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Representations of wild quivers. (English) Zbl 0863.16010

Bautista, Raymundo (ed.) et al., Representation theory of algebras and related topics. Proceedings of the workshop, Mexico City, Mexico, August 16-20, 1994. Providence, RI: American Mathematical Society. CMS Conf. Proc. 19, 65-107 (1996).
This is a brilliantly written version of a set of lectures on wild quiver algebras given by the author in various places. The lectures were largely self-contained, including proofs of most of the key facts, and so are these notes. The author studies the representation theory of the path algebras of wild quivers over an arbitrary commutative field, but the contents of this paper should be of interest for any specialist in the representation theory of Artin algebras.
The first three sections are introductory, with virtually no proofs: the author recalls a few facts on path algebras, Auslander-Reiten theory and Coxeter transformations, as can be found, for instance, in C. M. Ringel’s book “Tame algebras and integral quadratic forms” [Lect. Notes Math. 1099 (1984; Zbl 0546.16013)]. He then studies the morphisms between regular modules and elementary modules. This leads to the structure theorem for the regular components of the Auslander-Reiten quiver of a wild tilted algebra, and to the proof that exceptional modules (which the author calls stones) are bricks. The author then turns to torsion pairs and tilting theory and, after some considerations on standard wings, proves the existence of regular tilting modules for wild quiver algebras with at least three simple modules. All statements in this part are proven in detail, with short and elegant proofs. The author then reports (without proofs) about exceptional sequences and Ringel’s simplification procedure. He is then able to show that the perpendicular category of a quasi-simple regular stone is equivalent to the module category of a connected wild hereditary algebra. He ends with a report (with some proofs) on recent developments, like the graph of domination, or the bijection between the regular components of two wild quiver algebras.
For the entire collection see [Zbl 0836.00027].

MSC:

16G20 Representations of quivers and partially ordered sets
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16G60 Representation type (finite, tame, wild, etc.) of associative algebras

Citations:

Zbl 0546.16013
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