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By A. Jaffe (Chief Editor)

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We consider two types of polarized Hopf algebras. I. Let g be a simple Lie algebra, αi , i = 1, . . , r, simple roots, ei , fi , hi the corresponding Chevalley generators, g = n+ ⊕ h ⊕ n− the polar decomposition. Consider the polarized Hopf algebra A = U (g) with the Z-grading and polarizations defined by deg(ei ) = 1, deg(fi ) = −1, deg(hi ) = 0, A+ = U (n+ ), A− = U (n− ), A0 = U (h), A 0 = U (b+ ), A 0 = U (b− ), where b± = h ⊕ n± . In this ˆ A. case T = SpecA0 = h∗ . Fix on A the quasitriangular structure R = 1 ∈ A ⊗ II.

Define a bilinear from Bλ,V : V ⊗ ∗ V → C. For homogeneous v ∈ V , v ∗ ∈ ∗ V , with weights λv + λv ∗ = 0 set Bλ,V (v, v ∗ ) = 0. If λv + λv ∗ = 0, then define Bλ,V (v, v ∗ ) by the property (1⊗ < , >V ⊗∗ V ) ◦ v,v ∗ λ = Bλ,V (v, v ∗ ) IdMλ . ∗ is an intertwiner, hence it has the form: Const IdMλ . Notice that (1⊗ < , >V ⊗∗ V )◦ v,v λ The bilinear form Bλ,V is called the two point function. Lemma 27. Bλ,V (v, v ∗ ) =< v, K (λ)v ∗ >, where K (λ) is defined in (30). ∗ = Proof. Since v,v λ < v, K (λ)v ∗ >.

This construction allows us to define a natural embedding Arat,R → Arat,R . Clearly, the element D belongs to the image of the imbedding. SL(N ) Theorem 37. I. For F (SL(N)), the kernel of the epimorphism ϕ : Arat,R → F (SL(N )) of Corollary 34 is generated by the relation D = 1. SL(N ) II. For Fq (SL(N)), the kernel of the epimorphism ϕ : Arat,R → Fq (SL(N )) of Corollary 34 contains the ideal generated by the relation D = 1. Moreover, the kernel is generated by this relation for all q except a countable set.

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