LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 7
with pointwise multiplication
0102(A) = /i(A) /2(A) , (1.31)
and Lie-algebra
g = {X :X is a smooth loop from S to g£(N, (T),
with X(oo) real and diagonal, and satisfying (1.26)}
with pointwise commutator,
[XUX2}(\) = [AVA),-Y2(A)] . (1.32)
For X G, define 7r± : 0 —• g,
w+X(\) = lim lim / X(\')- , A iR , (1.33)
do r^oo Jlr X' - (A + e)
TT_X(A) = lim lim / X(\')—^ . A *JR . (1-34)
«io r^ooJ_ir A' _ ( A - e )
(Here and throughout the paper ?A denotes d\/2wi, etc.) Elements in Ran 7r± have
analytic continuations to Re A 0, Re A 0 respectively,
7r+ + 7 r _ = 7 , (1.35)
ir+X(oo) = TT_.Y(OO) = X(oo)/2 , (1.36)
and
R = *K+-K- (1.37)
is a classical i?-matrix on g satisfying the modified Yang-Baxter equation. Thus
[X^X2)R{\) = \{{XURX2] + [RX,,X2])
gives a second Lie-bracket on 0, and we denote the associated Lie-algebra and connected
Lie-group by g and G respectively. The Lie-Poisson structure on the coadjoint orbits of G
provides the underlying symplectic structure for the problems at hand.
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