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 workforce for sl2. the most categorification is received utilizing yes Harish-Chandra bimodules for the advanced Lie algebra gln. For the specific case of straightforward modules we evidently deduce a categorification through modules over the cohomology ring of yes flag types. extra geometric categorifications and the relation to Steinberg forms 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) general bases when it comes to projective, tilting, average and easy Harish-Chandra bimodules.
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Extra info for A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products
4(ii)] gives n−i−1 [P˜ (1i+1 0n−i )] = ˜ (1i 0n−i−1−k 10k k ]. [M k=0 Hence we finally get ˜ (1i 0n−i ) ∼ EM (45) = P˜ (1i 0n−i−1 1) n−i i n−i i n−i−k k ˜ ˜ and Ψ([EM (1 0 )]) = 10 k ]). On the other hand, we k=0 Ψ([M (1 0 n−i have to calculate △(E)va , where a = 1i 0n−i . We get △(E)va = k=0 qvak , where k i n−i−1+k k a = 10 10 . We get Ψ([EM ]) = EΨ([M ]). The relation Ψ([FM ]) = F Ψ([M ]) follows from analogous calculations. The existence of the desired isomorphism Φ follows. This proves part (a).
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Let again W = Sn with the Young subgroup Sd corresponding to the composition d. Let B d = Func(W/Sd ) be the algebra of (complex-valued) functions on W/Sd . 4) the subalgebras C i , C i,i+1 , C i,i−1 in the coinvariant algebra corresponding to W . The Weyl group W is acting on both B d and C. We set Hdi = (B d ⊗ C)Wi , Hdi,i+1 = (B d ⊗ C)Wi,i+1 , (51) where we take the Wi -invariants with respect to the diagonal action. For any w ∈ Wi \W/Sd , there is an idempotent fw = ew ⊗ 1, where ew (x) = ew (yx) = δw,x for any x ∈ Wi \W/Sd and y ∈ Wi .