A categori

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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Math. Z. 204 (1990), 209–224. C. Jantzen. Moduln mit einem h¨ ochsten Gewicht. Lect. Notes in Math. 750 (1979). R. Jones. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. 12 (1985), 103–111. H. Kauffman. State models and the Jones polynomial. Topology 26 (1987), 395–407. H. Kauffman and S. Lins. Temperley-Lieb Recoupling Theory and Invariants of 3manifolds. Ann. of Math. Studies 134, Princeton U. Press, Princeton, 1994. M. Khovanov. Graphical calculus, canonical bases and Kazhdan-Lusztig theory.

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. Corollary 6. The categories Ok,n−k Denote by Ξn,i the equivalence of categories i Ξn,i : Ok,n−k −→ Ok−1,n−k−1 given by the composition Ξn,i = νn ◦ Γ11 ◦ ε2 ◦ Γ12 ◦ . . , Ξn,i is the composition of equivalences of categories i Ok,n−k ∼ =✲ i−1 Ok,n−k ∼ =✲ ...

I does not lie on any other walls. Let Oξi be the subcategory of O(gln ) consisting of modules with generalized central character η(ξi ). Let Tµξii and Tξµii be translation functors from Oµi to Oξi and back. Then Tµξii takes the Verma module Mµi to the Verma module Mξi while Tξµii takes Mξi to Psi si+1 µi . Therefore, functor ℘ is isomorphic to the composition Tξµii Tµξii and µ i τi+1 τii+1 ∼ = Id ⊕Tξii Tµξii . 236 J. Bernstein, I. Frenkel and M. Khovanov Sel. , New ser. The category Oξi contains no U (pk )-locally finite modules other than the zero module.

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A categori
cation of the Temperley-Lieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors by Bernstein J.


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