LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 27

Then 0Ao is a AN2 — AN dimensional manifold given by

0Ao = {A = - ^ - + f^±

+

Ao+A1z: tr A? = tr(A?)*, tr AQA\ = tr A°(A?)fc,

2 — 1 2

tr A* ! = tr(A?.i)*, tr(A

p

- AoJA ^

= tr(Aj - A g X A i J * , 1 fc AT, and in the

case that N is even, and A\ and/or A?__x

has no real eigenvdalues, then, in addition,

P(Ai) = P(A?) and/or

P(A-

1

) = P(A°.

1

)}.

(2.78)

D

Rem£irk. The reduction from g* to the finite dimensional dual Lie-algebra g* does

not simplify the dynamical problem. In fact it is harder to solve the flows of interest on g*

than on g*. This is because the P-matrix structure is lost and the solution by factorization

is no longer at hand. As in QR (see [DLT], [DL]) the precise opposite is in fact the point:

one should lift the flows from the finite dimensional dual Lie-algebra to a loop algebra in

order to solve the problem.

(c) Commutin g integrals on a generic orbit 0A = 0Ap/{z-i)+A_l2-i+A0+Alz.

In this section we construct |(4iV2 — 47V) = 2N2 — 2iV commuting functionals on a

generic orbit OA. As we will see (subsection (d) below), these functionals provide integrals

for the flow that interpolates the Moser-Veselov algorithm.

From (1.39) we see that for the discrete Euler-Arnold equation the curve

det(M(t,A) -rj) = 0

is preserved in time. For A € g*. , this leads us to consider the curve det( (z — 1) A(z) —

77) = 0 with coefficients

ITk(A)=f I detdz-DMz)-,)-^^ (2.79)