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

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