Printed in the Netherlands. 299. Dual Moment Maps into Loop Algebras*. M. R. ADAMS. Department of Mathematics, University of Georgia, Athens, GA 30602, ...
Letters in Mathematical Physics 20: 299-308, 1990. 9 1990 Kluwer Academic Publishers. Printed in the Netherlands.
299
Dual Moment Maps into Loop Algebras* M. R. ADAMS Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. J. HARNAD Centre de Recherches Math$matiques, Universit$ de Montreal, CP 6128-.4, Montreal, Quebec H3C MT, Canada, and Department of Mathematics, Concordia University, Montreal, Quebec, Canada and
J. HURTUBISE Department of Mathematics, McGill University, Montreal, Quebec H3A 2K6, Canada (Received: 4 January 1990) Abstract. Moment maps are defined from the space of rank-r deformations of a fixed n • n matrix A to the duals ( 9 (r)+)% ( 9 (n)+)* of the positive half of the loop algebras ~ (r), ~ (n). These maps arc shown to give rise to the same invariant manifolds under Hamiltonian flow obtained through the Adler-Kostant-Symes theorem from the rings I( ~ (r)*), I( ~ (n)*) of invariant functions. This gives a dual characterization of integrable Hamiltonian systems as isospectral flow in the two loop algebras.
AMS subject daseiflcation (1980). 58F07.
O. Introduction A key to the understanding of a large number of integrable Hamiltonian systems, both finite- and infinite-dimensional, consists of realizing such systems as isospectral deformations of linear operators. In [2], it was shown how a natural embedding of a large class of finite-dimensional systems via a Poisson map into the dual ( g~ (r)+)* of the loop algebra of matricial polynomials led to such Lax representations through the Adler-Kostant-Symes theorem [3, 5, 13]. In many of the examples treated, e.g. the Neumann and Rosochatius systems, and other examples [4, 7, 8-11] different approaches have led to two alternative Lax representations. The first consisting of r x r matrix polynomials in the loop parameter 2 of fixed low degree (usually one or two) where r increases with the dimension of the phase space; the second involves polynomials of varying degree, with r -- 2. This suggests a 'dual' approach to such systems leading in two alternative ways to the same spectral invariants and essentially the same linearization map. The purpose of this Letter is to describe a simple approach, based on 'dual' moment maps (i.e. Poisson maps into loop algebras) defined on the 'generalized * This research partially funded by NSF grant DMS-8601995, U.S. Army grant DAAL03-87-K-0110, and the Natural Sciences and Engineering Research Council of Canada.
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Moser space' studied in [2], which explains the theory behind these alternate approaches.
1. Preliminaries Let M.,, denote the space of n x r matrices, with n/> r. The space M = M.., x Mn,, has a natural symplectic structure co(F, G) = tr(dF ^ dGr),
(F, G) e M.
(1.1)
Let A be a fixed n x n matrix. The general rank-r perturbation of A is given by L = A + FG r,
(F, G) ~ M.
(1.2)
In [2], a large class of integrable systems on M were derived by constructing a moment map into the dual of the positive part of a loop algebra. Let ~ (r) denote the semi-infinite formal loop algebra on gl(r, C), i.e. Z ( 2 ) e ~ ( r ) if X(2)= Era= _ ~ Xi2 i with X,- ~ gl(r, C). We use the vector space direct sum splitting ~ ( r ) = g~(r)+ (D g~(r)-,
(1.3)
where g~ (r) + denotes the space of r x r matricial polynomials and ~ (r)- is the space of strictly negative formal power series in g~ (r). Under the pairing
(1.4)
(X(2), Y(2) ) = tr(X(2) Y(2))_ ~,
where the subscript - 1 denotes taking the coefficient of 2 - t , there is a natural identification (~(r)+)
* ~
~(r)-.
(1.5)
Hence, gl(r)- carries a natural Poisson structure given by the Lie-Poisson structure of ( ~ (r)+)*. Let A e gl(n) be a fixed n x n matrix. Define (~l(r) +, to be the group of Gl(r)-valued functions in the complex 2 plane that are holomorphic inside a fixed disc D centered at 2 = 0 containing the spectrum of A. (The lie algebra of El(r) + may be viewed as a completion of ~ (r) +.) There is a symplectic action of (~l(r) + on M for each such A, defined by
g(~): (e, ~) -. (e., G.)
(1.O
where (A - ~,) - l E g - 1(,~) = ( a - - ~ ) - l F g g(~)~
-~- Fhol,
T
~(a - ,~)-' = a [ ( A - ,z) -' + ~ho,,
and Fhol, Ghol are holomorphic in D.
(1.7a) (1.7b)
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T H E O R E M 1.1. The El(r)+-action (1.6)/s Hamiltonian, with equivariant moment map: J / : M - , ( g~ (r)+) * defined by Y~(F, G) = - G r ( A - 2 ) - ' F .
(1.8)
Proof Since the action (1.6) is the cotangent lift of the linear action (1.72) on M,,r it is automatically Hamiltonian. The moment map aY~ must be given by (aY~(F, G), X(2)) = tr((G z)(~xca)(F))), where X(2) ~ g~ (r) + and ~xta)(F) denotes the value at F ~ M,,, of the vector field generating the group flow of exp tX(2). Differentiating (1.7a) yields (A - 2 ) - IExta)(F) = - ( A - 2 ) - IFX(2) + holomorphic. Hence tr(G rExta ) (F)) = tr(G r(A - 2) - ~-~xta)(F)) _ ~ = - tr(G r(A - 2) - IFX(2))_ 1. Since we identify (g] (r)+)* with g~ (r)- via (1.4) we conclude
L (r, G )
= - GT(,4 -- 2)-'F.
[]
Remark. This theorem was proved in [2] under the assumption that A is diagonalizable. Since the examples will use the result for more general A, we have included the general proof here. We now proceed analogously for the loop algebra g~ (n) + and its group (~l(n)+. Introduce a fixed matrix Y ~ gl(r, C) and define an action (~l(n, C) + : M - - ) M g(z) : (V, G) ~ (Fg, Gs),
(1.9)
where now g(z)F(Y - z) -1 = Vg ( Y - z) - ' + Vhol,
(1.10a)
( Y - z) - ' G r g - l ( z ) = ( Y - - z ) - l G r + Gro,
(1.10b)
and Fho~, Gho~ are holomorphic inside a disc centered at the origin of the complex z-plane containing Spec(Y). Then, as in Theorem 1.1, we have T H E O R E M 1.2. The action (1.9) is Hamiltonian with equivariant moment map: aye: M ~ (
g~ (n) +) *
defined by aY,r(F, G) = F ( Y - z) -IGr. Proof The proof is identical to that of Theorem 1.1.
( 1.1 l)
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We also have natural Hamiltonian actions of Gl(n) and Gl(r) on M defined by Gl(n) : M --. M, (1.12)
g: (F, G) ~ (gF, ( g r ) - ' G ) ,
Gl(r) : M -- M (1.13)
g: (F, G) ~-~ ( F g - ' , Gg r )
Denote by Gl(n)A and Gl(r)r the stabilizer under conjugation of A ~ gl(n) and Y ~ gl(r), respectively. The following lemma may be deduced immediately from the forms (1.8) and (1.11) of the moment maps J'~ and J'r LEMMA 1.3. The fibers for the maps 3ra : M ~ ( ~ ( r ) + ) are the GI(n)A and Gl(r)r orbits, respectively.
* anddTr: M - ~ ( ~ ( n ) + ) *
Let k
A(2) = det(A - 21) = 1-[ (~t~- 2) n,
(1.14)
i=1
be the characteristic polynomial of A. Denote by M A c M the submanifold consisting of pairs (F, G) on which GI(n)A acts freely. Assuming r >i n~, Vi, MA is open and dense in M and invariant under the G'~L(r)+-action. Projection to the quotient spaces MA ~MA/Gl(n)A and MA "-*MA/Gl(r)y determines induced Poisson structures on each. The moment maps J ' / a n d J r pass to the quotient spaces, defining injective equivariant moment maps that generate infinitesimal Hamiltonian actions of (~l(r) + and (~l(n) + on MA/Gl(n)A and MA/Gl(r)r, respectively. 2. Duality Let I( g~ (r)*) and I( ~ (n)*) denote the rings of Ad*-invariant polynomials on (r)* and g~ (n)*, respectively. Let N(2) =- Y + Gr(A - 2) -IF,
(2.1a)
M(z) - A + F ( Y - z) -IG
(2.1b)
be viewed as elements of g~ (r)* and ~I (n)*, respectively, by expanding in power series in 2 - l and z - ! (either formally, or analytically, outside discs centered at the origins of the complex 2 and z planes, containing Spec(A) and Spec(Y), respectively). Define two rings of functions #" r and #-A on M by .9~ r = {~b ~ C~176 [ ~b(F, G) = ~b(N), ~b ~ I( ~ (r)*) }
(2.2a)
~ a _ {~b E C~176 [ ~k(F, G) = ok(N), q~ ~ I( g~ (n)*)}
(2.2b)
THEOREM 2.1. The two rings ~rv and ~ra are equal, their elements Poisson commute and their Hamiltonian flows preserve the spectrum of N(2) and M(z). I f
DUAL MOMENT MAPS INTO LOOP ALGEBRAS
303
~b r ~ r = ~ a is of the form
~(F, G) = ~b,(N) = ~b2(M), with r ~ I( ~ (r)*),
r e I( ~ (n)*),
the integral curves (F(t), G(t)) of the corresponding Hamiltonian flow satisfy ~=
[(d~)+, N],
(2.3a)
dM ~ - -- [(d~b2)+,M]
(2.3b)
where (d~bt)+ and (d~2)+ denote the projections to the nonnegative parts of g~(r)* and g~ (n)*, respectively. Proof. The equality #-,4 = #- r follows from the identity det(A - 2) det(Y + Gr(A - 2)- ~F - z) = det(Y - z) det(A + F(Y - z) -
(2.4)
1 G T __ ~ ) ,
where expansion of the left-hand side in 2 and z provides the generators of ~ r as coefficients, and expansion of the fight-hand side provides the generators of ~A. Since the elementary spectral invariants of N and M are contained in these expansions, it is clear the flows will be isospcctral because the AKS theorem guarantees that the elements of a r r = #-,4 Poisson commute. The Lax form (2.3a,b) also proves isospectrality and follows from the AKS theorem, together with the fact that ]A and ~ r are Poisson maps. [] By Lemma 1.3, ]A is invariant under the action of GL(n)a and j r is invariant under the action of GL(r) r. Furthermore, ,Tffis equivariant for the action of GL(r) r on M and the natural action of GL(r)r on ~ (r)-, and the corresponding statement for ] r and GL(n)A is also true. Thus, we have the following commuting diagram (restricting to dense, open sets, all quotients are manifolds):
(n)-
M
,
-....
, g~(r)-
M/GL(nh
M /GL(r) r
g~ (n) -/GL(nlA ,
Y
M/(GL(r),. x aL(n).,)
, ~ (r) -/CL(r) y
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The bottom arrows are one to one Poisson maps, thus they give an identification of the symplectic leaves in M/(GL(n)A x GL(r)y) and certain symplcctic leaves in ~ ( r ) - / G l ( r ) y and ~(n)-/GL(n)A. Thus, certain symplectic leaves in ~ ( r ) - / G L ( 0 r , and the AKS flows thereon, are identified to certain symplectic leaves of ~(n)-/GL(n)A, and AKS flows there.
Remark. In [1] it is discussed how the Hamiltonian flows for 3t-r may be linearized on the Jacobi variety of the spectral curve S = {(2, z) [ det(Y + Gr(A - 2 ) - t F - z) = 0},
(2.5)
via the methods of Krichever [6]. Likewise, the flows for ~ A may be linearized on the Jacobi variety of the curve. = {(2, z) I det(A + F(Y - z) -~G r - 2) = 0}.
(2.6)
Equation (2.4) gives a birational equivalence of these curves.
3. Examples 3.1. THE ROSOCHATIUS SYSTEM The Rosochatius system is a Hamiltonian system on T'S"-~ described by constraining the Hamiltonian
H(x, y) = 89E y 2 + E #-~2+ ~, ~ x2
(3.1)
"%/'
on R ~ x R" to ~. x,.2 = 1,
~'. xiy, = 0.
(3.2)
The equations of motion arc thus given by
To explain this in terms of isospectral deformations, we must reduce the loop algebra ~ ( 2 ) to ~(1, 1) = {X(2) ~ g~(2); Xi ~ u ( l , 1))}. This is worked out in [2], the result is to consider W c M~.2 x M~,2, a symplectic subspace given by W = {(F, G); G = Fy}
(3.4)
with
Let A be the diagonal n x n matrix with jj term JY2A: M.,2 x M.,2--' g~(2)- by (1.8). Restricting J~ to W gives
Y~: w-,~(l, I)-.
and
define
(3.5)
DUAL MOMENT MAPS INTO LOOP ALGEBRAS
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This map is invariant under the action of the group H = U ( 1 ) x ... x U(1) (n times)
(3.6)
contained in GL(n). Performing the Marsden-Weinstein reduction of W by this H-action at the H-moment map value x//-Z-2(/~l. . . . , ~un)
(3.7)
gives an injective Poisson map J : R2~fi(1, 1)- given by J(x, y ) = _ 8 9 ,=,~ ~ l
5Fxiy` [L x2
l+ x / ' ~ P , [ 10 ~]--2/t2x2 [00
-x,y,/-"
10]}
= N(x)
(3.8)
Remark. J(x, y) is given by Gr(A - 2 ) - ~ F , where F and G have columns
F=
5
(x, p),
__l
G=~
( --/~, x)
(3.9)
with p, = - y , + x/-S2(p,/x,). A direct computation yields PROPOSITION 3.1 [2]. The function 9 e ~ r given by
o,x,
'0]))o
(3.10)
where a(2)= det(;t- A), agrees (up to an additive constant) with H(x,y) on the constraint space (3.2).
The proposition describes the Rosoehatius system as a constrained AKS flow in ~( I, 1)-. (The constraints (3.2) may be considered as a realization of the reduction of ~(1, 1) by the stabilizer group of Y.) From Theorem 2.1, we see that this flow is isospectral for 3'r = A + F ( Y - z)-~G r, where F and G are given by (3.9) and
Expanding ( Y - z)- ~ and using y2= 0 we get A + F(Y-z)-lGr=
A +~ ~ + ~ F
(3.12)
with xiyj -- xjy i --
\XiXJ "]- Xi
(3.13)
and F o = 89 xj.
(3.14)
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To get the n x n AKS flow and Lax pair, we should first work with the space W, since the H action is nontrivial on g~ (n)-. Using Equation (2.4), we have ~x,
_ F 2a(2) y) -
det(V - z + G r(a - ~)
-IF)]_1o.o
2z~ = 1 ~-:-~dct(A -A +F(Y-z)-IG
T]Ao.o
= tr(z2(a + F(Y- z)-'G~)2)o - 1,
(3.15)
where the subscript 0, 0 denotes taking the coefficient of A~ ~ Thus, we see that the 2 x 2 Lax pair given by (3.10) on W, namely
d (GT(A-~)-iF) ~ [ { ~ ( Y - ~ - G T ( A
-
~)-le)}+)Y-~-GT(A~)-Ie] (3.16)
is equivalent to the n x n Lax pair d dt (A + F ( Y - z) - ' G r) = [A + z - ' F G r, A + F ( Y - z) -'GT],
(3.17)
for (F, G) ~ W. The H reduction can be realized by constraining this Lax pair flow to be tangent to the subspace of g~ (n) - spanned by the set of A + ( 1/z)[~ + (1/z2)F with fl and r given by (3.13) and (3.14). This leads to the n x n Lax pair formalism of the Rosochatius system given by Ratiu [11]. 3.2. RIGID BODY MOTION IN n DIMENSIONS The motion of an n-dimensional rigid body is given by the geodesic spray of a left-invariant metric ( , ) on SO(n) which on Te SO(n) = so(n) is given by (X, Y) = - t r ( X r V ) ,
X, Y e so(n),
(3.18)
where r is an n x n diagonal matrix with positive eigenvalucs gl . . . . . gn (scc, e.g., [8, 12]). The Euler equations for this flow are given by d dt (rX + XV) = [FX + XF, X].
(3.19)
In 1976, Manakov [7] wrote this as a Lax pair with parameter d dt
- - L = [S, L]
(3.20)
with L = FX + x r ' +/~F 2,
B = - X +/xF.
(3.21)
Using the spectral invariants of L, this system was shown to be completely integrable by Ratiu [12], (cf. also [9]).
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DUAL MOMENT MAPS INTO LOOP ALGEBRAS
In 1980, Moser [10], (cf. also [4]) studied a rank-2 subsystem of (3.20) and (3.21) given as follows. Consider the map R 2 n ~ so(n) given by (x, y ) ~ X with
(3.22)
X O. - x i y J - x j y i
gi + gj
Then (3.20) and (3.21) pull back via this map to the Hamiltonian flow on R 2~ with Hamiltonian
H(x, y) = 1 ~, (x,yj - x y , ) 2 i
