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.