Coherent sheaves and categorical sl(2) actions

TL;DR

This paper introduces geometric categorical sl(2) actions, showing they induce strong categorical sl(2) actions, enabling applications to geometric contexts like derived categories of coherent sheaves on cotangent bundles of Grassmannians.

Contribution

It establishes a link between geometric and strong categorical sl(2) actions, expanding the framework for applying categorical representation theory to geometry.

Findings

Geometric categorical sl(2) actions induce strong categorical sl(2) actions.

Application to derived categories of coherent sheaves on cotangent bundles.

Construction of categorical sl(2) actions on Grassmannian cotangent bundles.

Abstract

We introduce the concept of a geometric categorical sl(2) action and relate it to that of a strong categorical sl(2) action. The latter is a special kind of 2-representation in the sense of Rouquier. The main result is that a geometric categorical sl(2) action induces a strong categorical sl(2) action. This allows one to apply the theory of strong sl(2) actions to various geometric situations. Our main example is the construction of a geometric categorical sl(2) action on the derived category of coherent sheaves on cotangent bundles of Grassmannians.