Categorification of the braid groups

TL;DR

This paper develops a categorification of braid groups linked to Coxeter groups within the homotopy category of Soergel bimodules, connecting algebraic and geometric representations.

Contribution

It introduces a new categorification framework for braid groups and constructs their representations on category O and constructible sheaves.

Findings

Constructed a categorification of braid groups in the homotopy category of Soergel bimodules.

Established representations of the categorified braid groups on category O and flag variety sheaves.

Proposed general methods for constructing self-equivalences as reflections.

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.