A ] 9 J an 1 99 8 QUANTIZATION OF LIE BIALGEBRAS , IV

Introduction This paper is a continuation of [EK3]. In [EK3], we introduced the Hopf algebra F (R) z associated to a quantum R-matrix R(z) with a spectral parameter defined on a 1-dimensional connected algebraic group Σ, and a set of points z = (z 1 , ..., z n) ∈ Σ n. This algebra is generated by entries of a matrix power series T i subject to Faddeev-Reshetikhin-Takhtajan type commutation relations, and is a quantization of the group GL N [[t]]. 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, trigono-metric, or elliptic R-matrices). This is also a Hopf algebra, which is a quantization of the group SL N [[t]]. This paper was inspired by [FR]. 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 [FR]. 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 [TUY], and for the elliptic R-matrix is a quantum analogue of the construction of [KT]. Our result is a generalization of the construction of Enriques and Felder [EF], 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. The results of this paper for the rational R-matrix (the Yangian case) can be directly generalized to the case of any simple Lie algebra g (what we do here corresponds to g = sl N). We did not include this generalization here since for a general g it is more difficult to write explicit formulas. We note that this paper does not use the results from [EK1,EK2] on the existence of quantization. Finally, we would like to explain the relationship between the present paper and the papers [FR,KS], which are devoted to the same subject. The papers [FR,KS] generalize to the quantum case the construction of Tsuchiya-Kanie ([TK]), which represents conformal blocks as intertwiners between a highest weight representation and the tensor product of …