, Volume 12, Issue 3-4, pp 379-431
Date: 13 Mar 2007

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.