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.

Cited by in corpus (24)

• Diagrammatics for Soergel categories
• Legendrian knots and constructible sheaves
• Higher-Dimensional Algebra VII: Groupoidification
• A geometric model for Hochschild homology of Soergel bimodules
• Hilbert schemes and $y$-ification of Khovanov-Rozansky homology
• TQFT with corners and tilting functors in the Kac-Moody case
• Annular Evaluation and Link Homology
• Sheaf quantization in Weinstein symplectic manifolds
• Braid group actions via categorified Heisenberg complexes
• Simple transitive 2-representations for two non-fiat 2-categories of projective functors
• Ozsvath-Szabo bordered algebras and subquotients of category O
• ${\mathbb{P}}^n$-functors
• The Hecke Bicategory
• On some $p$-differential graded link homologies
• Equivariant motives and geometric representation theory. (with an appendix by F. Hörmann and M. Wendt)
• Additive versus abelian 2-representations of fiat 2-categories
• Hochschild homology of certain Soergel bimodules
• Higher zigzag algebras
• Braid groups and Kleinian singularities
• Homotopical presentations of braid groups via reduced lifts
• Categorification of the Catalan monoid
• A note on Hecke patterns in Category O
• Symplectomorphism group of $T^{\ast }(G_{\mathbb{C}}/B)$ and the braid group I: a homotopy equivalence for $G_{\mathbb{C}}=S{L}_3(\mathbb{C})$
• Algebra and geometry of link homology