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Paris algebra seminar

Osamu Iyama, Nagoya

Tilting theory of contracted preprojective algebras and cDV singularities

Organizer's time: 2020-07-13 14:00 Europe/Paris

Duration: 1 hour

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A preprojective algebra of non-Dynkin type has a family of tilting modules associated with the elements in the corresponding Coxeter group W. This family is useful to study the representation theory of the preprojective algebra and also to categorify cluster algebras. In this talk, 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 associated with the chambers in the contracted Tits cone. They correspond bijectively with certain double cosets in W modulo parabolic subgroups. I will apply these results to classify a certain family of reflexive modules over a cDV singularities R, called maximal modifying (=MM) modules. We construct an injective map from MM R-modules to tilting modules over a contracted preprojective algebra of extended Dynkin type. This is bijective if R has at worst an isolated singularity. We can recover previous results (Burban-I-Keller-Reiten, I-Wemyss) as a very special case. This is joint work with Michael Wemyss.

Submitted by: Bernhard Keller