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
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