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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Extra resources for A categori
cation of the Temperley-Lieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors
Moreover, Vol. 5 (1999) Categorification of Temperley-Lieb algebra 237 this basis is related (see [FG]) to the Kazhdan-Lusztig basis in the Hecke algebra as well as to Lusztig’s bases in tensor products of Uq (sl2 )-representations (see [FK]). Proposition 18 implies that to an element of this basis, we can associate a projective functor from Ok,n−k to Ok,n−k , which is defined as a suitable composition of functors Ui , 1 ≤ i ≤ n − 1. We conjecture that these compositions of Ui ’s are indecomposable, and, in turn, an indecomposable projective functor from Ok,n−k to Ok,n−k is isomorphic to one of these compositions (compare with Theorem 4).
Similarly, let τii−1 be the functor from Oµi to Oµi−1 given by τii−1 = T i−1 ◦ Ti . Projective functors preserve subcategories of U (pk )-locally finite modules and thus k,n−k functors τii±1 restrict to functors from Oik,n−k to Oi±1 . Our categorification of the Temperley-Lieb algebra by projective functors is based on the following beautiful result of Enright and Shelton: k,n−k Theorem. Functors τii±1 establish equivalences of categories Oik,n−k and Oi±1 . Proof. 1 for a proof of a slightly more general statement.
Functor Ti takes the Verma module Mµi to the projective module Psi µ where si is the transposition (i, i + 1). On the Grothendieck group level, [Ti Mµi ] = [Mµ ] + [Msi µ ]. Let pk be the maximal parabolic subalgebra of gln such that pk ⊃ n+ ⊕ h and the reductive subalgebra of pk is glk ⊕ gln−k . Let Ok,n−k , resp Oik,n−k be the full subcategory of Oµ , resp. Oµi consisting of modules that are locally U (pk )-finite. From now on we fix k between 0 and n. Let τii+1 be the composition of Ti and i+1 T : τii+1 = T i+1 ◦ Ti .
cation of the Temperley-Lieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors by Bernstein J.