A categorification of finite-dimensional irreducible by Igor Frenkel, Mikhail Khovanov, Catharina Stroppel

By Igor Frenkel, Mikhail Khovanov, Catharina Stroppel

The aim of this paper is to review categorifications of tensor items of finite-dimensional modules for the quantum team for sl2. the most categorification is received utilizing yes Harish-Chandra bimodules for the advanced Lie algebra gln. For the detailed case of straightforward modules we obviously deduce a categorification through modules over the cohomology ring of convinced flag kinds. additional geometric categorifications and the relation to Steinberg forms are discussed.We additionally supply a specific model of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) regular bases when it comes to projective, tilting, normal and easy Harish-Chandra bimodules.

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Then we proceed to the geometric categorification of general tensor products using certain algebras of functions which generalise the cohomology rings and point towards the Borel–Moore homology of generalised Steinberg varieties. We conclude this section by formulating open problems related to the geometric categorification. 1. From algebraic to geometric categorification The categorification of simple Uq (sl2 )-modules we propose gives rise to a categorification of simple U(sl2 )-modules by forgetting the grading.

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Press, 1994. [CG97] N. Chriss and V. Ginzburg. Representation Theory and Complex Geometry. Birkh¨ auser Boston, Boston, MA, 1997. [CR07] J. Chuang and R. Rouquier. Derived equivalences for symmetric groups and sl2 -categorification. arXiv: mathRT/0407205, to appear in Ann. , 2007. [CPS88] E. Cline, B. Parshall, and L. Scott. Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391 (1988), 85–99. [CI89] D. H. Collingwood and R. S. Irving. A decomposition theorem for certain self-dual modules in the category O.

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