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))})