LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 21
Note that ((//v,0), (ljv,0)) is the identity in Go and
((g^^^h^v))-1 =(((g-\~g-1ug) Ah-K-h-'vh)) . (2.48)
The Lie-algebra a of Go is given by
g_0 = {((K,L)AU,V)) : K,L,U,Ve 9e(N,R)} (2.49)
with Lie-bracket
[((K,L),(U,V)),((K',L'),(U\V))}
(2.50)
= (([#', K], [K', L\ + [£', K}) , {[U',U], [U\ V] + [V, U])) .
Define the map $ : G GQ by
*(?) = (((ff+(0) ) _ 1 , (ff+(0) )-15V(°)) (»-(oo)_1, ff-(^)-1«7L(^))) , (2.51)
where g =
g+gZl
and #+(0), g'_(oo) are denned in (2.40). Note that 7+(t) is real for
0 * 1 by (2.15). As £+(2) is invertible for all \z\ 1, and as g+(l) 0 (see (2.14)), it
follows that det /+(0), and similarly det 7_(oo), is positive.
The map $ is a surjective homomorphism. Indeed for g\ = 7i+#^~i, #2 = 72+#2~-
*(9i)o$(
? 2
) = ((?i
+
(0r 1 ,Pi+(0)- 1 si
+
(0)),(?i-(~)- 1 ,ffi-(oo)- 1 ^_(^)) )
0 ((P2+C0)-1 , 52+(0)-1^+(0)) , (ff2_(oo)-\ 52_(«D)-1Sr2_(oo)))
=
(^(O-^i+CO)"1,
^ ( O r ^ + W
+ ff2+(0)-1ffi+(0)-1«/i+(0)72+(0)) ,
(32-(oo)_1yi_(oo)_1, 52-(°c)_1s2_(oo)
+
S2
-(oo)-1p1_(oo)-1^_(c»)32-(oo)) . (2.52)
But sn * 02 = ffi+^fi = #i+02+(s,i-32-)-1 and so
*(ffi * 92) = (((
5 l + ?
2
+
)-
1
(0), (g1+g2+rH0) (g'l+(0)92+(0) + 9i+(0)g'2+(0)) ) ,
((g1^g2-)~1(x)Agi-g2-)~l(oo)(g'1_(co)g2-{oo)
+ gi-(oo)g'2_(oo)))) ,
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