Tilted algebras (and generalised non-crossing partitions)

Claus Michael Ringel (Bielefeld, SJTU Shanghai, KAU Jeddah)

Abstract. The endomorphism ring B of a tilting module over a hereditary artin algebra A is called a tilted algebra. One can recover the category of B-modules using a (tilting) torsion pair in the category of A-modules. Tilted algebras play an important role in representation theory. They can be used in order to describe the structure of the indecomposable modules for any artin algebra of finite representation type, but they also povide important examples of non-trivial derived equivalences.

In the quiver case, a quite recent result of Ingalls and Thomas asserts that any tilting torsion pairs is linear: it is determined by a linear form f on the Grothendieck group, thus it determines a thick subcategory of the category of A-modules, namely the subcategory of all f-semistable modules. It turns out that these obeservations are true for a general hereditary artin algebra. We will report on these investigations, and outline in which way they furnish a categorification of the lattices of generalised non-crossing partitions for the Weyl groups.


Ringel
Last modified: Sep 5, 2014