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)
Previous Page Next Page