[Paper Review] Categorification of the braid groups

Raphaël Rouquier|arXiv (Cornell University)|Sep 30, 2004

Geometric and Algebraic Topology8 references48 citations

TL;DR

This paper introduces a categorification of braid groups via a strict monoidal category of complexes of Soergel bimodules, providing a higher categorical structure that lifts actions of braid groups on triangulated categories. It establishes a strong action of the braid group via monoidal functors to endofunctor categories, generalizing classical actions in representation theory and algebraic geometry.

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.

Motivation & Objective

• To construct a strict monoidal category ${\mathcal{B}}_W$ that categorifies the braid group $B_W$ of a Coxeter group $W$.
• To generalize classical actions of braid groups on triangulated categories to a stronger, monoidal categorical level.
• To provide a unified framework for actions of braid groups in representation theory, algebraic geometry, and category theory.
• To establish a genuine action of the braid group on derived categories via functors that lift known isomorphism-level actions.
• To prove that such actions lift to monoidal functors from ${\mathcal{B}}_W$ to the category of endofunctors of a triangulated category.

Proposed method

• Construct a strict monoidal category ${\mathcal{B}}_W$ as a full subcategory of the homotopy category of complexes of Soergel bimodules over a polynomial algebra.
• Use the theory of Frobenius categories and functorial cones in acyclic complexes of projectives to define natural transformations and cones between functors.
• Leverage the geometry of ${\mathbf{P}}^1$-fibrations and kernel transforms to construct self-equivalences categorifying reflections.
• Apply results from Soergel's theory of category $\mathcal{O}$ and Deligne's work on sheaves to lift braid group actions from $K_0$ to the derived category.
• Use adjunctions and base change isomorphisms in derived categories of sheaves to verify compatibility of functors and naturality of transformations.
• Verify commutativity of diagrammatic identities involving pullbacks, pushforwards, and tensor products to establish the monoidal structure.

Experimental results

Research questions

• RQ1 Can the braid group $B_W$ be categorified as a strict monoidal category acting on triangulated categories?
• RQ2 How can classical actions of braid groups on $K_0$ of triangulated categories be lifted to actual functors and natural transformations?
• RQ3 What is the role of Soergel bimodules in realizing a categorified braid group action?
• RQ4 Can the action of the braid group on derived categories of constructible sheaves or category $\mathcal{O}$ be lifted to a monoidal functor from a categorified braid group?
• RQ5 What is the relationship between categorified braid group actions and cohomology of Deligne-Lusztig varieties?

Key findings

• A strict monoidal category ${\mathcal{B}}_W$ is constructed that categorifies (conjecturally) the braid group $B_W$ via complexes of Soergel bimodules.
• The construction provides a self-equivalence of a triangulated category that categorifies a reflection, generalizing the notion of a Dehn twist via spherical objects.
• In category $\mathcal{O}$, the classical action of the braid group on $K_0$ lifts to a genuine action via functors, and further to a monoidal functor from ${\mathcal{B}}_W$ to $\mathrm{Hom}({\mathcal{C}},{\mathcal{C}})$.
• For flag varieties, the action of the braid group on the derived category of constructible sheaves lifts from isomorphism classes to a strict monoidal action via kernel transforms.
• The link with the cohomology ring of the flag variety provides an alternative proof of the braid group action and extends it to the stronger monoidal level.
• The paper establishes a framework for future work on presentations by generators and relations, homological vanishing, and connections to Deligne-Lusztig varieties.