Download PDF by A. Jaffe (Chief Editor): Communications in Mathematical Physics - Volume 205

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Phys. Lett. : The classical and quantum 6j-symbols. Math. Notes Princeton, NJ: Princeton University Press, 1995 [CP] Chari, V. : Quasi-Hopf Algebras. Leningrad Math. J. , Frenkel, I. : Lectures on representation theory and Knizhnik– Zamolodchikov equations. Providence, RI: AMS, 1998 [EK] Etingof, P. : Representation-theoretic proof of Macdonald inner product and symmetry identities. Comp. Math. 102, 179–202 (1996) [ES1] Etingof, P. : Algebraic integrability of Schrödinger operators and representations of Lie algebras.

Varchenko Let I ⊂ Arat,R be the ideal generated by relations (33). Consider the quotient Arat,R /I and the homomorphism ϕ¯ : Arat,R /I → Fq (G). One can prove as for GL(N ) that the homomorphism ϕ¯ is an isomorphism for q = 1 and for generic q if G = SP (2N ), and has kernel generated by D = 1 if G = SO(N ). Remark. If G = SO(N), then it is natural to denote the quotient Arat,R /I by Fq (O(N )) Remark. If q = 1, then in the limit γ → 0 we have J = 1. In this case relations (33) take the form L = (T ⊗ 1)(L−1 )t1 (T −1 ⊗ 1), which is the defining relation for the orthogonal and symplectic groups.

Theorem 45. I. This construction defines a tensor functor F : O0 (G, q) → Repf (Fq ) with a tensor structure ˜ (U ) → F (W ⊗ U ), JW,U : F (W )⊗F where JW,U is defined in (1). II. For generic q the map Hom O0 (G,q) (W, U ) → Hom Repf (Fq ) (F (W ), F (U )) defined by F is an isomorphism. Thus, F defines a tensor equivalence of O0 (G, q) onto a full subcategory of Repf (Fq ). Remark 1. e. that any object of Repf (Fq ) is in the image of F . Remark 2. We see that the representation category of Fq (G) is essentially the same as for Uq (g).

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