Categorification of Lie algebras [d'apres Rouquier, Khovanov-Lauda]

TL;DR

This paper surveys the development of categorification of Lie algebra representations, connecting algebraic, diagrammatic, and geometric approaches, highlighting recent advances and frameworks.

Contribution

It provides a comprehensive overview of the categorification of Lie algebras, integrating algebraic, diagrammatic, and geometric perspectives and recent theoretical frameworks.

Findings

Connections between algebraic and geometric categorification

Frameworks introduced by Rouquier and Khovanov-Lauda

Survey of developments linking quiver varieties and categorical representations

Abstract

Given a vector space with an action of a semi-simple Lie algebra, we can try to "categorify" this representation, which means finding a category where the generators of the Lie algebra act by functors. Such categorical representations arise naturally in geometric representation theory and in modular representation theory of symmetric groups. A framework for studying categorical representations was introduced by Rouquier and Khovanov-Lauda. Their definitions are algebraic/diagrammatic, but are connected to the topology of quiver varieties by the work of Rouquier and Varagnolo-Vasserot. In this paper, we give a survey of the above circle of ideas.