LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 15
and hence
f^i-i
=0
Ad tx,Tv t r . y l
e e
= ^
+
r , ; r
+
. Y ] - [ 7 r _ r , 7 r _ X ]
= i([iir,.Y] + [r,i?A'])
(2.18)
= [Y,X]R.
Thus # is the Lie-algebra of G.
(b) Dual Lie-algebras and coadjoint orbits
Notice first that under (2.1) the loop M(A) = r^ becomes ^ ~ + ^o +
1 A
Axz, where Ax = j2"^"\ A0 = ±±£- and A-X = A f = J 2 + M - I . One would hope
that elements of the form A-\z~x + Ao + A\z generated a coadjoint orbit consisting of
elements of the same form. As we will see, however, this is not the case and generically
elements of the form A-\z~x -f AQ -f A\z generate orbits consisting of elements of the form
——• + A-\z~x -f AQ + A\z with a polar singularity at z = 1 G S 1 .
Ay
z
Motivated by these considerations, we define
__ik
£*ng = {A : A(z) = ^ + Ateg(z) = A(z) gt(N, W) and
^reg(*) is a smooth loop on S 1 } .
In an obvious notation
^sing —pole reg
(2.19)
(2.20)
We will show below that g*. is invariant under the coadjoint action of G, and the coadjoint
orbits of interest will be found to be subsets of g*
Elements A in g*. induce linear functionals on g_ through the non-degenerate pairing
(A,X)=j txA(z)X(z)^- (2.21)
Note that on g x g this pairing is ad-invariant,
(X,\Y,Z]) = -([Y,X),Z). (2.22)
Previous Page Next Page