Sincere stable tubes
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)].
scientific article; zbMATH DE number 1539375
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