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Tilting sets on cylinders. (English) Zbl 0583.16020

This paper gives a comprehensive generalization of the theory of tilting modules and tilting algebras introduced by the author and C. M. Ringel in 1982 [see Trans. Am. Math. Soc. 274, 399-443 (1982; Zbl 0503.16024)]. The main new idea is the definition of the so called root algebras, a concept of combinatorial origin (as it is presented). In the first place, it is given the notion of a tilting set (which, in some sense, replaces the notion of a tilting module) in the mesh category M(\(\Delta)\) of some quotient \({\mathbb{Z}}\Delta /G\) of the translation quiver (\({\mathbb{Z}}\Delta,\tau)\) associated to a Dynkin diagram \(\Delta\). By definition, the root algebras of type \(\Delta\) are the algebras of endomorphisms of the tilting sets \(T: End_{M(\Delta)}T.\)
The main result is the following. Theorem. A cycle-free root algebra is an iterated tilting algebra, and conversely. - Another interesting characterization of the latter is through Cartan matrices. If \(C_ A\) is the Cartan matrix of a simply connected algebra A, in order that there exists a hereditary algebra \(B\) of type \(\Delta\) such that \(C_ A\sim C_ B\) it is necessary and sufficient that A is an iterated tilting algebra of type \(\Delta\).
The paper is very well written and gives enough basic material to be easily read by a non specialist. Several examples are given, together with the complete classification of root algebras of type \(A_ n\) and also of all iterated tilting algebras of type \(E_ 6\).
Reviewer: H.A.Merklen

MSC:

16Gxx Representation theory of associative rings and algebras
16P10 Finite rings and finite-dimensional associative algebras
16S50 Endomorphism rings; matrix rings

Citations:

Zbl 0503.16024
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