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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Additional info for A categori
cation of the Temperley-Lieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors
The proof for Γi+1 is the same. 2. Equivalences of categories. Let εi be the inclusion functor Ok,n−k → Ok,n−k . Consider a pair of functors i−1 i Γ1i−1 εi : Ok,n−k → Ok,n−k i−1 i Γ1i εi−1 : Ok,n−k → Ok,n−k . These two functors are exact, take generalized Verma modules to generalized Verma modules and the compositions Γ1i−1 εi Γ1i εi−1 and Γ1i εi−1 Γ1i−1 εi are identities on generalized Verma modules. i Theorem 5. Functors Γ1i−1 εi and Γ1i εi−1 are equivalences of categories Ok,n−k i−1 and Ok,n−k .
Bernstein, I. Frenkel and M. Khovanov Sel. , New ser. We next state the parabolic analogue of Conjecture 2. It is convenient to suppress parameter k in the definition of ∩i,n and ∪i,n by summing over k and passing to categories On = ⊕k Ok,n−k . We switch to derived categories and extend functors ∩i,n , ∪i,n and Ui to derived functors ∩i,n : Db (On ) −→ Db (On−2 ) ∪i,n : Db (On ) −→ Db (On+2 ) Ui : Db (On ) −→ Db (On ). 2. To the tangle (a) associate functor ∩i,n given by the formula (61). To the tangle (b) associate functor ∪i,n given by the formula (62).
Let Y be the one-dimensional gl2 ⊕ gln−2 -module with weight n−3 2 (e1 + e2 ) relative to gl2 and −(e1 + · · · + en−2 ) relative to gln−2 . Let p be the maximal parabolic subalgebra of gln that contains gl2 ⊕ gln−2 and the subalgebra of upper triangular matrices. Then Y ⊗ (L2 ⊗ M ) is naturally a p-module with the nilradical of p acting trivially. Now we parabolically induce from p to gln and define ςn (M ) = U (gln ) ⊗U(p) (Y ⊗ L2 ⊗ M ). 1 Let νn be the functor from Ok,n−k to On−1,n−k−1 defined as follows.