By Bernstein J.

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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. I. Grojnowski and G. Lusztig. On bases of irreducible representations of quantum GLn . In Kazhdan-Lusztig theory and related topics, Chicago, IL, 1989, Contemp.

From Proposition 15 Cij M = M 0 if j = 2 if j = 2. Therefore, Ci = Id[−2] and we have the isomorphism (53). 3. A realization of the Temperley-Lieb algebra by functors. Define functors Vi , 1 ≤ i ≤ n − 1 from Db (Ok,n−k ) to Db (Ok,n−k ) by Vi = εi ◦ RΓi [1]. (54) Vol. 5 (1999) Categorification of Temperley-Lieb algebra 231 Theorem 6. There are natural equivalences of functors (Vi )2 ∼ = Vi [−1] ⊕ Vi [1] Vi Vj ∼ = Vj Vi for |i − j| > 1 Vi Vi±1 Vi ∼ = Vi . (55) (56) (57) Proof. Isomorphism (55) follows from Proposition 16.

Where, we recall Mi (a1 . . an−2 ) = M (a1 . . ai−1 10ai . . an−2 )/M (a1 . . ai−1 01ai . . an−2 ). Therefore, [RΓi ◦ εi (Mi (a1 . . an−2 ))] = [Mi (a1 . . an−2 )] ⊕ [Mi (a1 . . an−2 )] and [(RΓi ◦ εi )M ] = [M ] ⊕ [M ] i for any M ∈ Ok,n−k . On the other hand, [(RΓi ◦ εi )M ] = [Γ0i εi M ] − [Γ1i εi M ] + [Γ2i εi M ] = [M ] − [Γ1i εi M ] + [M ]. i Thus, [Γ1i εi M ] = 0 for any M ∈ Ok,n−k and, hence, Γ1i εi M = 0 for any M ∈ i Ok,n−k . i Proposition 16. Restricting to the subcategory Db (Ok,n−k ), we have an equivalence of functors RΓi ◦ εi ∼ (53) = Id ⊕ Id[−2].