A categorification of finite-dimensional irreducible representations of quantum $${\mathfrak{sl}\_2}$$ and their tensor products

The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for sl2. The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra gln. 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.