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)))) ,