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 acquired utilizing sure Harish-Chandra bimodules for the advanced Lie algebra gln. For the targeted case of straightforward modules we clearly deduce a categorification through modules over the cohomology ring of definite flag types. extra geometric categorifications and the relation to Steinberg types 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) typical bases by way of projective, tilting, general and straightforward Harish-Chandra bimodules.
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Additional info for A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products
We get an isomorphism of functors d Ei ∼ = Ei d : n n i i C -mod → C -mod. g. 3]) some ki ∈ Z such n n that d Ei ∼ = Ei d ki : i=0 C i -gmod → i=0 C i -gmod. 4 we get isomorphisms of graded vector spaces Ei (Si ) ∼ = C i,i+1 ⊗C i C −n + i + 1 n−i+1 ∼ = C 2r − n + i − 1 r=0 ∼ = C n − i − 1 ⊕ C n − i − 3 ⊕ · · · ⊕ C −n + i + 3 ⊕ C −n + i + 1 ∼ = d Ei (Si ). Hence ki = 0 and so d E ∼ = E d. The arguments establishing d F ∼ = F d are analogous. 2. The relations (47) follow directly from the definitions.
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 . f (x) = g(πj (x))f (x) for x ∈ W/Wi,i+1 . The B i ’s are commutative, hence we get a left and a right module structure.