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Details can be found in [Soe00] and [BGS96]. In [Str03a] it is proved that projective, simple and Verma modules are gradable, and a graded shift is unique up to isomorphism and overall shift in the grading. We take their standard graded lifts to be determined by requiring that the simple head is concentrated in degree 0, and by a slight abuse of notation we will denote them again by L(λ), M (λ) and P (λ). In short P (x·λ), particular notice that we have a decomposition of graded modules Aλ ∼ = x∈ Sn and the idempotent projecting onto P (x · λ) is homogeneous of degree 0.

To define T on morphisms, it suffices to consider elementary pieces of webs. 12) a+b T = a b V(a + b) Φa,b V(a) ⊗ V(b) Chapter 3. 3. 13) define a dense full monoidal functor T : Web → Rep. Proof. First, we have to check that T satisfies the relations defining Web. 2d). 7). 9c). The functors T is dense since, by definition, the objects of Rep are exactly the V(a) for all sequences a of positive integer numbers. We prove now that T is full. 2) the map HomWeb (n, n) → HomRep (V ⊗n , V ⊗n ) induced by T is surjective.

0 and (vηa , Evγa )a = (vηa , [ar ]q −ar+1 −···−a vηa )a = [ar ]q −ar+1 −···−a β1η + · · · + β η β1η , . . , β η . 0 Since βiη = βiγ for i = r while βrη = βrγ + 1 = ar , we have (vηa , Evγa )a [β1η + · · · + β η ] β1η +···+β η +β η +···+β η −a1 −···−ar−1 −ar+1 −···−a r−1 r+1 q q = [a ] r (F vηa , vγa )a [βrη ] = [β1η + · · · + β η ]0 q 1−a1 −···−a , which proves the claim. 4. 21) E=q q 2h1 − 1 E K −1 q2 − 1 then we get an adjuction between F and E . 22) (vηa , vγ♣a )a = 1 if η = γ, 0 otherwise.

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