A categorification of finite-dimensional irreducible by Igor Frenkel, Mikhail Khovanov, Catharina Stroppel

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 received utilizing convinced Harish-Chandra bimodules for the complicated Lie algebra gln. For the precise case of easy modules we clearly deduce a categorification through modules over the cohomology ring of convinced flag types. additional geometric categorifications and the relation to Steinberg types are discussed.We additionally supply a express model of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) average bases when it comes to projective, tilting, typical and straightforward Harish-Chandra bimodules.

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Note that qq−q = for j ∈ Z>0 . Therefore, if −1 k=0 q 2i − n > 0, we use the notation Kn−i − Kn−i −1 Ki − Ki−1 = − q − q −1 q − q −1 2i−n−1   Id 2i − n − 1 − 2k = k=0   0 if 2i − n > 0, (46) if 2i − n = 0, (as endofunctors of ni=0 C i -gmod). 2 (The categorification of Vn ). Fix n ∈ Z>0 . (a) The functors E, F and K, K −1 satisfy the relations KE = 2 EK, KF = −2 F K, KK −1 = Id = K −1 K, (47) 420 I. Frenkel, M. Khovanov and C. Stroppel Ki − Ki−1 Ei−1 Fi ∼ = Fi+1 Ei ⊕ q − q −1 Ki − Ki−1 Fi+1 Ei ∼ = Ei−1 Fi ⊕ − q − q −1 Sel.

Let n B i -mod . Cfunc := i=0 For technical reasons, if i > n or i < 0, let B i -mod denote the category consisting of the zero C-module. We define the following endofunctors of B: Vol. 12 (2006) Categorification of representations of quantum sl2 425 n • Efunc = i=0 Ei , where Ei : B i -mod → B i+1 -mod is the functor B i,i+1 ⊗B i • if i < n and the zero functor otherwise. • Ffunc = ni=0 Fi , where Fi : B i -mod → B i−1 -mod is the functor B i,i−1 ⊗B i • if i > 0 and the zero functor otherwise.

Ii) va✸ − va ∈ b=a q −1 Z[q −1 ]vb . Given the canonical basis, the dual canonical basis is defined by n b va✸ , v♥ δai ,bn−i+1 . ) On the other hand, the permutation module Mi also has several distinguished C[q, q −1 ]-bases, namely • • • • • the the the the the standard basis {Mxi = 1 ⊗ Hx ⊗ 1 | x ∈ W i }, (positive) self-dual basis {M ix | x ∈ W i }, ˜ ix | x ∈ W i }, (negative) self-dual basis {M i “twisted” standard basis {(Mxi )Twist := q l(w0 ) Hw0 Mxi | x ∈ W i }, “twisted” positive self-dual basis i {(M ix )Twist := q l(w0 ) Hw0 M ix | x ∈ W i }, • the “twisted” negative self-dual basis ˜ ix | x ∈ W i }.

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