[Paper Review] Braid groups and geometric categorical Lie algebra actions

Sabin Cautis, Joel Kamnitzer|arXiv (Cornell University)|Jan 5, 2010

Algebraic structures and combinatorial models3 citations

TL;DR

This paper introduces geometric categorical Lie algebra actions on derived categories of coherent sheaves, proving they induce braid group actions via a general construction. The method applies to both geometric and strong categorical actions, with a key example showing a braid group action on derived categories of coherent sheaves over cotangent bundles to partial flag varieties.

ABSTRACT

We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov-Lauda and Rouquier also lead to braid group actions. As an example, we construct a braid group action on derived categories of coherent sheaves on cotangent bundles to partial flag varieties.

Motivation & Objective

• To define and formalize geometric categorical Lie algebra actions on derived categories of coherent sheaves.
• To establish a general mechanism by which such actions yield braid group actions.
• To extend the result to strong categorical actions in the sense of Khovanov-Lauda and Rouquier.
• To provide a concrete geometric realization of braid group actions via derived categories on cotangent bundles of partial flag varieties.

Proposed method

• Define geometric categorical Lie algebra actions using derived categories of coherent sheaves on algebraic varieties.
• Utilize the structure of the Lie algebra to construct functors that satisfy categorical action axioms.
• Demonstrate that the braid group relations emerge naturally from the categorical action's defining properties.
• Apply the same framework to strong categorical actions, showing isomorphic braid group actions via identical proof techniques.
• Construct explicit functors on derived categories of coherent sheaves over cotangent bundles to partial flag varieties.
• Verify that the resulting functors satisfy braid group relations through categorical commutativity and coherence conditions.

Experimental results

Research questions

• RQ1 How can geometric categorical Lie algebra actions be defined on derived categories of coherent sheaves?
• RQ2 What conditions ensure that such actions induce braid group actions?
• RQ3 To what extent do the same results hold for strong categorical actions in the Khovanov-Lauda-Rouquier framework?
• RQ4 Can a braid group action be explicitly constructed on derived categories of coherent sheaves over cotangent bundles to partial flag varieties?
• RQ5 What is the relationship between the geometric structure of the variety and the resulting braid group action?

Key findings

• Geometric categorical Lie algebra actions on derived categories of coherent sheaves naturally induce braid group actions.
• The same proof technique applies to strong categorical actions in the sense of Khovanov-Lauda and Rouquier, yielding isomorphic braid group actions.
• A concrete braid group action is constructed on the derived category of coherent sheaves over the cotangent bundle of a partial flag variety.
• The construction relies on the categorical structure of the action and the geometric properties of the underlying variety.
• The braid group action arises from the interplay between the Lie algebra's root system and the derived category's triangulated structure.
• The result establishes a new link between geometric representation theory and braid group representations via categorical actions.