9 J an 1 99 8 QUANTIZATION OF LIE BIALGEBRAS , III

Introduction This article is the third part of the series of papers on quantization of Lie bial-gebras which we started in 1995. However, its object of study is much less general than in the previous two parts. While in the first and second paper we deal with an arbitrary Lie bialgebra, here we study Lie bialgebras of g-valued functions on a punctured rational or elliptic curve, where g is a finite dimensional simple Lie algebra. Of course, the general result of the first paper, which says that any Lie bialgebra admits a quantization, applies to this particular case. However, this result is not sufficiently effective, as the construction of quantization utilizes a Lie asso-ciator, which is computationally unmanageable. The goal of this paper is to give a more effective quantization procedure for Lie bialgebras associated to punctured curves, i.e. a procedure which will not use an associator. We will describe a general quantization procedure which reduces the problem of quantization of the algebra of g-valued functions on a curve with many punctures to the case of one puncture, and apply this procedure in a few special cases to obtain an explicit quantization. The main object of study in this paper are Lie bialgebras associated to rational and elliptic curves with punctures, which can be described as follows. We work over an algebraically closed field k of characteristic zero. Let Σ be a 1-dimensional algebraic group over k (i.e. G a , G m , or an elliptic curve), and u be an additive formal parameter near the origin. Let r ∈ g⊗g(Σ) be a rational g⊗g-valued function on Σ with the Laurent expansion near 0 of the form α X α ⊗X * α /u+O(1), satisfying the classical Yang-Baxter equation (2.1) (here {X α }, {X * α } are dual bases of g with respect to the half-Killing form). Such a function is called a classical r-matrix. Consider the vector space a = t −1 g[t −1 ]. Let τ (u) = α m≥1 (X α ⊗X * α t −m)u m−1 ∈ g ⊗ a[[u]]. Define a Lie bialgebra structure on a by the formulas [τ 13 (u), τ 23 (v)] = [r 12 (u − v), τ 13 (u) + τ 23 (v)], δ(τ (u)) = [τ 12 (u), τ 13 (u)]. (1) It is convenient to understand the classical r-matrix as a bilinear form β on …