LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 29
where Er(-) is the elementary symmetric function of degree r in the entries of M, Er(tM) =
trEr(M)
for all t G R. Thus
Er(-z-lA-i
+ (AP+A-1 - A
0
) + ^(Ao - A i ) + *
2
Ai) = ] T Jr*2* . (2.88)
k=-r
(Note that the notation is consistent: J, ,
= 1
Ji„|
= 1
IrkZkrjn~r
n-?u p ^ r = /rib-) It follows
that the number of nontrivial integrals is given by
N
] T ( 3 r + 1) = 3iV(iV + l)/ 2 + AT
r = l
Simple computations show that the coadjoint invariants {tr
AfL11
tr(A
p

Ao)AIL1}ik,k,N a r e
equivalent to {Ir,-r, Ir',-r'+i}ir,r'N•
a n
d the coadjoint invariants
{tr Af, tr AoAf }ijt,fc'N are equivalent to {/r,2r, -f"r',2r'-i}ir,r'N- We thus obtain
(3N(N + l)/ 2 + iV) - 4iV = 3N(N - l)/ 2
nontrivial integrals,
{^r*}lri V , - r + 2 f c 2 r - 2 ( 2 . 8 9 )
on the orbit OA- We construct the remaining
(
2~ commuting integrals from the entries
of Ap. By (2.86) it is enough to construct N(N l)/ 2 combinations of the entries of Ap
that commute amongst themselves.
Note first that (Ap)ij can be rewritten in the form
,z*-l.
(A,)ij = *y(A ) =j ( ^ 3 ^ ) t r ( A (
2
) E
J
, ) ^
2
where £^;i is the elementary matrix tjtj. By differentiation
(2.90)
dQij(A) = {£-r^)Ejieg, (2.91)
and we find
Rd$ij{A) =
(z-z-1)2Eji
. (2.92)
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