A categori cation of the Temperley-Lieb algebra and Schur by Bernstein J.

By Bernstein J.

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cation of the Temperley-Lieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors PDF

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cation of the Temperley-Lieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors

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1. On and off the wall translation functors in parabolic categories. Let µ be an integral dominant regular weight and µi , i = 1, . . , n − 1 an integral dominant subregular weight on the i-th wall. Let Oµ and Oµi be the subcategories of O(gln ) of modules with generalized central characters η(µ) and η(µi ). Then Oµ is a regular block of O(gln ) and Oµi is a subregular block of O(gln ). Verma modules Mµ and Mµi with highest weights µ − ρ and µi − ρ are dominant Verma modules in the corresponding categories.

5 (1999) Categorification of Temperley-Lieb algebra 233 We next state a conjecture on a functor realization of the category of tangles in R3 . Consider the following 3 elementary tangles (a) 1 2 i-1 i n (b) 1 2 i-1 i n (c) 1 2 i i+1 n Every tangle in 3-space can be presented as a concatenation of these elementary tangles. We associate to these 3 types of tangles the following functors: To the tangle (a) associate functor ∩i,n given by the formula (58). To the tangle (b) associate functor ∪i,n given by the formula (59).

An−2 )] [RΓi (M (a1 . . ai−1 01ai . . an−2 ))] = − [Mi (a1 . . an−2 )] 230 J. Bernstein, I. Frenkel and M. Khovanov Sel. , New ser. where, we recall Mi (a1 . . an−2 ) = M (a1 . . ai−1 10ai . . an−2 )/M (a1 . . ai−1 01ai . . an−2 ). Therefore, [RΓi ◦ εi (Mi (a1 . . an−2 ))] = [Mi (a1 . . an−2 )] ⊕ [Mi (a1 . . an−2 )] and [(RΓi ◦ εi )M ] = [M ] ⊕ [M ] i for any M ∈ Ok,n−k . On the other hand, [(RΓi ◦ εi )M ] = [Γ0i εi M ] − [Γ1i εi M ] + [Γ2i εi M ] = [M ] − [Γ1i εi M ] + [M ]. i Thus, [Γ1i εi M ] = 0 for any M ∈ Ok,n−k and, hence, Γ1i εi M = 0 for any M ∈ i Ok,n−k .

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