Categorification of the braid groups

Abstract

We construct a categorification of the braid groups associated with Coxeter groups inside the homotopy category of Soergel's bimodules. Classical actions of braid groups on triangulated categories should come from an action of this monoidal category. We construct representations of this monoidal category on category O of a complex semi-simple Lie algebra and on constructible sheaves over flag varieties. We also consider general constructions of self-equivalences as reflections around another category.