Sincere stable tubes
type: Item
instance of: scholarly article
title: Sincere stable tubes
zbMATH Open document ID: 0970.16013
published in: Journal of Algebra
publication date: 2001-07-05T00:00:00Z
review text: Let \(\Lambda\) be an indecomposable finite-dimensional algebra over an algebraically closed field. Recall that \(\Lambda\) is called concealed-canonical if it is the endomorphism algebra of a tilting bundle over a weighted projective line [see \textit{H. Lenzing} and the reviewer, in Representation theory of algebras. Seventh international conference, August 22-26, 1994, Cocoyoc, Mexico. American Mathematical Society. CMS Conf. Proc. 18, 455-473 (1996; Zbl 0863.16013)]. The class of concealed-canonical algebras includes the class of tame hereditary, tame concealed and tubular algebras. The main result of the paper under review is the following. If the Auslander-Reiten quiver \(\Gamma\) of \(\Lambda\) has a sincere family of pairwise orthogonal standard stable tubes \(\mathcal C\) such that there is no sequence \(X\to Y\to Z\) of nonzero nonisomorphisms between indecomposable \(\Lambda\)-modules with \(X\) and \(Z\in{\mathcal C}\) and \(Y\notin{\mathcal C}\) then \(\Lambda\) must be concealed-canonical. Furthermore, as a consequence it is shown that if \(\Gamma\) has a sincere standard stable tube \(\mathcal C\) such that there is no sequence \(X\to Y\to X\) of nonzero nonisomorphisms between indecomposable modules, with \(X\in{\mathcal C}\) and \(Y\notin{\mathcal C}\), then \(\Lambda\) is concealed-canonical. The main result improves the characterization of concealed-canonical algebras given by \textit{H. Lenzing} and \textit{J. A. de la Peña} [Proc. Lond. Math. Soc., III. Ser. 78, No. 3, 513-540 (1999; Zbl 1035.16009)] and generalizes a result for tame algebras obtained by \textit{A. Skowroński} [Proc. Am. Math. Soc. 127, No. 7, 1941-1949 (1999; Zbl 0942.16017)].
reviewed by: Hagen Meltzer
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 1539375
zbMATH Keywords: finite-dimensional algebras
zbMATH Keywords: concealed-canonical algebras
zbMATH Keywords: tame hereditary algebras
zbMATH Keywords: tame concealed algebras
zbMATH Keywords: tubular algebras
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: stable tubes
Wikidata QID: Q105554194
author: Idun Reiten
author: Andrzej Skowroński
MaRDI profile type: Publication
full work available at URL: jabr.1999.8386
OpenAlex ID: W2029220672
cites work: Q4318498
cites work: On Tame Algebras and Bocses
cites work: On tubes for blocks of wild type
cites work: Q3795770
cites work: Q3994341
cites work: Q4215067
cites work: Q3890860
cites work: Hereditary categories with tilting object
cites work: Q4288691
cites work: Tilting in abelian categories and quasitilted algebras
cites work: DTr-invariant modules
cites work: Q4888472
cites work: Concealed-Canonical Algebras and Separating Tubular Families
cites work: Quasi-tilted algebras of canonical type
cites work: Degrees of Irreducible Maps and the Shapes of Auslander-Reiten Quivers
cites work: Auslander-Reiten components for concealed-canonical algebras
cites work: Short Chains and Short Cycles of Modules
cites work: Tame algebras and integral quadratic forms
cites work: On the Composition Factors of Periodic Modules
cites work: Q4888484
cites work: On the structure of periodic modules over tame algebras
cites work: The Structure of Stable Components
DOI: 10.1006/JABR.1999.8386
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description: scientific article; zbMATH DE number 1539375