International Journal of Mathematics and Mathematical Sciences

Categorification in Representation Theory

Publishing date
15 Sep 2010
Submission deadline
15 Mar 2010

Lead Editor

1Department of Mathematics and Statistics, University of Ottawa, ON, Canada

2Department of Mathematics, Columbia University, New York, NY, USA

3Department of Mathematics, Stanford University, Palo Alto, CA, USA

Categorification in Representation Theory


The term “categorification” was introduced by L. Crane and I. Frenkel to describe the process of realizing certain algebraic structures as shadows of richer higher ones. Their original motivation was to combinatorially understand geometric constructions of quantum groups in order to produce four-dimensional topological quantum field theories, replacing the algebraic structure of a quantum group with a categorical analog, or “categorified quantum group.” In the past 15 years, it has become increasingly clear that categorification is actually a broad mathematical phenomenon with applications extending far beyond these original considerations. Approaches to categorification vary but generally involve replacing set-theoretic statements by their category-theoretic analogues. Sets become categories, functions become functors, and equations become natural isomorphisms. In many cases, this process leads to the appearance of structures not previously observed. Representation theory provides an especially fertile ground for categorification. The development of categorification in representation theory owes much to the geometric methods which hint at the existence of higher mathematical structure. At their best, categorifications in representation theory improve our understanding of the original algebraic structure, explaining positivity and integrality properties, canonical bases, symmetries, and nondegenerate bilinear forms. Important examples of categorification in representation theory include the geometric categorifications of Ginzburg, Lusztig and Nakajima, which realize quantum groups and their representations in a geometric framework, and Khovanov's categorification of the Jones polynomial. This special issue will focus on categorification in the context of representation theory, in the spirit of these important examples. We invite authors to submit original research articles as well as exceptional review articles. Topics to be considered include, but are not limited to:

  • Geometric categorifications, including geometric realizations of crystals, representations of quantum groups, braid group actions, and derived equivalences
  • Combinatorial categorifications of quantum groups, Hecke algebras, cluster algebras, and relations between combinatorial constructions and geometric categorifications
  • Diagrammatic categorifications, especially diagrammatic interpretations of geometric categorifications
  • Categorified link invariants with representation theoretic origins

Before submission authors should carefully read over the journal's Author Guidelines, which are located at Articles published in this special issue do not require any Article Processing Charges. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at according to the following timetable:

International Journal of Mathematics and Mathematical Sciences
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