14 P. DEIFT, L. C. LI, AND C. TOMEI

which is the modified Yang-Baxter equation. Thus [X, Y]R following (1.37) is a Lie-

bracket,and the set g together with the structure [•,•]# forms a Lie-algebra which we

denote by g as in Section 1 (note that [X,Y]R(Z = 1) = 0, which is diagonal).

Define

G = {g 6 G : g has a factorization g(z) = g+(z)g-(z), where the invertible

loops g±(-) have invertible analytic continuations to

{\z\ 1} , {1 \z\ oo}respectively, which are smooth up to (2.14)

the boundary and contractible (preserving analyticity) to the

identity. Also g+(l) = #_(1) = V s O ) 0} .

Note that the factorization g±, if it exists, is unique. Indeed if g+g- = g+gL are two

such factorizations, then (g+(z) )~xg+(z) = g'_(z)(g-(z) ) ~ 1 is bounded and analytic in

the entire z plane, and hence is constant. Evaluating at z = 1 we see that the constant

must be the identity. As g(z) = g(z), uniqueness implies

9±{z)=g±(z) (2.15)

Note also that the requirement of contractibility, preserving analyticity, is not in-

dependent and follows by a standard argument from the other properties of g(z) in the

definition of G.

For g, h G G, define the multiplication

g*h = g+hg- = (g+h+)(h-g-) (2.16)

One checks directly that (G, *) is a group. Note that the *-inverse of g = g+g- is g+ gZ .

For X e g, et7r+x

e

t 7 r - x is a curve in G with derivative ir+X + n-X = I a t the identity.

Thus

A d -* - 1 ,

g *

C

«*+A- * 0- 1

(= 0

g+e**+xg?glie**-Xg- ( 2 ' 1 ? )

_ d_

~ dt

= g+{n+X)g-1 -f gZl{^-X)g- ,

t=o