Selecta Mathematica

, 12:379

A categorification of finite-dimensional irreducible representations of quantum \({\mathfrak{sl}_2}\) and their tensor products

  • Igor Frenkel
  • Mikhail Khovanov
  • Catharina Stroppel

DOI: 10.1007/s00029-007-0031-y

Cite this article as:
Frenkel, I., Khovanov, M. & Stroppel, C. Sel. math., New ser. (2007) 12: 379. doi:10.1007/s00029-007-0031-y


The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for \({\mathfrak{sl}_2}\) . The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra \({\mathfrak{gl}_n}\) . For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur–Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.

Mathematics Subject Classification (2000).

Primary 20G4217B10Secondary 14M1516G10


Categorificationquantum groupsLie algebrascanonical basesflag varieties

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Igor Frenkel
    • 1
  • Mikhail Khovanov
    • 2
  • Catharina Stroppel
    • 3
  1. 1.Department of MathematicsYale UniversityNew HavenUSA
  2. 2.Department of MathematicsColumbia UniversityNew YorkUSA
  3. 3.Department of MathematicsUniversity of GlasgowGlasgowUnited Kingdom