebra \(A\) is tame and \(\beta(A)\leq 2\). An alternative proof of this result using new covering techniques, has been recently obtained by the reviewer and \textit{P. Dowber} in the paper ``Galois coverings of representation-infinite algebras'' [Comment. Math. Helv. 62, 311--337 (1987; Zbl 0628.16019)].
reviewed by: Andrzej Skowroński
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16D80
zbMATH DE Number: 3904786
zbMATH Keywords: basic connected finite dimensional algebras
zbMATH Keywords: bounden quiver algebras
zbMATH Keywords: representation-finite special biserial algebras
zbMATH Keywords: non-projective-injective indecomposable direct summands
zbMATH Keywords: Auslander-Reiten sequences
MaRDI profile type: Publication
cites work: Representation theory of artin algebras iii almost split sequences
cites work: Representation theory of artin algebras IV1)Invariants given by almost split sequences
cites work: Q3901625
cites work: The indecomposable modular representations of certain groups with dihedral Sylow subgroup
cites work: Q3889229
cites work: INDECOMPOSABLE REPRESENTATIONS OF THE LORENTZ GROUP
cites work: Indecomposable modules for finite groups
cites work: Q3934560
cites work: Almost Split Sequences for Group Algebras of Finite Representation Type
cites work: The indecomposable representations of the dihedral 2-groups
cites work: Q3900207
cites work: Q4749976
cites work: Eine Klasse von Quasi-Frobenius-Algebren und ihre unzerlegbaren Moduln
full work available at URL: 0021-8693(85)90119-x
OpenAlex ID: W2038746329
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description: scientific article; zbMATH DE number 3904786