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

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 team for sl2. the most categorification is received utilizing yes Harish-Chandra bimodules for the complicated Lie algebra gln. For the specific case of straightforward modules we clearly deduce a categorification through modules over the cohomology ring of convinced flag forms. additional geometric categorifications and the relation to Steinberg forms are discussed.We additionally provide a express 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, regular and straightforward Harish-Chandra bimodules.

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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. 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 .

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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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