Braiding via geometric Lie algebra actions

TL;DR

This paper introduces geometric categorical Lie algebra actions on derived categories, demonstrating they induce braid group actions, with applications to cotangent bundles of partial flag varieties.

Contribution

It establishes that geometric categorical Lie algebra actions lead to braid group actions and connects strong categorical actions to braid group representations.

Findings

Geometric categorical Lie algebra actions induce braid group actions.

Strong categorical actions also produce braid group actions.

Constructed explicit braid group actions on derived categories of cotangent bundles.

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.