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 staff for sl2. the most categorification is got utilizing yes Harish-Chandra bimodules for the advanced Lie algebra gln. For the targeted case of straightforward modules we obviously deduce a categorification through modules over the cohomology ring of sure flag kinds. 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) common bases when it comes to projective, tilting, regular and easy Harish-Chandra bimodules.
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Extra resources for A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products
4 we get isomorphisms of graded C i -modules Fi+1 Ei (C) ∼ = C i,i+1 ⊗C i+1 C i,i+1 ⊗C i C −n + 1 i n−i−1 ∼ = C 2k + 2l −n + 1 , l=0 k=0 i,i−1 Ei−1 Fi (C) ∼ =C ⊗C i−1 C i,i−1 ⊗C i C −n + 1 n−i i−1 ∼ = C 2l + 2k −n + 1 . l=0 k=0 If n − 2i ≥ 0, then n−i−1 G G FiG = EiG − Ei−1 Fi+1 G (2k + 2i) (1 − n) k=0 i−1 − G (2(n − i) + 2k) 1 − n k=0 422 I. Frenkel, M. Khovanov and C. Stroppel 0 = Sel. , New ser. if n − 2i = 0, n−2i−1 G (2i + 2k) 1 − n if n − 2i > 0, k=0 = [n − 2i] Id . If n − 2i < 0, then i−1 G G G EiG ∼ FiG − Fi+1 Ei−1 = (2(n − i) + 2k) (−n + 1) k=0 n−i−1 − G (2k + 2i) 1 − n k=0 2i−n−1 = G (n − 2i + 1 + 2k) k=0 = [2i − n] Id .
J. Amer. Math. Soc. 3 (1990), 421–445. [Soe92] W. Soergel. The combinatorics of Harish-Chandra bimodules. J. Reine Angew. Math. 429 (1992), 49–74. Vol. 12 (2006) [Soe97] [Str03] [Str05] [Str06] [Sus05] [Zhu04] Categorification of representations of quantum sl2 431 W. Soergel. Kazhdan–Lusztig polynomials and a combinatorics for tilting modules. Represent. Theory 1 (1997), 83–114. 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.
4). We believe that this is a necessary ingredient of a more substantial and general construction which will be considered in the next subsection. Let W = Sn be the symmetric group of order n! with subgroup Wi as above. For any 0 ≤ i ≤ n let B i = Func(W/Wi ) be the algebra of complex-valued functions on the (finite) set W/Wi . Similarly, for 0 ≤ i, i + 1 ≤ n let B i,i+1 = Func(W/Wi,i+1 ) be the algebra of functions on W/Wi,i+1 . e. ew (x) = (i) δw,x . In fact, the ew , w ∈ W/Wi , form a complete set of primitive, pairwise orthogonal, idempotents.
A categorification of finite-dimensional irreducible representations of quantum sl2 and their tensor products by Igor Frenkel, Mikhail Khovanov, Catharina Stroppel
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