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Mat. Ser. III 32(52) (1997), 179–199. A. Rocha-Caridi. Splitting criteria for g-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite dimensional, irreducible g-module. Trans. AMS 262 (1980), 335–366. Yu. G. Turaev. Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys. 127 (1990), 1–26. G. Turaev. Quantum Invariants of Knots and 3-Manifolds. De Gruyter Studies in Mathematics 18 (1994). V. Zelevinski. Small resolutions of singularities of Schubert varieties.

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 . This is a functor from Oµi to Oµi+1 . 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.

This is very much in line with the factorization of the element Ui of the Temperley-Lieb algebra as the composition ∪i,n−2 ◦ ∩i,n of morphisms ∪i,n−2 and ∩i,n of the Temperley-Lieb category. We now offer the reader a conjecture on realizing the Temperley-Lieb category via functors between parabolic categories Ok,n−k . Let ζn : O1k,n−k ∼ =✲ Ok−1,n−k−1 (60) be the Enright-Shelton equivalence of categories. Introduce functors ∩i,n :Ok,n−k −→ Ok−1,n−k−1 ∪i,n :Ok,n−k −→ Ok+1,n+1−k given by i−2 ∩i,n = ζn ◦ τ21 ◦ τ32 ◦ · · · ◦ τi−1 ◦ τii−1 ◦ T i ∪i,n = Ti ◦ i τi−1 ◦ i−1 τi−2 ···◦ τ23 ◦ τ12 ◦ −1 ζn+2 .

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