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 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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The formula = is obvious from the bilinearity of the form. The proposition follows. 6. 3 generalises to arbitrary tensor products Vd such that the standard basis corresponds to standard modules, the dual standard basis to dual standard modules, the canonical basis to tilting modules and the dual canonical bases to simple modules. 6 is also true in the graded setup. 3. Categorification dictionary Quantum sl2 and its representations ring Z[q, q −1 ] Functors and categories category C-Vect of graded vector spaces multiplication by q grading shift up by 1 representation Vn n category C ∼ = i=0 C i -gmod, where C i is the cohomology ring of a Grassmannian weight spaces of Vn semilinear form < , > : Vn × Vn → C(q) canonical basis of Vn dual canonical basis of Vn representation V1⊗n weight spaces of V1⊗n standard basis of V1⊗n summands of C ∼ = n i=0 C i -gmod bifunctor HomC (∗, ∗) indecomposable projective modules in n i i=0 C -gmod simple modules n i=0 C i -gmod n graded version of i=0 i;n O, certain blocks of the category O(gln ) blocks in the graded version of n i=0 i;n O standard (= Verma) modules Vol.

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.

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