Volume 127, number1 PHYSICSLETTERSA 1 February1988

SOME INTEGRABLE SYSTEMS RELATED TO AFFINE LIE ALGEBRASAND HOMOGENEOUS SPACES

Ian MARSHALLMathematicsDepartment,UniversityofManchester,InstituteofScienceand Technology,P.O. Box88, ManchesterM60 JQD, UK

Received11 May 1987; revisedmanuscriptreceived15 October1987; acceptedfor publication4 November1987Communicatedby A.P.Fordy

In anattempt to understandhow a resulton hamiltoniansystemsrelatedto hermitiansymmetricspacescan beseenasanapplicationof theAdler—Kostant—Symestheorem,it is foundthatquitesubstantialextensionsto thisresultcanbemade,

1. Introduction

In the paperby Fordy, WojciechowskiandMarshall [1] it is shownhow to generatehamiltoniansystemsfrom hermitiansymmetricspacesandto give a Lax pair for thesesystems.Whenone is in sucha happysit-uationoneusuallyfeelsjustifiedin calling one’ssystemintegrable.In factthe authorsof ref. [1] only did the(typically very tedious)calculationsnecessaryto provethe functional independenceof the integralsfor oneexampleof the generalclass,although the involutivity of the integralsis shown moregenerally — usingagainhoweverratherlong-handtechniques.

In thepresentarticlewe will seehow,by theuseoftheAdler—Kostant—Symestheorem,describedforexamplein refs. [3—5]the classof hamiltoniansystemstreatedin ref. [1] canbe dealtwith much moresimply andindeedcanbe quite substantiallyextended.Theconclusionis arrivedat by the straightforwardsuperpositionof theresultsof FordyandKulish andthework of bothAdler andvanMoerbeke,andReymanandSemenov-Tian-Shansky[2,6,71.It is anticipatedthat thecompleteintegrabilityof all thesystemscontainedherecanbeaccountedforby theapplicationof the rathersophisticatedmethodsexplainedin ref. [71,seealso refs. [6,8].

2. The AKS theorem

Let g bea finite dimensionalLie algebra.Supposethatg canbewrittenasthevectorspacedirectsum,g = k+ 2,andsupposethatk and Q are not only subspacesof g butthat theyare also subalgebras.Let G be a connectedLie group associatedwith g; let L andK be subgroupsassociatedrespectivelywith 2 andk. Supposethat wehavea nondegenerate,ad-invariant,bilinearform < , > : gxg—+C, by meansof which we can identify g withits dual spaceg*; with this identification we havek*_~Q±and Q*~.~k±where (k’, k> =0= <21, 2> andg=k-’- +2--. Thebilinearform < , > inducesa bilinearform on k-- xQ by restriction,we write this as < ,

Every function in C00(g*) can be identified with one in C00(g). Each FeC*(g) inducesan elementofC (k--) ‘~C°°(Q*) by restriction,wewriteP= FJ k’~ Let 174, III, fl~±,17~±be thecanonicalprojectionson g rel-ative to the decompositionsg=k+2 andg=k’ +Q~.For ØeC~’(k’)we write V~Øfor the gradientof 0, i.e.

This paperwaswrittenduringa visit to themathematicsdepartmentof LinkopingUniversity,Linkdping, Sweden,

0375-9601/88/$03.50© ElsevierSciencePublishersB.V. 19(North-HollandPhysicsPublishingDivision)

Volume 127, number 1 PHYSICSLETTERSA 1 February1988

d<y,V~O(x)>j=~—Ø(x+yt)j,=0 Vx,yek-’-

It is easyto checkthat

V1P’(x)=JJjVF(x) foranyFeC°°(g),xek--_~Q*, (1)

whereV is the usualgradient,i.e.

<~,VF~))= ~F(ii+~t)I,=0 V,~,~eg.

Also, if we denotethe actionof L on k~(wherewe think of k’ Q*) by AdL~andthe actionof G on g byAd, we have

AdL~aa114~Ada-i c~for aek~Lc~g, aeLc G. (2)

We definethe set of ad-invariantfunctionson g* by

,f(g*) = {feC~~(g*)(ad~ct)(Vf(a))=0 Vaeg*,xeg} Ca0(g*)

andwe canidentify 5(g*) ‘..~I(g) C~(g)in the obviouswaywhere

I(g)={FeC~(g)I<ad~à,VF(á)> =0 Váeg,Vxeg}. (3)

With theseconditionson g andthe abovedescribednotation,we havethefollowing:Theorem(AKS).Let ~~ k-i- 2~be anycoadjointorbit of L in k’, andlet w ~, be the Kirillov two-formon

this orbit. Thentheset1(g) k’ c C°°(k--) --‘ C~(2*) of functionsin1(g) restrictedto k1 isan involutive system

on the sympleticmanifold (~2,a~~); thehamiltonianflow correspondingto a functionHeC~(k’) is given by

x=X,,(x)=ThLEI7iVH(x),x] =1741[—J14VH(x), x]= —[174 VH(x),xl forxek

1 , (4)

kk --1 ~ k’-, asis easyto check.(Noticethat k-- andQ* areconsideredinterchangeablefor practicalpurposes.)~ is the usualHamilton vectorfield definedby H.

3. Someapplications to systemson homogeneousspaces

We now cometo theapplicationof this schemeto buildingnewintegrablesystems.It is a matterof choosingg, k, 2 and < , > andthen consideringa particularchoiceof orbit of L in Q*_~k~.In what follows we chooseto work with matricesin orderto simplify varioustechnicalproblems.Thus let t~be a complexsemi-simpleLie algebra(therebyguaranteeingthe existenceof a nondegenerate,ad-invariantinner product)with CartansubalgebraX. Let g be a representationof ~ in which the Cartansubalgebrais I).

Let ~=g®C[~~., A ‘] be the affine Lie algebrarelatedto g. TheLie bracketon ~ is given by

rs I ~l s I

I ~ ~,A’, ~ iiA~1= ~ ~ [~~.])~i~J ~L,=r j=k J ,=~j=k

TheCartansubalgebraof~is ~=I)®C[A, 2~]. Define

drsspan{~ ~NI~eg, ~1=0fori<r, or

Now set

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Q=~d~1, k=~nd0~,.

It is easyto checkthatk and Q are subalgebras.Define the (ad-invariant,nondegenerate)inner producton

Is I \ fs I \~ ‘i~2J,)= ~ tr(~1~)=tr(~ ~ ~i1Ai) (5)

i=r j—k i+j= — I \j=r j=k —

where( )I means“take the )~ component”.With this inner product k=k-1- and 2=2’. The groups

G= exp(~),L = exp(2), K=exp( k) canbetakenfor connectedgroupscorrespondingrespectivelyto ~, k, 2.We haveto be carefulnow in applyingthe AKS theoremas ~is not finite dimensional;for this reasonthe

identifications~ 2-- -.~k”,k-- _.Q* are not valid.Let uslook at anorbit of L in k’ wherewecanat leastthink of k’ asbeingcontainedin Q*~Fix an element

aek’, thenaed0~,for someve7L÷.Underthe AdL~actionof L on k’ the set Loa is still in d0~,moreover

LoaL~oawhereL~=exp(2rid_~_1,_~);in fact we canargueas follows: /~=2r~d_~_i,_1is a lie algebraifwe definethe Lie bracketto be thestandardonewith the truncationrule (seeref. [4]). We cantakethe con-nectedLie groupL~= exp(l~)J.,, where 4 denotestruncation,to bethat correspondingto /~.Usingthe inner

productdefinedabovein (5) wecanidentify l~with d,,. We find that, in the usualway, if we denotetheaction of L~on d0,~—~l~by Ad”’~’,wehave

Ad~’artThiAda-irt for aeL~.

As a set,L~c G, so on theleft-handsideof the aboveexpressiona is thoughtof asan elementof L~,on theright-handsideas an elementof G.

ClearlythenL~oa=L~oa=Loa.In otherwords, if wedenotethe Ad~”~’orbit of L~through a in dç1,. by9~~’)andthe AdL~orbit of L througha in k’ by ~ the two orbits areequal as sets.By the generaltheory[91 2~’~is a symplecticmanifold and thus &,~canbegivena symplecticstructurein a straightforwardway.

Now by reexaminingthe proofof the AKS theorem,we canseethat evenwith the abovemodificationswecan still say that the set of functionsin I( ~) (which is now of coursenot identified with 5(~)) inducesa setof functionsin involution in C00(t~) andso in C~(!2~’))by restriction.

Whatwe find thenis that for any given aek’ we canfind an involutive systemon thecoadjointorbit ofL through aby usingthe AKS theorem,the only thingwhich is specialis that to apply the theoremwe mustthink of this orbit as the orbit ofL~througha for someveZ÷,where v dependson a. We cancheckthe

Proposition.Forany polynomialf in onevariable,andfor any integersk,j the following function is an ele-ment of I(~):

(6)

By the AKS theoremthe set of functions{tr(A”f (x1~))I ~ j, k integers;fa polynomial in onevariable} isin involution with respectto the Kirillov symplecticform on any orbit of L in k’.

Let A be an elementof b anddefine

t=C5(A)={BegI[AB]=0}. (7)

Setm=vectorspacecomplementoft in g, so thatg=t+m. We canidentify m with the tangentspaceat somepoint of thehomogeneousmanifold f~/X’where ~ is a Lie group associatedto g and1’ is the subgroupas-sociatedto t. With this construction,

[tt]ct and [tm]cm, (8)

i.e. f~/Y(is a reductivehomogeneousspace.

Proposition.Let

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Volume 127, number 1 PHYSICSLETTERSA 1 February1988

F ={A +QA”~+Xp2 IQem,x~_2edo,~_2}

Thenfor any aeF, (9a~F.Proof Follows directly from L=exp(Q) and [gA] c m.Chooseany orbit ~ in F andlet xc ~ (so that ~= (9k). Write

x=A~+QA~~+x~,with Qem. (9)

Let

~ . (10)

We get then that the hamiltonianflow of H on &~is givenby

(11)

as

174(xA’ ~=174(AA+Q+A~ ‘x~-2) =AA+Q.

The functionsFeC~(k--) of the form givenby eq. (6) form an involutive set of integralsof this flow.It shouldbenotedthat this is notyet sufficientto assertcompleteintegrabilityof the systems;to do this we

shouldineachcaseshowthatthesubmanifoldon which all theelementsof I (~) k’ areconstantsis a lagrangiansubmanifold.

Thecasev= 2 includesthe stationaryflows of all of the equationsin the so-calledN-wavehierarchylookedat by FordyandKulish in ref. [21. Whenin additionto (8), t andm satify

[mt] ~t, (12)

sothat ~4/1 is a hermitiansymmetricspace,we havethe systemsdiscussedin ref. [1]. What is good aboutthe presentwork is that it gives us the meansto write an extremelylargeclass of dynamicalsystemsasham-iltoniansystems.With themorenaiveformalismbroughtto bearon theproblemin ref. [1] this isby no meanspossible— at leastnototherthan by a flash of inspiration;or morelikely by meansof a verylucky guess.ByusingtheAKS theoremandusingthe setupdescribedby ReymanandSemenov-Tian-Shansky,andby AdlerandvanMoerbeke[7,6] the questionof finding integralsin involution for thesesystemsis reducedto thecal-culations(10) and(11) and the proofof the proposition!

4. Example — a new system

Thesimplestkind of new examplewould beto takev=3, g=su(n+l) andA=idiag(n, —1, —l —1).Thissystemis then relatedto the hermitiansymmetric spaceSU(n+ l)/S(U(l) xU(n)); the correspondingv=2 casegives the Gamiersystem.

We can show that, for

a=23A+ 1 A

(n+l)2

a typical elementof ~ hasthe form

x=~3A+22Q+~(P_~[QQ+])+T+[SQ]+ (~1)2A~ (13)

whereA is an arbitraryconstantmatrix in su( n + 1). Q, P, T, Sare all elementsof m and

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S= —~1(P÷—P_)+cQ

(cc~ is arbitrary). We canassumethatAct by incorporatingits rn componentsin T.The notationin (13) is the sameasthat of ref. [2]. The±subscriptsare usedto denoteelementsof the

subspacesm+, wherern~is the i(n+ 1) eigenspaceofadA andrn is the —i(n+ 1) eigenspaceofadA. In factm+ = mt.. in this example,andrn + has(complex)dimensionn. Hencethemanifold &~,hascomplexdimension3n.

Let us demonstrate(13): X=(a~[AA3+(n+l)2A]a)+ for someacE

3, ( ).~. is the sameas H4i. Nowa=exp(,+,22

2+~3A

3+~A4) for some~, ~, ~, ,j4esu(n+l). So we get

X_—23A+22[A,iil +).([A?1

2] +~[A~~]‘ii]) + [A~] +~[Ai~]~i2] +~[Aj2]~1]

~ (n+l)2~L

Use [Ag] m, [rn±rn+]=0= [rn_rn_], [rn+t] c rn~,[rn_t] m, all of which are easyto show. Set

Q=[A,~1], P=[A~2}+~[Q~~11]

T= [Ai~3]+ ~[Arn ]‘72i] + ~[Ai~2]~1~]+ ~[Q’i~~Pi11]+ ~[Q’lin,j’liml

S — ~12m + ~ [~11Q 1— ~ ,~11Q+]+cQ= —~-~ (P±—P_)+cQ,

whereccP is arbitrary. (Subscriptsrn andt denotecomponentsof rn andt respectively.)Clearly Q, P, T, Sare all in rn andwe get (13).

Let us now substitute(13) in (11) andcomparecoefficientsof powersof ~%as in refs. [2,11, we find thaton trajectoriesof (11) P, SandTare givenby

P=~~(~÷—~_), (14)

T= (+1)2 Q+ (~~)2 [Q+[Q~Q+]1~(n+l)~ (15)

S= (+1)2 (~++~)+c(Q++Q), (16)

it in no way affectsour resultsto set c=0.Now we cansay then:On the symplecticmanifold ((ca,w ia) the hamiltonianflow correspondingto the hamiltonianfunction

H(x) = — ~tr(x2A2) — in (10), and with initial condition

X(0)=~3A+A2Qo+2(Po__~-j-[QoQo+])+To+[SoQo]+(n~l)2A~ (17)

(s0= _-~-~(P0+_P0))

generatesthesystemofthird order ode’s in C”, havingtheLax representation

L=[L1+Q,L],

where

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L=~3A+~2Q+~(_n+l ~ n+1 [Q~Q+I)_ (fl+l)2

+ (~~)2 ([Q+[Q~Q+]]-[Q~[Q~Q+]]+[QQ]+A). (18)

The integrabilityofthe hamiltoniansystem(~2a,w ~ H, X(0)) impliesthe integrability ofthe Laxsystemgivenby (18) with the appropriateinitial condition.

We canwrite (18) in coordinatesby recognisingthatQ hasthe form

(0 qt~ k,—q 0

q is a complexn X 1 matrix.We get

dt3 q,=~ q~i)_3~i~~ q~j)_Aooqi+ > ~ (19)

Initial conditionsare

q,(0)=a,, 4,(0)=i(n+l)ir,, ~1(0)=—(n+1)

2r,+2IaI2a,

where

Qo(° ~),~0(° ~t), T0(° ‘a). (20)

Letuswrite (19) in termsof therealvariablesx,y whereq=x+iy, settingA=E+ixwith£T= —landXT=X:

d3—~ x, = —3~~ (x~+y1)—3x, ~ (x~+y

1j’,)+3y, ~ (x~—~y~)—iA00y,+ ~ (E,~x—x,~y~),dt J1 J=

1 j=l

d3 “

—i- y = — 3j~,~ (xi +y1) — 3y, ~ (x,i, +y~$~,)— 3x, ~ (x~$’,—.~y1)+iA00x1 + ~ (L’,~y1+x,y1).

dt 1=! 1=l

(21)

Wecango further:It is clear from (21) thatwhenA00 andx arebothzerowe cansety=0, which correspondsto theimaginary

partsof a, n, ‘r all beingzero. We thenhavethe systemof third order ode’son P~:

d3 ‘I

~ x1—3x1 ~ x~i,+~ I,~x,. (22)

dt 1=l

If now 1=0, wecan integrate(22): multiply by x, andusex13~~=(d/dt)(x,ii1—~±~)to get

~

y,=const,i=l, ..., n. Now use~ setting~ we find that in the case1=0,(22) givesus

(23)

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Volume 127, numberI PHYSICSLETTERSA 1 February1988

Thisis certainlya new dynamicalsystemandis very likely integrable.The readermaylike to apply the ideato the following case:

v=2, g=su(n+2), A=idiag(a,/J;—(a+fl),...,—(a+fl)), a�fl.

This systemis relatedto the homogeneousspaceSU(n+2)/S(U(l)xU(l) XU(n)). It is convenientto set�1=(n+l)a+/J,�2=a+(n+l)fl. Write (9=orbit throughA

2+A with AeC~(A).We find in this casethatwe cansay:On the symplectic manifold (~2,W~) the Hamilton flow correspondingto the hamiltonian function

H(x) = — ~tr(x2..~.”~)I _ i generatesthesecondorder systemin C” havingtheLax representation

L=[.~A+Q,L],

where

/0 a* xtQ=(—a 0 yt

—y 0

and -

/ 0 a* xt\ / 0 i(�i_�2)_Ia* ...j~jlj~t

L=12A+A( —a 0 yt J + ( —i(�1—�2)’â 0

—y oJ \ —i�1~’~ —i�~~’~ o

0 OT\ (—xix 0 OT\ i/o 0 OT\ /A_~ 0 OT+ ( 0 al

2 OTJ+( 0 0 OT J+—(o _yty OT J+( 0 A0 OT

~l’~2 \ o 0 oJ ~ \ 0 0 xxtJ ~2 \~ 0 yyt) \, 0 0 A (24)

a, XEC, x, y are nXl complexmatrices,A0,A_1eu(n),trA+A0+A_~=0.In coordinates this systemis givenby

—i ~ 2i ~�l’�2 �2 El’’E2

1 ày— ~ aj+ ~ laI2x+2 -~- (x~x)x+-~‘- (ytx)y+Ax_A_ ~X,

~I ~l ‘“~2 �2 El “‘~2 El �2

1 á*x+~a*i_ ~ ~�2 Ei �2 Ei �2 �2

It is interestingto identify the canonicalcoordinatesfor all thesesystems— i.e. thosewhich diagonalisethesymplecticform w ~. An accountof how to do this is in preparation.

A similar pieceof work by Reymanrelatingto the resultsof ref. [1] hasbeenrecentlybroughtto my at-tention,seeref. [10].

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Acknowledgement

I amgratefulto manyof theparticipantsof theLondonMathematicalSocietyconferenceon IntegrableFieldTheoriesheldat Durhamin July 1986,andin particularto JohnHamad,for helpful discussions.I wouldalsolike to thank my collaboratorsin ref. [1].

References

[1] A. Fordy, S. WojciechowskiandI. Marshall,Phys.Lett. A 113 (1986)395.[2] A. FordyandP. Kulish,Commun.Math.Phys.89(1983)427.[3] M. Adler, Invent.Math. 50(1979)219.[41B. Kostant,Invent.Math.34 (1979) 195.[5] W. Symes,Invent.Math.59 (1980) 13.[6] M. Adlerand P. vanMoerbeke,Adv. Math. 38 (1980) 318.[71 A. ReymanandM. Semenov-Tian-Shansky,Invent.Math.63 (1981)423.[8] MR. Adams,J. HamadandE. Previato,GeneralisedMosersystemsandmomentmapsinto loop algebras,lAS. preprint.[9] B. Kostant,Lecturenotesin mathematics,Vol. 170. Quantizationandunitary representations(Springer,Berlin, 1970) pp.87—208.

[10] A. Reyman,An orbit interpretationof hamiltoniansystemsofharmonicoscillatortype,Proc.SteklovInstituteLeningrad,Vol. 155(1986).

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