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

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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.

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Correspondence to Igor Frenkel.

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Frenkel, I., Khovanov, M. & Stroppel, C. A categorification of finite-dimensional irreducible representations of quantum \({\mathfrak{sl}_2}\) and their tensor products. Sel. math., New ser. 12, 379 (2007). https://doi.org/10.1007/s00029-007-0031-y

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  • DOI: https://doi.org/10.1007/s00029-007-0031-y

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