Quantization of Lie bialgebras, IV

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Abstract: This paper is a continuation of "Quantization of Lie bialgebras, III" (q-alg/9610030, revised version). In QLB-III, we introduced the Hopf algebra F(R)_z associated to a quantum R-matrix R(z) with a spectral parameter, and a set of points z=(z_1,...,z_n). This algebra is generated by entries of a matrix power series T_i(u), i=1,...,n,subject to Faddeev-Reshetikhin-Takhtajan type commutation relations, and is a quantization of the group GL_Nt. In this paper we consider the quotient F_0(R)_z of F(R)_z by the relations qdet_R(T_i)=1, where qdet_R is the quantum determinant associated to R (for rational, trigonometric, or elliptic R-matrices). This is also a Hopf algebra, which is a quantization of the group SL_Nt. This paper was inspired by the pioneering paper of I.Frenkel and Reshetikhin. The main goal of this paper is to study the representation theory of the algebra F_0(R)_z and of its quantum double, and show how the consideration of coinvariants of this double (quantum conformal blocks) naturally leads to the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. Our construction for the rational R-matrix is a quantum analogue of the standard derivation of the Knizhnik-Zamolodchikov equations in the Wess-Zumino-Witten model of conformal field theory, and for the elliptic R-matrix is a quantum analogue of the construction of Kuroki and Takebe. Our result is a generalization of the construction of Enriques and Felder, which appeared while this paper was in preparation. Enriques and Felder gave a derivation of the quantum KZ equations from coinvariants in the case of the rational R-matrix and N=2.

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