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 rst 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 nite dimensional simple Lie algebra. Of course, the general result of the rst paper, which says that any Lie bialgebra admits a quantization, applies to this particular case. However, this result is not suuciently eeective, as the construction of quantization utilizes a Lie asso-ciator, which is computationally unmanageable. The goal of this paper is to give a more eeective 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 eld k of characteristic zero. Let 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 2 gg(() be a rational gg-valued function on with the Laurent expansion near 0 of the form P i X i X i =u+O(1), satisfying the classical Yang-Baxter equation (2.1). Such a function is called a classical r-matrix. Consider the vector space a = t ?1 gt ?1 ]. Let (u) = P P m1 (X X t ?m)u m?1 2 g au]], where fX g; fX g are dual bases of g with respect to the half-Killing form. Deene a Lie bialgebra structure on a by the formulas The classical r-matrix deenes a bilinear form on a with values in k((u)), by the rule (x; y)(u) = Res v;w=0 < x(v) y(w); r(v ? w ? u)i, where h; i is the invariant form on g, and r(v ? w ? u) is understood as an element of g gv; …