LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 13

Changing variables in (1.34), we have for z €.

S1

*-X(z) = lim [- I X(z)-^-+ / X(z)-^-] . (2.6)

Substituting (2.3) in (2.6) yields

TT-A(Z)

= X.(z) + (A+(1) + A0 - A_(l) )/2 (2.7)

where

A

+

(.) = £ A (2.8)

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and

*-(* ) = £ * ; * ' . (2.9)

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Similarly we find

*+X(z) = X+(z) + (A_(l) + A0 - A

+

(l) )/2 . (2.10)

Formulae (1.35) and (1.36), and the fact that 7r± map £—£, are now imme-

diate from these formulae. From (1.36), for A, Y € g, 7r_[7r+X, n+Y] is a constant given

by ([7r+X, ir+Y](l))/2 = [AT(l),F(l)]/4 = 0, as both matrices are diagonal. Similarly

7r+[7r_X,7r_y] = 0. Now R = 7r+ — 7r_, so

7T+ = (1 + R)/2 , 7T. = (1 - iZ)/2 , (2.11)

which lead to the formulae

(l-R)[(l+R)X, (l + R)Y) = 0,

(2.12)

(1 + #)[(1 - i?)X, (1-R)Y) = 0.

Adding and multiplying out yields

[RX, RY] - R([RX, Y] + [X, RY}) = -[X, Y) (2.13)