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 team for sl2. the most categorification is received utilizing yes Harish-Chandra bimodules for the complicated Lie algebra gln. For the specific case of straightforward modules we clearly deduce a categorification through modules over the cohomology ring of convinced flag forms. additional geometric categorifications and the relation to Steinberg forms are discussed.We additionally provide a express model of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) general bases when it comes to projective, tilting, regular and straightforward Harish-Chandra bimodules.

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For any 0 ≤ i ≤ n let B i = Func(W/Wi ) be the algebra of complex-valued functions on the (finite) set W/Wi . Similarly, for 0 ≤ i, i + 1 ≤ n let B i,i+1 = Func(W/Wi,i+1 ) be the algebra of functions on W/Wi,i+1 . e. ew (x) = (i) δw,x . In fact, the ew , w ∈ W/Wi , form a complete set of primitive, pairwise orthogonal, idempotents. The algebra B i is semisimple with simple (projective) (i) i modules Sw = B i ew . On the other hand, B i,i+1 is both a B i -module and a B i+1 module as follows: Because Wi,i+1 is a subgroup of Wi and of Wi+1 , we have surjections πi : W/Wi,i+1 → W/Wi and πi+1 : W/Wi,i+1 → W/Wi+1 .

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### A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products by Igor Frenkel, Mikhail Khovanov, Catharina Stroppel

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