A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products

By I. Frenkel, Mikhail Khovanov, C. Stroppel. 2005

103 citations recorded by the source corpus.

The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1aea0b8043d7fd8b4be2b5567a0dd72e09834f3e

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

Categorified Quantum sl(2) and Equivariant Cohomology of Iterated Flag Varieties (2008, background): “It was already known that these iterated flag varieties categorify irreducible representations of Uq(sl2) [1, 4, 7].”
A diagrammatic approach to categorification of quantum groups II (2008, unknown): “This model was given a categorical interpretation by Grojnowski and Lusztig [15] and later reinterpreted, for N = 2, via translation and Zuckerman functors in [6], with various generalizations constructed in [13], [44], [48], and in a very recent striking work [49].”
Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality (2013, unknown): “Theorem D gives a new instance of based categorifications in the context of category O, but now connecting canonical bases of Hecke algebras with canonical bases of quantum symmetric pairs instead of quantum groups as for instance in [BS10], [FKS06], [Sar13], [Web10], see [Maz12] for an overview.”
A categorification of quantum sl(n) (2008, unknown): “This 2-category or its mild modifications is expected to act on parabolic-singular blocks of highest weight categories for slN in the context of categorification of sln representations [7], [10], [31], on derived categories of coherent sheaves on Kronheimer–Nakajima [20] and Nakajima [26] quiver var”
Categorification of (induced) cell modules and the rough structure of generalized Verma modules (2007, unknown): “For completeness we would like to formulate here a categorical version of the Schur–Weyl duality generalising the approach of [21].”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “Categorifications associated with category O were initiated by Bernstein, Frenkel, and Khovanov [3] and were further developed in [66, 24].”