Preprojective algebras of quivers are important objects in representation theory. In extended Dynkin case, they give non-commutative resolutions of simple singularities of dimension two. Motivated by studying non-commutative resolutions of cDV singularities, I will discuss tilting theory of a contracted preprojective algebra, which is a subalgebra eAe of a preprojective algebra A given by an idempotent e of A. It has a family of tilting modules which correspond bijectively with the chambers in the contracted Tits cone, and mutation of these tilting modules correspond to wall-crossing of chambers. Moreover, they are parametrized by certain double cosets in the Coxeter group W modulo parabolic subgroups, and mutation can be described in terms of these double cosets by using longest elements. I will explain some results and conjectures on the derived equivalence classes of contracted preprojective algebras. This is a joint work with Michael Wemyss.
talk was part of the Birthday Colloquium for Bill Crawley-Boevey on Sept 10th 2020.