Bihamiltonian structure of the KP hierarchy and the WKP algebra

Phy51c5Letter5 8 266 ( 1991 ) 298-302 N0rth-H011and

PHY51C5 LE77ER5 8

81ham11t0n1an 5tructure 0f the KP h1erarchy and the WKp a19e6ra J056 M. F19Uer0a-0•Farr111 a,~,2, JaV1er M a 5 6,3 a n d E d U a r d 0 R a m 0 5 a,4 a 1n5t1tuut v00r 7he0ret15che Fy51ca, Un1ver51te1t Leuven, Ce1e5t1jnen1aan 200D, 8-3001 Hever1ee, 8e191um 6 Departament0 de F151cade Part1cu1a5 E1ementa1e5, Facu1tad de F151ca, Un1ver51dad de 5ant1a90, E- 15 706 5ant1a90 de C0mp05te1a, 5pa1n

Rece1ved 6 June 1991

We c0n5truct the 5ec0nd ham11t0n1an 5tructure 0f the KP h1erarchy a5 a natura1 exten510n 0f the 6erfand-D1ckey 6racket5 0f the 9enera112edKdV h1erarch1e5.7he f1r5t5tructure - wh1ch ha5 6een recent1y1dent1f1eda5 W t+00- 15c00rd1nated w1th the 5ec0nd 5tructure and ar15e5a5 a tr1v1a1 (9enera112ed) c0cyc1e.7he 5ec0nd 5tructure 91ve5r15et0 a n0n 11neara19e6ra, den0ted Wr,~, w1th 9enerat0r5 0f we19ht5 1, 2, .... 7he reduced a19e6ra 06ta1ned 6y 5ett1n9 the we19ht 1 f1e1dt0 2er0 c0nta1n5 a center1e55 V1ra50r0 5u6a19e6ra, and we ar9ue that th1515 a un1ver5a1W a19e6ra fr0m wh1ch aU Wn a19e6ra5are 06ta1ned thr0u9h reduct10n.

1.1ntr0duct10n 7 h e K P h1erarchy [ 1 ] ha5 6eC0me a U614U1t0U5 5U6jeCt 1n 5tr1n9 the0ry ever 51nCe the 5Urpr151n9 d15c0very 0 f the c0nnect10n 6etween the matr1x m0de1 f0rmu1at10n 0f tw0-d1men510na1 9rav1ty and the 9enera112ed K d V h1erarch1e5 (5ee e.9. ref. [ 2 ] ) . Neverthe1e55 the K P h1erarchy- a5 we11 a5 1t5 reduct10n5 ha5 6een a r 0 u n d f0r much 10n9er 1n the f1e1d 0 f c1a551ca1 1nte9ra61e 5y5tem5.1t5 exact 1nte9ra6111ty pr0pert1e5 and 1t5 re1at10n w1th free ferm10n the0r1e5 v1a 1nf1n1te d1men510na1 9ra55mann1an5 are 6y n0w we11 under5t00d; a5 we11 a51t5 ham11t0n1an nature w1th re5pect t0 the natura1 K1r1110v 6racket 0n a c0adj01nt 0r61t 0f a f0rma1 L1e a19e6ra 0f p5eud0-d1fferent1a1 0perat0r5 [ 3,4 ], wh1ch ha5 6een recent1y 1dent1f1ed [ 5,6 ] w1th the W1 +~ a19e6ra. 7he5e re5u1t5 n0tw1th5tand1n9, a 61ham11t0n1an 5tructure f0r the K P h1erarchy ha5 n0t, t0 the 6e5t 0f 0ur kn0w1ed9e, 6een c0n5tructed. 1t 15 the purp05e 0f th15 1etter t0 d0 50. A5 a 6ypr0duct we w1115ee that the c0n5erved char9e5 06ey ( L e n a r d ) recur510n re1at10n5. A5 a further 6ypr0dAddre55after 0ct06er 1991: Phy51ka115che51n5t1tutder Un1ver51tf1t80nn, W-5300 80nn 1, FR6. E-ma11addre55: f96da 11•61eku111.817NE7. E-ma11addre55:jama5•eu5cvx.decnet.cern.ch. E-ma11addre55:f96da06 •61eku111.817NE7. 298

uct, wh1ch 15 1ntere5t1n9 1n 1t5 0wn r19ht, we f1nd a n0n11near a19e6ra 9enerated 6y f1e1d5 0f we19ht5 1, 2, ... fr0m wh1ch W~+~0 can 6e rec0vered a5 a tr1v1a1 (9enera112ed) c0cyc1e. Furtherm0re, 5ett1n9 the f1e1d 0f we19ht 1 e4ua1 t0 2er0 y1e1d5, up0n ham11t0n1an reduct10n, a n0n11near a19e6ra c0nta1n1n9 a center1e55 V1ra50r0 5u6a19e6ra - 1.e., c0nta1n1n9 a 5u6a19e6ra 150m0rph1c t0 the L1e a19e6ra 0f D1ff 51. 1t 15 rea50na61e t0 expect and, 1n fact, we 5ha11 ar9ue that th15 15 the ca5e, that th15 new n0n11near a19e6ra 15 a un1ver5a1 W a19e6ra 1n the 5en5e that a11 W , a19e6ra5 [ 7 ] can 6e 06ta1ned fr0m 1t thr0u9h ham11t0n1an reduct10n. 7 h e K P h1erarchy 15 def1ned a5 the 1505pectra1 pr061em f0r the p5eud0-d1fferent1a10perat0r ( W D 0 ) A=0+

~ 0-1u1. 1=0

(1)

7 h e K P f10w5 are 91ven 6y 011

~[A~+,A1 .

(2)

7 h e nth-9enera112ed K d V h1erarchy 15 06ta1ned 6y 1mp051n9 the c0n5tra1nt A ~ = 0.1t 15 we11 kn0wn that the f10w5 1n ( 2 ) pre5erve th15 c0nd1t10n a n d that they are 61ham11t0n1an; that 15, ham11t0n1an re1at1ve t0 tw0 c00rd1nated P01550n 6racket5: the f1r5t and 5ec0nd

0370-2693/91/$ 03.50 • 1991 E15ev1er5c1ence Pu6115her5 8.v. A11r19ht5 re5erved.

v01ume 266, num6er 3,4

PHY51C5 LE77ER5 8

6e1•fand-D1ckey 6racket5 [ 8 ]. Furtherm0re, 1t 15 6y n0w c0mm0n kn0w1ed9e that the 5ec0nd 6racket 91ve5 r15e t0 a c1a551ca1 rea112at10n 0f the E , a19e6ra, and that the f1r5t 6racket can 6e 06ta1ned fr0m the 5ec0nd a5 a tr1v1a1 (9enera112ed) c0cyc1e. 1t 15 theref0re a natura1 4ue5t10n t0 a5k whether the 61ham11t0n1an 5tructure 0f the nth KdV h1erarchy 15 1nher1ted under reduct10n fr0m a 51m11ar 5tructure 1n the KP h1erarchy. We part1a11y an5wer th15 4ue5t10n aff1rmat1ve1y 6y pr0v1n9 that the KP h1erarchy 15 1n fact 61ham11t0n1an. M0re0ver we pre5ent 50me ev1dence that 1ead5 u5 t0 the c0njecture that the reduct10n actua11y w0rk5. 7he tw0 ham11t0n1an 5tructure5 0f the KP h1erarchy are 06ta1ned 6y a 5tra19htf0rward m0d1f1cat10n 0f the ham11t0n1an f0rma115m f0r 9enera112ed Lax 0perat0r5. Lack 0f 5pace f0r61d5 a11 6ut a 5ketchy de5cr1pt10n 0fthe f0rma115m and hence we refer th05e reader5 n0t fam111ar w1th the f0rma1 ca1cu1u5 0f p5eud0-d1fferent1a1 0perat0r5 a n d / 0 r the 6a51c5 0f1nte9ra61e 5y5tem5 t0 D1ckey•5 c0mprehen51ve treatment [ 9 ].

0AF=

29 Au9u5t 1991

F[A+~]

~ ak ~=0 ~

k=0

(4) ~

•

where the Eu1er var1at10na1 der1vat1ve 15 91ven 6y

8F

8uk

•

0f " ~ (~0)10U~1)

(5)

1=0

N0t1ce that 60th 5um5 are actua11y f1n1te f 0 r f a d1fferent1a1 p01yn0m1a1 and thu5 the act10n 0f0A 15 we11def1ned 0n 0ur c1a55 0f funct10n5. 7he 5pace 6P 0f~D0~5 0fthe f0rm ~1~0 6101- ~parametr12e5 the 0ne-f0rm5. N0t1ce that 6y def1n1t10n the a60ve 5um 15 f1n1te 51nce ~D0~5 are f0rma1 Laurent 5er1e5 1n 0 -~. Dec0mp051n9 the r1n9 ~ 0f ~D0~5 a5 = ~+ 09 ~ , where the ~ are re5pect1ve1y the 5u6r1n95 0f d1fferent1a1 and 1nte9ra1 0perat0r5, we 5ee that 6e ~- ~ / 0 - ~~ . 7he dua1 pa1r1n9 6etween vect0r5 and 0ne-f0rm515 natura11y 91ven 6y the Ad1er trace 7 r def1ned a5 f0110w5: f0r any ~ D 0 P = ~ p ~ 0 -~, 7 r P = fre5P--fp~. 7hu5, 1 f A ~ Y and X ~ then (0A, X ) = 7r(AX). 7he reader can check that the pa1r1n9 15 n0nde9enerate. 1f F=ff 15 any funct10n 0n M, we def1ne 1t5 9rad1ent d F 6 y 0AF=(0A, dF) whence 1t f0110w5 that

2.81ham11t0n1an 5tructure 0f the KP h1erarchy

dF= k~0~ We want t0 def1ne P01550n 6racket5 1n the 5pace M 0 f K P 0perat0r5 A 0fthe f0rm ( 1 ). 1n 0rder t0 def1ne P01550n 6racket5 we need t0 def1ne 5evera19e0metr1c 06ject5: the c1a55 0f funct10n5 0n wh1ch we w15h t0 def1ne the P01550n 6racket5, the vect0r f1e1d5 and 0nef0rm5, and a map 5end1n9 (the 9rad1ent 0f) a funct10n t0 1t5 a550c1ated ham11t0n1an vect0r f1e1d. We w111def1ne P01550n 6racket5 0n funct10n5 0fthe f0rm

0k-1 .

1t 15 06v10u5 that f0r any funct10n 1n the c1a55 def1ned a60ve, 1t5 9rad1ent 15 a 0ne-f0rm. 7he 1a5t 1n9red1ent re4u1red t0 def1ne P01550n 6racket5 15 a 11near map 12 fr0m 0ne-f0rm5 t0 vect0r f1e1d5 wh1ch take5 the 9rad1ent 0 f a funct10n t0 1t5 a550c1ated ham11t0n1an vect0r f1e1d. 7h15 map 15 natura11y 1nduced fr0m a 11near map J: 6 e ~ 9 - 6y 12(X) =0j~x). 7he P01550n 6racket 15 then 91ven 6y {F, 6 } = (K2(dF), d 6 )

F[A1 = ~f(u),

(3)

w h e r e f ( u ) 15 a d1fferent1a1 p01yn0m1a11n the u~, and the 1nte9ra1 519n 5tand5 f0r any rea50na61e 11near map ann1h11at1n9 perfect der1vat1ve5 50 that we can ••1nte9rate 6y part5••. 7he tan9ent 5pace 3- t0 M at A 15150m0rph1c t0 the 1nf1n1te51ma1 def0rmat10n5 0f A, wh1ch are c1ear1y 91ven 6y 4~D0~5 0f the f0rm ~ ,%0 0-1a1. Each A e J 91ve5 r15e t0 a vect0r f1e1d 0A wh05e act10n 0n a funct10n F = f f 15 91ven 6y

(6)

=7r(J(dF)d6).

(7)

Demand1n9 that th15 6e ant15ymmetr1c and 06ey the Jac061 1dent1ty, c0n5tra1n5 the a110wed map5 J. We ca11 the a110wed map5 ham11t0n1an. We n0w pr0ceed t0 c0n5truct a 0ne-parameter fam11y 0f 5uch map5. Let 2 6y any c0n5tant parameter and def1ne .,~=A +2. 7hen def1ne J 6 y

J(X) = (AX) + J - , 4 ( X A ) + =3( x2)• - (2x)•3,

(8)

f0r x a n y w D 0 . Fr0m the f1r5t e4ua11ty 1t f0110w5 that 299

V01ume 266, num6er 3,4

PHY51C5 LE77ER5 8

J ann1h11ate5 0-1 ~ , hence 0n1y the 1ma9e 0f X 1n 6~ rea11y matter5, wherea5 fr0m the 5ec0nd 1t f0110w5 that J(X) 11e51n 9-; 1n 0ther w0rd5, Jdef1ne5 a map 6e--. J . 7he pr00f that the P01550n 6racket5 def1ned 6y J are ant15ymmetr1c and 06ey the Jac061 1dent1ty f0110w5 c105e1y the pr00f0f6e1•fand and D1ckey [ 8 ] f0r the1r ana1090u5 re5u1t (cf. a150 ref. [9 ] ) and thu5 we 0m1t 1t here. Mak1n9 the dependence 1n 2 man1fe5t we can wr1te J(X) =J2(X) +2J1 (X) where J~ (X) = [X+, A]•,

J2( X) = (AX) + A - A ( XA) . .

(9)

(10)

80th J~ and J2 def1ne (c00rd1nated) P01550n 6racket5 wh1ch, a5 we w111n0w 5h0w, are the f1r5t and 5ec0nd ham11t0n1an 5tructure5 0f the KP h1erarchy 0f e4uat10n5. N0t1ce that J1 15 06ta1ned fr0m J2 6y 5h1ft1n9 u0~u0+2. 7he fundamenta1 P01550n 6racket5 c0mputed fr0m J~ have recent1y 6een 5h0wn t0 6e 150m0rph1c [ 5,6 ] t0 W~ +00. Let u5 def1ne funct10n5 Hk= ( 1/k)7rA k. 7hey are c1ear1y c0n5erved 4uant1t1e5 f0r the Lax f10w5, 51nce the Ad1er trace ann1h11ate5 c0mmutat0r5. M0re0ver they are n0ntr1v1a1 f0r a11 k. U51n9 the cyc11cpr0perty 0f the trace 0ne f1nd5 that 0AHk=7r(AA ~-~) fr0m where the 9rad1ent5 1mmed1ate1y f0110w:

dHk=A k-1 m0d 0 - 1 ~ .

(11)

29 Au9u5t 1991

3. Ham11t0n1an reduct10n f0r u0= 0

7he free term (.u0) 1n the KP 0perat0r d0e5 n0t ev01ve under any 0f the KP f10w5 91ven 6y (2) and thu5 the 5u6man1f01d 1Q10f KP 0perat0r5 w1th u0= 0 15 pre5erved under the KP dynam1c5.7he5e dynam1c5 can 6e de5cr16ed 1ntr1n51ca11y re1at1ve t0 a 61ham11t0n1an 5tructure 0n 1{,11nduced 6y that 0n M and wh1ch we n0w pr0ceed t0 de5cr16e. A5 we w111 n0w 5h0w, the f1r5t 5tructure re5tr1ct5 tr1v1a11y t01V151nce u015 centra1 re1at1ve t0 1t; wherea5 re1at1ve t0 the 5ec0nd 5tructure, 1{,1 15, at 1ea5t f0rma11y, a 5ymp1ect1c 5u6man1f01d 0f M and the 1nduced 6racket5 are the D1rac 6racket5.70 5ee th15 we need t0 1ntr0duce a few ca1cu1at10na1 t0015. 1f FA-7 r AA, F8---7r 8A are 11near funct10n5 0n M - 1.e., A, 8 e 3e 1ndependent 0fA - 50 that dFA =A and dF8 = 8, then the1r P01550n 6racket re1at1ve t0 a ham11t0n1an map J 15 91ven 6y

{Fa, F8}=7r J(A )8= --7rAJ(8) .

(13)

1fA=~1~0a,01-1 and 8=Y~1>~06,01-L,then (13) can 6e rewr1tten a5

{FA,F5}=- ~ ~ a,(12u~6j) ,

(14)

where the t21j are d1fferent1a1 0perat0r5 def1ned 6y

J(8) = ~1a 0-1(121j~6j). 1n term5 0fthe C00rd1nate5 u, 0n M, the fundamenta1 P01550n 6racket5 are 91ven 6y

61ven a ham11t0n1an map J, there 15 a way t0 a550c1ate a f10w t0 a funct10n H 6y 0A/0t=J(dH). Ch0051n9./2 and Hk we f1nd

J2(dHk)=J2(A k-~ ) =[Ak+,A] -

011 0tk

(fr0m ( 2 ) ) ,

50 that the KP f10w5 (2) are ham11t0n1an re1at1ve t0 the 5ec0nd ham11t0n1an 5tructure. M0re0ver, 51nce [Ak+, A]=J1(dHk+1) we 5ee that they are a150 ham11t0n1an w1th re5pect t0 the f1r5t 5tructure. 7hu5 the KP f10w5 are 61ham11t0n1an, a5 5ummar12ed 6y the Lenard re1at10n5 re1at1n9 the c0n5erved char9e5 0,4 =J~ (dHk+.) =J2 ( d H k ) . 0tk 300

( 12 )

{U1(X), Uj(y)} =--ff21j•6(X--y),

(15)

where 12015 taken at the p01nt x. Fr0m the expre5510n (9) f0r the f1r5t ham11t0n1an 5tructure we 5ee that 51nce 0n1y A + c0ntr16ute5, t20~= 0 f0r a11 1. Hence u0 15 centra1 a5 c1a1med. Fr0m the expre5510n (10) f0r the 5ec0nd ham11t0n1an 5tructure we 5ee that 1200= - 0 and hence 1t 15 f0rma11y 1nvert161e. H0wever, the D1rac 6racket5, 1nv01v1n9 a ~-0 ~ c0u1d 6e p0tent1a11y n0n10ca1. We w111 5ee, h0wever, that th1515 n0t the ca5e. 7he fundamenta1 6racket5 0n 1~1

{U•(X), Uj(y) }= --~1j~6(x--y)

(16)

are 91ven fr0m th05e 0n M v1a the Ce1e6rated D1raC f0rmu1a

~u = ~u - f2~0~Q6-0~f20j.

(17)

V01ume266, num6er 3,4

PHY51C5 LE77ER5 8

7h15 f0rmu1a 1nduce5 6racket5 0n funct10n5 F, 6 0n 1V106ta1ned fr0m th05e 0f M a5 f0110w5. We f1r5t extend the funct10n5 t0 M wh1ch we a150 den0te F, 6. 51nce the exten510n 15 n0t un14ue, the term5 8F/8u0, 86/8u0 1n the expre5510n f0r the 9rad1ent5 are rendered undef1ned. 7h15 am619u1ty 15 f1xed 6y demand1n9 that the a550c1ated ham11t0n1an vect0r f1e1d 6e tan9ent t0 1V1.1n 0ther w0rd5, we f1x 8F/8u0 6y the re4u1rement that J2(dF) 5h0u1d have n0 free term. Let the 9rad1ent 0f F 6e 91ven 6y d F = ~1>~0 Xt 01-1. 7hen X1=8F/8U~ f0r 1> 0 and X015 t0 6e determ1ned 6y demand1n9 that J 2 ( d F ) = Y~.j~>00--~(92¢Xj) have n0 free term. 1n 0ther W0rd5, X0=-

2 126-0~£20j~XJ,

j>0

(18)

whence the expre5510n f0r the D1rac 6racket5 f0110w5 at 0nce after rewr1t1n9 J2(dF) 1n term5 0f the X~ f0r 1>0: J2(dF)=

2

0--1(~U~XJ ) "

(19)

1,j> 0

7h15 c0nd1t10n 0n X0 can 6e wr1tten m0re 1nvar1ant1y a5 f0110w5. 7he free term 0f J2(dF) 15 91ven 6y re5 J2 (dF) 0- ~. 5ett1n9 th15 t0 2er0 we f1nd 0 = r e 5 J 2 ( d F ) 0 -1

=re5 ( A ( d F A ) • 0 - 1 - ( A d F ) ~ A 0 -1 ) =re5 ( d F A ( A 0 - 1 ) + - - A d F ( A 0 - 1 ) + ) =re5 [dF, A] .

(20)

A5 an e4uat10n 0n X0, th15 51mp1y 5ay5 that X~ 15 t0 6e e4ua1 t0 the Ad1er re51due 0 f a c0mmutat0r, wh1ch 15 a1way5 a perfect der1vat1ve. 7heref0re X0 can a1way5 6e 501ved a5 a d1fferent1a1 p01yn0m1a1 1n the X, and hence the 1nduced 6racket5 0n 1V1are 10ca1. N0t1ce that 1f we a5519n we19ht5 [ 0 ] = 1 and [u~] = 1 + 1, then the KP 0perat0r A 1n (1) 15 h0m09ene0u5 0fwe19ht 1.1t 15 ea5y t0 ver1fy that 1fwe def1ne [f•] = [f] + 1 f0r any h0m09ene0u5 d1fferent1a1 p01yn0m1a1f the mu1t1p11cat10n 1n the r1n9 0fWD0~5 pre5erve5 the we19ht and thu5 6ec0me5 a 9raded r1n9. 51nce the ham11t0n1an 5tructure5 are def1ned u51n9 0n1y the r1n9 5tructure 0f the WD0~5 (up t0 50me harm1e55 pr0ject10n5), 1t 15 c1ear that the d1fferent1a1 0perat0r5 5~20 (and a150 ~0) are h0m09ene0u5 0f we19ht 1+j+ 1.7hu5 the c0eff1c1ent5 0f920 (and a150 0f~1j) can 6e a pr10r1 d1fferent1a1 p01yn0m1a15 1n the

29 Au9u5t 1991

u~, u2, ..., u1+j; a1th0u9h 5ymmetry c0n51derat10n5 actua11y f0r61d the appearance 0f u~+j. A 5tra19htf0rward c0mputat10n y1e1d5 the f1r5t few /20:

~00 =

-

0,

92,1 =0u~ + u ~ 0 , ff~20 ~---/,/10 ,

ff212 =20U2 +U2 0-- 02/21 ,

K222=(u~+4u3-2u~2)0+u1u~1 --u•2•+2u•3,

(21)

w1th 12j~=~2~ and a11 0ther 120 6e1n9 2er0.7he D 0 f0110w fr0m (17): ~ 1 =0u, + u 1 0 , ff~12 =20U2"~1-~/2 0 - - 02//1 ,

~22 = (2u~ +4u5-2u~2)0+2u~u~ - u ~ + 2 u ~ 3 ,

(22)

where a5 6ef0re ~j~ = ~ . 7he f1r5t e4uat10n exh161t5 exp11c1t1y a 5u6a19e6ra 150m0rph1c t0 the L1e a19e6ra 0f D1ff 5 ~. 7he 5ec0nd e4uat10n 5ay5 that u 2 - •u•t 15 a D1ff 5 ~ten50r 0fwe19ht 3. Pre5uma61y th15 rema1n5 true f0r the uj> 2 and they can 6e m0d1f1ed 6y add1n9 d1fferent1a1 p01yn0m1a15 0f the 10wer uj that make them 1nt0 ten50r5. 7he P01550n 6racket 0f u2 w1th 1t5e1f 1nv01ve5 u~ n0n11near1y wh11e a150 the f1e1d u3 5h0w5 up. 7heref0re the fundamenta1 P01550n 6racket5 0n 1V1 re1at1ve t0 the 5ec0nd 5tructure def1ne a n0n11near exten510n 0f the L1e a19e6ra 0f D1ff 5 ~ 6y f1e1d5 0fwe19ht5 3, 4, ....

4. C0nc1u510n5

7he 9enera112at10n 0f the 6e1•fand-D1ckey 6racket5 t0 the 5pace 0 f K P 0perat0r5 ha5 a110wed u5 t0 d15p1ay the KP h1erarchy a5 a 61ham11t0n1an 1nte9ra61e h1erarchy. 7he fundamenta16racket5 c0m1n9 fr0m the f1r5t 5tructure have 6een recent1y 1dent1f1ed w1th the W1 +~ a19e6ra [ 5,6], wherea5 the 0ne5 c0m1n9 fr0m the 5ec0nd 5tructure y1e1d a n0n11near a19e6ra wh1ch we den0te WKp. 7he fundamenta1 6racket5 1nduced 0n the 5u6man1f01d 0f KP 0perat0r5 w1th0ut free term 6y re5tr1ct1n9 the 5ec0nd ham11t0n1an 5tructure, y1e1d a n0n11near exten510n 0fthe L1e a19e6ra 0fD1ff 5 ~6y ten50r5 0fwe19ht5 3, 4, .... 301

v01ume 266, num6er 3,4

PHY51C5 LE77ER5 8

1mP051n9 the c0n5tra1nt A ~ = 0 reduce5 the K P h1erarchy t0 the nth 0rder 9enera112ed K d V h1erarchy wh1ch 15 61ham11t0n1an re1at1ve t0 the 6e1•fandD1ckey 6racket5. 7he5e are 6racket5 def1ned 0n the 5pace Y 0f Lax (d1fferent1a1) 0perat0r5 L = 0n + .... Let N c M den0te the 5u6man1f01d 0f K P 0perat0r5 06ey1n9 the c0n5tra1nt A ~ = 0 . 7 h e n N 15150m0rph1c t0 X , the 150m0rph15m 6e1n9 exp11c1t1y 91ven 6y A ~ A ~ . 7 h e funct10n5 H19enerat1n9 the K P f10w5 re5tr1ct t0 N and can 6e pu11ed 6ack t0 Y . M0re0ver the Lax f10w5 ( 2 ) pre5erve N a n d 1nduce f10w5 0n ~/" wh1ch are 61ham11t0n1an re1at1ve t0 the 6e1•fandD1ckey 6racket5 and are 9enerated 6y the funct10n5 1nduced 0n Y 6y the H1. N0w, N can 6e 91ven a 61ham11t0n1an 5tructure f0r the KdV f10w51n tw0 d1fferent w a y 5 . 0 n the 0ne hand, we can pu11 6ack the 6e1•fand-D1ckey 6racket5 0n X t0 N. 1t 15 5tra19htf0rward t0 der1ve rea50na61y exp11c1t f0rmu1a5 f0r the 1nduced 6racket5. 0 n the 0ther hand, 0ne expect5 that N 1nher1t5 a 61ham11t0n1an 5tructure fr0m the 0ne 0n M that we have c0n5tructed 1n th15 paper. 1t 15 natura1 t0 c0njecture that the5e tw0 5tructure5 c01nc1de. 1t 15 n0t en0u9h that the Lax f10w5 c0rre5p0nd, 51nce the vect0r f1e1d5 c0rre5p0nd1n9 t0 the Lax f10w5 c 0 m m u t e and thu5 0n1y 5pan a (p055161y max1ma1) 150tr0p1c 5u65pace 0f the tan9ent 5pace. H0wever, th15 c0rre5p0ndence pre5ent5 5tr0n9 ev1dence f0r 0ur c0njecture. 7 h e d1ff1cu1ty 1n pr0v1n9 the c0njecture ar15e5 1n der1v1n9 an expre5510n f0r the 6racket5 1nher1ted 6y N fr0m M. 7 h e c0n5tra1nt A ~ = 0 tran51ate51nt0 an 1nf1n1te n u m 6 e r 0f c0n5tra1nt51nv01v1n9 the c00rd1nate5 u1 - name1y 5ett1n9 each uj>~n e4ua1 t0

302

29 Au9u5t 1991

a d1fferent1a1 p01yn0m1a1 1n the u1<, - wh1ch make5 the D1rac pre5cr1pt10n d1ff1cu1t t0 1mp1ement. W0rk 0n th15151n Pr09re55.

Ackn0w1ed9ement E.R. 15 9ratefu1 t0 the D e p a r t a m e n t 0 de F151ca de Part1cu1a5 E1ementa1e5 0f the Un1ver51dad de 5ant1a90 de C0mp05te1a and J.M.F. t0 the 1n5t1tute 0f 7he0ret1ca1 Phy51c5 at 5t0ny 8 r 0 0 k f0r the1r h05p1ta11ty dur1n9 the f1na1 5ta9e5 0f th15 c011a60rat10n.

ReferenCe5 [ 1] E. Date, M. J1m60, M. Ka5h1wara and 7. M1wa, Pr0c. Jpn. Acad. 5c1. 57A (1981) 387; J. Phy5. 50c. Jpn. 50 (1981) 3866. [2] M. D0u91a5, Phy5. Len. 8 238 (1990) 176. [3]A.6. Re1man and M.A. 5emen0v-7yan-5han5k11, J. 50v. Math. 31 (1985) 3399. [4] Y. Watana6e, Lett. Math. Phy5. 7 (1983) 99. [5] K. Yama915h1, A ham11t0n1an 5tructure 0f KP h1erarchy, W~+~ a19e6ra, and 5e1f-dua1 9rav1ty, Lawrence L1verm0re prepr1nt (January 1991 ). [6] F. Yu and Y.-5. Wu, Ham11t0n1an 5tructure, (ant1-) 5e1fadj01nt f10w51nKP h1erarchyand the W~+~ and W~ a19e6ra5, Utah prepr1nt (January 1991 ). [7] V.A. Fateev and 5.L. Lykyan0v, 1ntern. J. M0d. Phy5. A 3 (1988) 507. [8] 1.M. 6e1•fand and L.A. D1ckey, A fam11y 0f ham11t0n1an 5tructure5 c0nnected w1th 1nte9ra61e n0n11near d1fferent1a1 e4uat10n5, 1PM AN 555R, M05c0w prepr1nt 136 (1978). [9]L.A. D1ckey, Lecture5 1n f1e1d the0ret1ca1 La9ran9eham11t0n1an f0rma115m, unpu6115hed; 1nte9ra61e e4uat10n5 and ham11t0n1an 5y5tem5 (W0r1d 5c1ent1f1c, 51n9ap0re), t0 appear.