International Journal of Mathematics and Mathematical Sciences

Categorification in Representation Theory


Publishing date
15 Sep 2010
Status
Published
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

Description

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 http://www.hindawi.com/journals/ijmms/guidelines/. 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 http://mts.hindawi.com/ according to the following timetable:


Articles

  • Special Issue
  • - Volume 2010
  • - Article ID 592359
  • - Editorial

Categorification in Representation Theory

Alistair Savage | Aaron Lauda | Anthony Licata
  • Special Issue
  • - Volume 2010
  • - Article ID 978635
  • - Research Article

Diagrammatics for Soergel Categories

Ben Elias | Mikhail Khovanov
  • Special Issue
  • - Volume 2010
  • - Article ID 326247
  • - Research Article

Universal Verma Modules and the Misra-Miwa Fock Space

Arun Ram | Peter Tingley
  • Special Issue
  • - Volume 2010
  • - Article ID 530808
  • - Research Article

A Diagrammatic Temperley-Lieb Categorification

Ben Elias
  • Special Issue
  • - Volume 2010
  • - Article ID 896879
  • - Research Article

Integral HOMFLY-PT and s l ( 𝑛 ) -Link Homology

Daniel Krasner
  • Special Issue
  • - Volume 2010
  • - Article ID 468968
  • - Research Article

The Diagrammatic Soergel Category and sl(N)-Foams, for N4

Marco Mackaay | Pedro Vaz
  • Special Issue
  • - Volume 2010
  • - Article ID 892387
  • - Research Article

The Khovanov-Lauda 2-Category and Categorifications of a Level Two Quantum SL(N) Representation

David Hill | Joshua Sussan
  • Special Issue
  • - Volume 2010
  • - Article ID 612360
  • - Research Article

The Diagrammatic Soergel Category and sl(2) and sl(3) Foams

Pedro Vaz
International Journal of Mathematics and Mathematical Sciences
 Journal metrics
Acceptance rate12%
Submission to final decision38 days
Acceptance to publication29 days
CiteScore0.900
Impact Factor-
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