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By Igor Frenkel, Mikhail Khovanov, Catharina Stroppel

The aim of this paper is to check categorifications of tensor items of finite-dimensional modules for the quantum staff for sl2. the most categorification is acquired utilizing sure Harish-Chandra bimodules for the advanced Lie algebra gln. For the specific case of straightforward modules we evidently deduce a categorification through modules over the cohomology ring of convinced flag kinds. extra geometric categorifications and the relation to Steinberg kinds are discussed.We additionally provide a specific model of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) common bases when it comes to projective, tilting, usual and straightforward Harish-Chandra bimodules.

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Extra info for A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products

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Bernstein, I. Frenkel, and M. Khovanov. A categorification of the Temperley–Lieb algebra and Schur quotients of U (sl2 ) via projective and Zuckerman functors. Selecta Math. ) 5 (1999), 199–241. [BGG76] I. N. Bernˇste˘ın, I. M. Gel’fand, and S. I. Gel’fand. A certain category of g-modules. Funktsional. Anal. i Prilozhen. 10 (1976), no. 2, 1–8 (in Russian). [BG80] J. N. Bernstein and S. I. Gel’fand. Tensor products of finite- and infinitedimensional representations of semisimple Lie algebras. Compos.

C. Stroppel was supported by CAALT and EPSRC grant 32199. References [AJS94] H. H. Andersen, J. C. Jantzen, and W. Soergel. Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p. Ast´erisque 220 (1994). [AL03] H. H. Andersen and N. Lauritzen. Twisted Verma modules. In: Studies in Memory of Issai Schur, Progr. Math. 210, Birkh¨ auser, 2003, 1–26. [AS03] H. H. Andersen and C. Stroppel. Twisting functors on O. Represent. Theory 7 (2003), 681–699.

Funktsional. Anal. i Prilozhen. 10 (1976), no. 2, 1–8 (in Russian). [BG80] J. N. Bernstein and S. I. Gel’fand. Tensor products of finite- and infinitedimensional representations of semisimple Lie algebras. Compos. Math. 41 (1980), 245–285. [BB85] W. -L. Brylinski. Differential operators on homogeneous spaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals. Invent. Math. 80 (1985), 1–68. [CP94] V. Chari and A. Pressley. A Guide to Quantum Groups. Cambridge Univ. Press, 1994.

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