LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 19
Our goal is to show (see Corollary to Theorem 2.76 below) that for generic Ap, A_i,
AQ and Ai, the coadjoint orbit
OA = {Adg(^1 + ^-+A0+A1z):g€G} (2.41)
has dimension 4N2 4iV. But first we consider briefly an abstract question. Let Gi and G2
be Lie-groups, with Lie-algebras £ and £ and dual Lie-algebras g* and £*, respectively.
Let $ be a homomorphism from G\ to G2, with derivative 0 at the identity,
f = &'(ci) :gx -+ g2 ,
«*)
=
A
(2.42)
We say that F : g* —• (F is smooth on £* if F is smooth with derivative dF(a) E 7 for
all a £ £*. In finite dimensions the derivative trivially belongs to #; in infinite dimensions
this is an additional assumption. Recall that dF(a) is defined as a linear functional on g*
through
F(a + 0t) = dF(a)(0)
d_
dt
lt=o
for all /?€£* .
L e m m a 2.43.
(i) ^ w a Lie-algebra homomorphism from g to g .
(ii) 0* 23 a Poisson map from g* to g* equipped with the Lie-Poisson structures.
(hi) Ad^(5) o 4 = / o Ad^.
(iv) 0*o A d ^
}
= A d * o ^ .
Proof: This lemma is standard. We prove only (ii). For F , G smooth on £*, we must
show that
{F op\ Go f*} = {F, G} o j* , (2.44)
or
a([d(F o p)(a) . d(G o *•)(*)]) = om(a) ( [ W ( : r ) ) , dG(cj*(a))})
Previous Page Next Page