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 bought utilizing definite Harish-Chandra bimodules for the advanced Lie algebra gln. For the specified case of easy modules we certainly deduce a categorification through modules over the cohomology ring of sure flag types. 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 by way of projective, tilting, common and easy Harish-Chandra bimodules.

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Ix )Twist := q l(w0i ) Hw M {(M 0 These bases were defined by Kazhdan, Lusztig and Deodhar (see [KL79], [Deo87]). We use here the notation from [Soe97], except that we have the upper index i to indicate that Mxi ∈ Mix and we also use q instead of v. The bases can be characterised as follows ([KL79] in the notation of [Soe97]): Let Ψ : H → H be the Z-linear involution given by Hx → (Hx−1 )−1 , q → q −1 . It induces an involution on any Mi . Then the M ix are uniquely defined by (i) Ψ(M ix ) = M ix , (ii) M ix − Mxi ∈ y=x vZ[v]Myi .

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