A brief review of abelian categorifications

TL;DR

This paper reviews abelian categorifications of semisimple representations, illustrating how they relate to link homology and providing a simple definition with examples involving symmetric groups and Lie algebra modules.

Contribution

It offers a clear, accessible overview of abelian categorifications, including a simple definition and illustrative examples relevant to representation theory and link homology.

Findings

Provides a simple definition of abelian categorification

Includes examples with symmetric groups and Lie algebra modules

Connects categorification to link homology extensions

Abstract

This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various representations of the symmetric group and its Hecke algebra via highest weight categories of modules over the Lie algebra sl(n). The review is intended to give non-experts in representation theory who are familiar with the topological aspects of categorification (lifting quantum link invariants to homology theories) an idea for the sort of categories that appear when link homology is extended to tangles.