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Tame triangular matrix algebras. (English) Zbl 0978.16014

Let \(A\) be a finite dimensional \(k\)-algebra for \(k\) algebraically closed, such that the triangular matrix algebra \(T_2(A)\) is tame. It is known that in this case, \(A\) is of finite representation type and standard. In this paper, the authors describe, in terms of full, convex subcategories of \(\widetilde A\) (the universal Galois covering of \(A\)) all algebras \(A\) such that \(T_2(A)\) is tame. Moreover, there is also a complete classification of these algebras in finite representation type, domestic type, of polynomial growth, and tame type.
As a consequence there is also a classification for the Auslander algebras \({\mathbf A}(A)\). For this purpose, they list the families \(\mathbf W\), \(\mathbf{NPG}\), \(\mathbf{ND}\) and \(\mathbf{IT}\). So, \(T_2(A)\) is tame if and only if \(\widetilde A\) does not contain a finite full convex subcategory of the \(\mathbf W\) family and in a similar way, for the other families. The proof is done by a case by case inspection of all possible finite full convex subcategories of \(T_2(A)\).

MSC:

16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16S50 Endomorphism rings; matrix rings
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16G30 Representations of orders, lattices, algebras over commutative rings
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