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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

Example text

Xn ∈ Z and ρn is the half-sum of the positive roots of gln . This direct sum is a gln−2 -module in a natural way. Define νn (M ) as the tensor product of this module with the one-dimensional gln−2 -module of weight e1 + · · · + en−2 . 232 J. Bernstein, I. Frenkel and M. Khovanov Sel. , New ser. Proposition 17. Functors ςn and νn are mutually inverse equivalences of cate1 gories Ok−1,n−k−1 and Ok,n−k . We omit the proof as it is quite standard. i , 1 ≤ i ≤ n−1 and Ok−1,n−k−1 are equivalent.

Frenkel and M. Khovanov. Canonical bases in tensor products and graphical calculus for Uq (sl2 ). Duke Math J. 87 (1997), 409–480. B. Frenkel, M. Khovanov and A. Kirillov, Jr. Kazhdan-Lusztig polynomials and canonical basis. Transformation Groups 3 (1998), 321–336. K. M. Green. Monomials and Temperley-Lieb algebras. J. Algebra 190 (1997), 498–517. B. Frenkel and F. Malikov. Annihilating ideals and tilting functors. Preprint qalg/9801065. I. Grojnowski. The coproduct for quantum GLn (1992), Preprint.

Let T i , Ti be translation functors on and off the i-th wall T i :Oµ −→ Oµi Ti : Oµi −→ Oµ . These functors are defined up to an isomorphism by the condition that they are projective functors between Oµ and Oµi and 1. Functor T i takes the Verma module Mµ to the Verma module Mµi . 2. 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 .

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