A categorification of finite-dimensional irreducible representations of quantum \mathfraksl2{\mathfrak{sl}_2} and their tensor products |
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Authors: | Igor Frenkel Mikhail Khovanov and Catharina Stroppel |
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Institution: | (1) Department of Mathematics, Yale University, 10 Hillhouse Avenue, PO Box 208283, New Haven, CN 06520-8283, USA;(2) Department of Mathematics, Columbia University, New York, NY 10027, USA;(3) Department of Mathematics, University of Glasgow, 14 University Gardens, Glasgow, G12 8QW, United Kingdom |
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Abstract: | The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group
for
. The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra
. 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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Keywords: | Primary 20G42 17B10 Secondary 14M15 16G10 |
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