Selecta Mathematica

, 12:379 | Cite as

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

  • Igor Frenkel
  • Mikhail Khovanov
  • Catharina Stroppel
Article

Abstract.

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 20G42 17B10 Secondary 14M15 16G10 

Keywords.

Categorification quantum groups Lie algebras canonical bases flag 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

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