×

Auslander algebras of tame representation type. (English) Zbl 0861.16010

Bautista, Raymundo (ed.) et al., Representation theory of algebras. Seventh international conference, August 22-26, 1994, Cocoyoc, Mexico. Providence, RI: American Mathematical Society. CMS Conf. Proc. 18, 475-486 (1996).
Let \(\Lambda\) be a basic and connected finite dimensional \(k\)-algebra, where \(k\) is a fixed algebraically closed field. Suppose there exist finitely many nonisomorphic indecomposable \(\Lambda\)-modules, say \(M_1,\dots,M_t\) (that is \(\Lambda\) is representation-finite). Then \({\mathcal A}(\Lambda)=\text{End}_\Lambda(M_1\oplus\cdots \oplus M_t)\) is called the Auslander algebra of \(\Lambda\). The representation-finite Auslander algebras have been described by Igusa-Platzeck-Todorov-Zacharia. In this paper, the authors give a classification of the Auslander algebras which are of tame type, that is, the indecomposable \({\mathcal A}(\Lambda)\)-modules occur, in each dimension, in a finite number of discrete and a finite number of one-parameter families. Moreover, they also provide lists for the Auslander algebras of polynomial growth and domestic type. There are no proofs of the stated results, the authors refer to an unpublished paper of them [Z. Leszczyński and A. Skowroński, Triangular matrix algebras of tame representation type (in preparation)].
For the entire collection see [Zbl 0837.00015].

MSC:

16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16G30 Representations of orders, lattices, algebras over commutative rings
PDFBibTeX XMLCite