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**

**Example text**

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).

C. Stroppel. Category O: Gradings and translation functors. J. Algebra 268 (2003), 301–326. C. Stroppel. Categorification of the Temperley–Lieb category, tangles, and cobordisms via projective functors. Duke Math. J. 126 (2005), 547–596. C. Stroppel. TQFT with corners and tilting functors in the Kac–Moody case. RT/0605103. J. Sussan. In preparation, 2005. B. Zhu. On characteristic modules of graded quasi-hereditary algebras. Comm. Algebra 32 (2004), 2919–2928.

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 .