Constrained KP models as integrable matrix hierarchies

H. Aratyn, L. A. Ferreira, J. F. Gomes, and A. H. Zimerman

Citation: Journal of Mathematical Physics 38, 1559 (1997); doi: 10.1063/1.531908 View online: http://dx.doi.org/10.1063/1.531908

H. Aratyn

Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607-7059

L. A. Ferreira, J. F. Gomes, and A. H. Zimerman

Instituto de Fı´sica Teo´rica-UNESP, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Brazil

~Received 3 April 1996; accepted for publication 12 August 1996!

We formulate the constrained KP hierarchy~denoted bycKP K11,M) as an affine

sl

d(M1K11) matrix integrable hierarchy generalizing the Drinfeld–Sokolov hier-archy. Using an algebraic approach, including the graded structure of the general-ized Drinfeld–Sokolov hierarchy, we are able to find several new universal results valid for thecKPhierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac– Moody current algebra. An explicit example is given for the casesld(M1K11), for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimplenon-regularelementE ofsl(M1K11) and the content of the center of the kernel ofE. © 1997 American Institute of Physics.

@S0022-2488~97!00202-8#

I. INTRODUCTION

The constrained KP~cKP!hierarchy occupies one of the central positions in the current study of integrable hierarchies. This is mainly due to the fact that it represents a direct generalization of the KdV models and includes an impressive list of partial differential soliton equations1–7. Also important are the relationships of the cKP hierarchy to several physically relevant models~like Toda models and discrete matrix models!.

Let us recapitulate the most general form of the Lax operator belonging to thecKPhierarchy:

L5DK11

1

(

l50

K21

ulDl1

(

i51

M

F_iD21_C

i, ~1.1!

and subjected to the following flow evolution equations:

]L ]tn5

@~Ln/~K11!_!

1,L#. ~1.2!

We will denote the hierarchy defined by~1.1!and~1.2!ascKPK11,M. There are several different parametrizations ~obtained by acting with various Miura maps! of the coefficients ul,Fi,Ci in

~1.1!, defining various reformulations of thecKPK11,M hierarchy. It is, for instance, known that the Lax operator from~1.1!can be rewritten as a ratioL5LM1K11/LMof two purely differential operatorsLM1K11 andLM of ordersM1K11 and M, respectively.

K50,M5N has been treated in Ref. 14 and shown to correspond to the generalized NLS hierarchy,15 which in turn generalizes the AKNS hierarchy16–18for whichK50,M51.

The paper is organized as follows. In Section II the connection between the generic matrix eigenvalue problem we are interested in and the pseudo-differential Lax operator of thecKPtype is established. Section III provides the algebraic foundation, within the generalized Drinfeld– Sokolov hierarchy for our model, with subsection III A dealing with the example of the

sl

d(M1K11) algebra. Section IV examines the Zakharov–Shabat equation for the problem and provides the construction of the recurrence operator. In Section V the second bracket of the cKPK11,Mhierarchy is obtained as a Dirac bracket, where the matrix hierarchy is considered as a constrained system. We conclude with Section VI, suggesting few possible applications and ex-tensions of our results.

II. MATRIX EIGENVALUE PROBLEM ANDcKPLAX OPERATORS

Consider the matrix eigenvalue problem

LC5~D1A1lE!C50 ~2.1!

for the (M1K11)3(M1K11) Lax matrix operatorL5D1A1lE given by

L5

1

D 0 • • • _{0 q1 0} • • • • • • • • • ₀

0 D 0 • • • _{q2 0} • • • • • • • • • ₀

A A ₀ • • • • • • • • • A

0 D qM 0 • • • • • • • • • 0

r1 r2 • • • rM D2v₁ l 0 • • • • • • 0

0 • • • _{0 0 D2}v₂ l 0 • • • A

0 • • • _{0 0 0 D2}v₃ l • • • A

A • • • _{0 0} • • • _{0 0}

A • • • _{0 0} • • • _{0 l}

0 • • • _{0 l 0} • • • ₀ • • • _D2v_K

11

2

~2.2!

and acting in~2.1!on the (M1K11) columnC such thatCT

5(c1,c2, . . . ,cM1K11).D is a

differential operator which acts on the function f according to @D,f#5f

8

. We impose the condi-tion (_i51

K11

v_i50. Similar matrix operators have appeared in, e.g., Refs. 12 and 19.

We write explicitly the linear problem~2.2!as

]c_i1q_ic_M1150, i51, . . . ,M,

(

i51

M

rici1~]2v₁!c_M

111lcM1250,

~2.3!

~]2v_r!c_M

1r1lcM1r1150, r52, . . . ,K,

lc_M111~]2vK11!cM1K1150.

Equation~2.3!gives rise toK11 scalar Lax eigenvalue equations,

LjcM1j5~2l !K11cM1j; j51, . . . ,K11, ~2.4!

L_r5~D2v

r21!~D2vr22!• • •~D2v2!

S

D2v12

(

i51

M

r_iD21_q

i

D

~D2vK11!• • •~D2vr!,

~2.5!

L15~D2vK11!~D2vK!• • •~D2v2!

S

D2v12

(

i51

M

r_iD21_q

i

D

.

For all K11 values of j the corresponding Lax operator Lj can be cast in the form of the Lax operator~1.1!incKPK11,M hierarchy. All of the Lax operators~2.5!can be associated with the one-matrix eigenvalue problem~2.2!. The question therefore arises whether the above reduction of the matrix eigenvalue problem determines uniquely the scalar Lax operator. We will now answer this problem of potential ambiguity by showing the equivalence between all the Lax operators from~2.5!in the sense of the Darboux–Ba¨cklund~DB!symmetry.

The following similarity transformations connect the neighboring Lax operators from~2.5!:

~D2v_r

21!21Lr~D2v_r

21!5Tr221 1

LrTr215Lr21, r53, . . . ,K12, ~2.6!

with the ‘‘periodic condition’’ L_K125L1. In Eq. ~2.6! we have introduced the operator T_j5F_jDF2_j 1

with F_j[exp(*v_j) to emphasize the Darboux–Ba¨cklund character of the above

similarity transformation. In addition we have the following eigenvalue equation holding for each Lax operatorL_j:

LjFj50, j52, . . . ,K11. ~2.7!

Assume now thatLjsatisfies the Lax flow equation~1.2!. Applying it to the equation~2.7!we find that (]_t

n2(Lj

n/(K11)

)₁)F_j is annihilated byL_j and we therefore expect, without loosing general-ity, to have the following identity (]t_n2(Lj

n/(K11)

)₁)F_j5a(t2,t3, . . . )Fj, where the propor-tionality coefficienta depends only on timesti withi>2. Comparing both sides of this identity we find thata50 and therefore F_j is an eigenfunction ofLj, meaning that

]t_ne~ * v_j!

5~Ln_j/~K11!

!₁e~ *v_j!

. ~2.8!

Recall now that the DB transformation L→TLT21, whereT5FDF21 with an eigenfunction Fpreserves the form of the Lax equation~1.2!, i.e., the DB transformed Lax operator satisfies the same evolution equation as the original Lax operator ~see, e.g., Refs. 20, 6, and 7!. Since

Lj5Tj21Lj21Tj221 1

and we have Eq.~2.8!, we conclude that all the Lax operators from~2.5!are equivalent belonging to the same ‘‘multiplet’’ from the DB symmetry point of view.

III. CONSTRUCTION OF HIERARCHIES

In this section we provide the basic ingredients for the construction of the type of integrable hierarchies we are going to consider. The discussion is based on the method of references 8–12. LetGˆ _{be an affine Kac–Moody algebra, and}G _{be the finite dimensional simple Lie algebra}

associated to it. The integral gradations ofGˆ _{are defined by vectors}s_{5(s0,s1, . . . ,s}

r),21where

si are non-negative relatively prime integers, andr[rankG. The corresponding grading operator is given by

Qs[

(

a51

r

sa

2l_a•_H0

a_a2 1Nsd, ~3.1!

whereH_a0,a51,2, . . . ,r, are the Cartan subalgebra generators ofG_,l_aits fundamental weights satisfying 2l_a•a_b/a

b

2

Gˆ_{, responsible for the homogeneous gradation of} Gˆ_{, corresponding to} s_{hom5(1,0,0, . . . ,0). In}

addition, we have,Ns[ (_ir₅₀ s_im_i

c

,c5(_ar₅₁ m_aca_a,m0

c

51, wherec is the highest positive root of G_{. So, we have}

Gˆ₅%

nPZGˆ_n~s!, @Qs,Gˆ_n~s!#5nGˆ_n~s!, @Gˆ_m~s!,Gˆ_n~s!#,Gˆ_m1n~s!. ~3.2!

Introduce the Lax matrix operator

L[]x1E1A, ~3.3!

where E is a semisimple element of Gˆ_{, lying in} Gˆ_{1, and A is a potential belonging to the}

subalgebra Gˆ_{0. The construction works equally well with E belonging to any subspace} Gˆ_n,

n.0, andAhaving grade components ranging from 0 ton21. However, such general setting will not be needed in what follows.

The fact thatE is semisimple means thatGˆ _{can be decomposed into the kernel and image of}

the adjoint action ofE

Gˆ_5Ker~adE)1Im~ad E!. ~3.4!

As a consequence of Jacobi identity one has

@Ker~adE!, Ker~adE!#, Ker~adE!, @Ker~adE!, Im~adE!#,Im~adE!. ~3.5!

Using the fact that E is semisimple, one can perform a gauge transformation to rotate the Lax operator into Ker~adE). Consider

L0[ULU21[]x1E1

(

j52`

0

K~j!_[]

x1E1K0, ~3.6!

where U is an exponentiation of negative grade generators, U5exp(`_j51T(2j)

, with T(2j) P Gˆ2j(s). Decomposing ~3.6! into Qs eigensubspaces, we get equations of the form

K(j)52@E,T(j21)

#1X(j), whereX(j) depends onT(m)s form.j21. Therefore, starting from the highest grade component, one can choose T(j21)

to exactly cancel the component of X(j) in Im~adE). Consequently, one gets K(j) P Ker~adE). Notice that the choice of T(j21) is not

unique, since its component in Ker~adE) is not relevant in the cancellation of the Im~adE) component ofX(j). In addition,T(j21)

is determined as a local polynomial of the components of the potentialA and itsx-derivatives.

The flow equations for the hierarchies are constructed in a quite simple way.9,10 Consider a constant elementb(N), with grade N (N.0), belonging to the center of Ker~adE). Then, from the considerations above one gets thatb(N) commutes withL0, and so, from~3.6!one has

@L,U21

b~N!U#50 ~3.7!

or

@L,~U21

b~N!U!>0#52@L,~U21b~N!U!,0#, ~3.8!

where (•_{)>0 and (}•₎

,0 mean non-negative and negative grade components, respectively. One

observes that the l.h.s. of~3.8!has components with grades varying from 0 toN11, and the r.h.s. has grades varying from2`to 0. Consequently, both sides of~3.8!have to lie on the subalgebra

dL dtb~N!5

dA

dtb~N![@L,Bb

~N!#, ~3.9!

where

Bb~N![~U21b~N!U!_>0[

(

j50

N

B_b~j~!N!, B~_bj~!N!_PGˆ_j~s!. ~3.10!

Notice thatB_b(N) is a polynomial of the components ofA and itsx-derivatives.

From~3.6!and~3.9!one gets

dL0

dtb~N!5@L0,B

˜

b~N!#, with B˜_b~N![UB_b~N!U211

dU dtb~N!U

21

. ~3.11!

In fact,B˜b(N) lies in Ker~adE). In order to see that, we denoteB˜_b(N)5B˜

b(N)

K

1B˜_b(N)

I

, withB˜_b(N)

I

PIm~adE) andB˜b(N)

K

P Ker~adE). Then, splitting~3.11!in its Ker~adE!and Im~adE)

com-ponents one gets

dK0 dt_b~N!2]xB

˜

b~N!

K

5@K0,B˜_b~N!

K

# ~3.12!

and

]xB˜b~N!

I

1@E1K0,B˜_b~N!

I

#50. ~3.13!

The highest grade component of~3.13!is@E,(B˜_b(N)

I

)N#50, with (B˜b(N)

I

)N[˜Bb(N)

I

ùGˆ

N(s). Since there is no intersection between Ker~adE) and Im~adE), it follows that (B˜_b(N)

I

)N50. Following this reasoning one concludes thatB˜_b(N)

I

50, and soB˜b(N) given in~3.11!lies in Ker~adE).

Notice that if Ker~adE) is Abelian~as is a case whenE is regular!, then~3.12!constitutes a local conservation law.

The flows defined in ~3.9! commute, as a consequence of the fact that B˜b(N)_PKer~adE),

and that b(N) _{belongs to the center of Ker~adE). Indeed, those facts imply that} @d/dtb(N)2B˜_b(N),b(M)#50. Conjugating with U, one gets (d/dt_b(N)) (U21b(M)U) 5@Bb(N),U21b(M)U#. Taking the positive grade part and subtracting the same relation with b(M) andb(N) interchanged, one gets

dB_b~M!

dt_b~N! 2

dB_b~N!

dt_b~M!5@Bb

~N!,U21b~M!U#_>02@B_b~M!,U21b~N!U#_>0. ~3.14!

But@Bb(N),U21b(M)U#_>05@B_b(N),B_b(M)#1@B_b(N),(U21b(M)U)_,0#_>0. Since b(N) andb(M)

com-mute, it follows that@(U21

b(M)U)>0,U21b(N)U#52@(U21b(M)U),0,U21b(N)U#. Taking the

positive grade part of it one gets@Bb(M),U21b(N)U#_>05@B_b(N),(U21b(M)U)_,0#_>0. Therefore one

concludes that

dB_b~M!

dt_b~N! 2

dB_b~N!

dt_b~M!1@Bb

~M!,B_b~N!#50 ~3.15!

and due to Eq.~3.9!that is a sufficient condition for the flows to commute:

F

d

dtb~M!,

d

Notice that the gauge transformations

L→eSLe2S, Bb~N!_→eSB_b~N!e2S,

dS

dtb~N!50, SP

K[Gˆ_{0ù Ker~adE!} ~3.17!

are symmetries of the flows equations ~3.9!, in the sense that they preserve the form of the Lax operatorL. Associated to such symmetries we have conserved quantities. Indeed, the component of zero grade on the r.h.s. of~3.8!is@E,(U21

b(N)U)₂1#. But that implies that the l.h.s. of~3.8!,

and consequently both sides of~3.9!, have no components in Ker (adE). Then

dAK

dtb~N!50, A

K

[AùKer~adE!. ~3.18!

Therefore, if we choose AK50 at tb(N)50, it will remain zero for all times. That is a

reduction procedure which we will use below to obtain the constrained KP hierarchies from the above formalism. We shall decompose the potentialA _PGˆ_0as

A[A01A

K

~3.19!

with AK

P K, and A0 lying in the complement M of K in Gˆ0. A0 contains therefore the

dynamical variables of the integrable hierarchy.

Since we are working with loop algebras~vanishing central term! it is useful to work with finite matrix representations. The commutation relations for Gˆ _{can be written as}

@T_am,T_bn#5f_abc T_cm1n

, @d,T_am#5mT_am, ~3.20!

where T_a0[Ta, a51,2, . . . ,dimG, are the generators of the finite simple Lie algebra G, and

f_abc are its structure constants. If one has a ~finite! matrix representation of G_{, then one can}

construct a representation ofGˆ _{by replacing}

T_am→zmTa, ~3.21!

wherez is a complex parameter. However, in some calculations we will be interested in another representation of such type, where the powers of the complex parameter count the grade w.r.t.

Qsdefined in~3.1!. Accordingly we replace

T_am→llTa, l5ga1mNs, ~3.22!

wherel is a complex parameter, and

F

_b

(

51

r

sb

2l_b•_H0

a_b2 ,Ta

G

5gaTa. ~3.23!

Notice thatga take values between2Ns11 andNs21.

In the representation~3.21!one hasd[z(d/dz). Therefore if@Qs,X#5xsX, withQsgiven by ~3.1!one has@Qs,zX#5(xs1Ns)zX. Now, ifb(N)is an element of the center of Ker~adE), so is

zb(N). We shall denoteb(N1Ns)_[

zb(N). Therefore,zU21

b(N)U5U21 b(N1Ns)

U, and so

z~U21_b~N!_U!

>01z~U21b~N!U!,05~U21b~N1Ns!U!>01~U21b~N1Ns!U!,0, ~3.24!

where (•_{)>0and (}•₎

,0mean the non-negative and negatives-grade components, respectively. But

since multiplication byzincreases thes_{-grade byNswe have that the positive part of~3.24!leads}

B_b~N1Ns!5zB_b~N!1z

(

j51

Ns

~U21_b~N!_U!

2j. ~3.25!

Notice the second term on the r.h.s. of ~3.25!have components withs_{-grades varying from 0 to} Ns21. But, analyzing~3.1!, one concludes that any generator having positive grade w.r.t. d, has

necessarily s_{-grade greater or equal to Ns. Therefore, we conclude that the quantity} z(_j

51

Ns

(U21_b(N)_U

)₂j isz independent in the representation~3.21!.

A. The case ofsld(M1K11)

We now apply the above formalism to the example of the affine Kac–Moody algebra

Gˆ_5sld_{(M1K11), (A}

M1K

(1)

) without a central term ~i.e., a loop algebra!, furnished with the fol-lowing gradations_{and corresponding grading operatorQs:}

~3.26!

We will denote the simple roots ofsld(M1K11) byaj, j50,1, . . . ,M1K, witha0[2c, and c being the highest positive root of G_{5sl(M1K11), which is the subalgebra of}

sl

d(M1K11) commuting withd. Since sld(M1K11) is simply laced we normalize the roots such thata_j2

52. The ordering of the roots is such that forj Þ k,aj•ak52dj,k61 (modM1K11), j,k50,1, . . . ,M1K. The fundamental weights l_j satisfy l_j•a_k5d_{j,k. We choose the} semi-simple elementE to be

E5

(

j5M11

M1K

Ea~_j

0!

1E_{2~ a}

M111• • •1aM1K!

~1!

. ~3.27!

One can check that each generator inE has grade one w.r.t.Qs. Hence

@Qs,E#5E. ~3.28!

The zero grade subalgebra is

Gˆ₀[$G₀~0![sl~M11!; aj•H~0!, j5M11,M12, . . . ,M1K%, ~3.29!

whereG₀(0)_{is the sl(M11) subalgebra of}G_{5sl(M1K11), with simple roots a1, . . . ,a}

M. ForE defined in~3.27!we have

Ker~adE!5$Kˆ0[sld~M!%Uˆ~1!,Hˆ K%, ~3.30!

wheresld(M) is the affine Kac–Moody subalgebra of Gˆ₅d_{sl(M1K11) with simple rootsaj,}

j51,2, . . . ,M21, anda052(a11a21 . . .1aM21). The Kac–Moody algebraUˆ(1) is

gener-ated by l_M•_H(k)_{, k P} Z_{. In addition} Hˆ_{K is the principal Heisenberg subalgebra of sl}d_(K11)

E_l1~K11!n5E~an_M!₁₁1a_M121. . .1a_M1l1Ea~n_M!₁₂1a_M131. . .1a_M1l111. . .

1E~an_M!_1K2l111a_M1K2l121. . .1a_M1K211a_M1K1E2~ aM111aM121. . .1aM1K2l11!

~n11!

1E_{2~ a}

M121aM131. . .1aM1K2l!

~n11!

1 . . .1E_{2~ a}

M1l1aM131. . .1aM1K!

~n11! _~

3.31!

withl51,2,3, . . . ,K, and so

@Qs,E_l1~K11!n#5~l1~K11!n!El1~K11!n. ~3.32!

Notice that E1[E. In addition, since we are working here with the case of loop algebra

(c50),

@EN,EN₈#50. ~3.33!

In fact, in the representation~3.22!of the loop algebra one has

E5lE˜5l

S

(

j5M11

M1K

Ea_j1E2~ a_M111• • •₁a_M1K!

D

, ~3.34! and in the defining representation ofsl(M1K11), one has

E

˜₅

S

0 0

0 ˜e

D

; ˜e5

S

0 1

0 1

0 1

1 0

D

, ~3.35!

whereE˜ and˜e are (M1K11)3(M1K11) and (K11)3(K11) matrices, respectively. Ele-ments of this type are among the generators of the nonequivalent Heisenberg subalgebras, see, e.g., Appendix of Ref. 13 for the cases ofsl(3) andsl(4).

In addition, one has

El1~K11!n5ll1~K11!n~E˜!l. ~3.36!

Also forc50 we have

center Ker~adE!5$Uˆ~1!,Hˆ

K%, ~3.37!

where Uˆ(1) is generated by l_M•_H(k)_{,k P} Z_{. Notice that} @Qs,l_M•H(k)#5k(K11)l_M•H(k).

Therefore the center of Ker~adE), while having generators of arbitrary grade, has one and only one generator of a given grade. Then the choices we have for the elements b(N), introduced in

~3.7!, are

b~N!5EN, N51,2, . . . ,K mod~K11!, ~3.38!

b~k~K11!!

5l_M•_H~k!_{, kP}Z_{. ~3.39!}

According to~3.9!each of the generators from the center of Ker~adE!appearing in~3.38!–~3.39!

The gauge symmetries of the model are then given by the transformations~3.17!, where S

belongs to the subalgebra

K[Gˆ_{0ù Ker~adE!5$sl~M!,l}

M•H~0!%, ~3.40!

wheresl(M) is the subalgebra ofG_{5sl(M1K11) with simple roots a1,a2, . . . ,aM}

21.

The generators of the complementM _of K _inGˆ_{0 are}

M_5$P

6i5E~6~ a_i1a_i111. . .1a_M!

0!

,i51,2, . . . ,M, and a_a•_H~0!_{,a5M11,M12, . . . ,M1K%.}

~3.41!

We then parametrizeA0, defined in~3.19!, as follows:

A05

(

i51

M

~qiPi1riP2i!1

(

a5M11

M1K

Ua~aa•H~0!!, ~3.42!

whereqi, ri andUa are fields of the model.

As we have shown in ~3.18!, AK is constant in time. Therefore, we will work with the submodel defined by

AK50. ~3.43!

The flow equations~3.9!, in this case, become

dA0

dt_b~N!2]xBb~N!

0

5@E1A0,B_b~N!

0

#, ~3.44!

whereB_b(N) 0

is the constrainedBb(N), i.e.,

B_b~N!

0

5Bb~N!u_AK

50. ~3.45!

The effective potential of our submodel lies therefore, on the tangent plane of the coset space

Gˆ_0/K_[(sl(M11) % _U(1)K_)/(sl(M) % _U(1)

M).U(1)Mis generated bylM•H(0), and conse-quently is a linear combination of the generators of the Cartan subalgebra ofsl(M11) and also of all the generators ofU(1)K. Remember thatl_i5K_{i j}21a_j, and the inverse of the Cartan matrix

K_{i j}21

ofAn has no vanishing elements. Those facts preventGˆ0/K from being a symmetric space.

Indeed, one can verify that

@P_j,P_2j#5aM•H~0!1

(

i5j M21

a_i•_H~0!_{, P}K₁M_{, P}

j,P2jPM, ~3.46!

sinceaM•H(0)have components on bothK andM. However, one has

@Pj,P2k#PK for jÞk, @Pj,Pk#5@P2j,P2k#50 for any j,k. ~3.47!

1. The case K50

In this case we have Gˆ_5sld_{(M11). The relevant gradation is the homogeneous one,}

s_{5(1,0,0, . . . ,0), and soQs[d. The semisimple elementE is now given by}

This example was discussed in detail in Ref. 14, and here we just give a brief description of it to make contact with the model discussed above.

The grade zero subalgebra is

Gˆ_05$G_{[sl~M11!% ~3.49!}

and

Ker~adE)5$sld~M!%_Uˆ_~1!% ~_3.50!

withUˆ(1) being generated byl_M•_H(k)_,kP Z_.

The center of Ker (adE) is just the homogeneous Heisenberg subalgebra of

Gˆ_5sld_{(M11), namely,}

center Ker~adE!5$l_i•_H~k!_{,i51,2, . . . ,M,kP}Z_%. ~3.51!

Therefore, we can introduce a flow for each element@see~3.7!#

b_i~k![l_i•_H~k!_{, k being a positive integer.} ~_3.52!

The gauge subalgebraK _{introduced in}~3.17!is

K_5$sl~M!%_{U~1!%, ~3.53!}

where sl(M) is the subalgebra of G_{5sl(M11) with simple roots a1,a2, . . . ,a}

M21, and U(1) is generated byl_M•_H(0)_{. We write the potentialAP}Gˆ_0asA5AK_{1A0, withA}K_PK_and

A0 lying in the complementM ofK inGˆ0. Then, we parametrizeA0 as

A05

(

i51

M

~qiPi1riP2i!, ~3.54!

whereP_6iwere introduced in~3.41!. Comparing with~3.42!, we notice an absence ofU_a’s fields in this special example. Also Gˆ_0/K _{is now a symmetric space.}

IV. ZAKHAROV–SHABAT EQUATION AND RECURSION OPERATOR

Recall that within the models considered here the semisimple element E is given for

G_{5sl(M1K11) by ~3.34!. We first notice that E commutes with its conjugated counterpart}

E† and therefore, although not Hermitian, may be diagonalized.22 As a consequence, the Lie algebraG _{under consideration can be decomposed into graded subspaces, i.e.,}

G₅%_sG_{s; @E,}G_s#5sG_s, ~4.1!

wherescan be in general a complex number (sP C). This is a crucial property allowing to solve

the Zakharov–Shabat ~ZS! equation in the manner shown below. Consider, namely, vs the ZS equation

]_mA02]Bm1l@E,Bm#1@A0,Bm#50, ~4.2!

whereA0 defined in Section III lies, as described there, in subspace orthogonal to Ker~adE).

Decomposing Bm5(sBm

(s)

2]B_m~0!

1

(

x1y50

@A~x!,B~_my!#50, ~4.3!

]mA~s!2]Bm~s !

1lsB_m~s!1

(

x1y5s

@A~x!,B~_my!#50, ~4.4!

where in the last equation the summation is overx,y_P C_{and includesy50,x5s. Equations~4.3!} and~4.4! contain components of ~4.2! in Ker~adE) and Im~adE)5%

sPc_2$0%Gs. We now as-sume the following expansions:

B_m~s!5

(

i50

m21

B~_ms!~i!li_{, sÞ0, ~4.5!}

for all nonzero gradation components ofBm, while for the zero component we find from~4.3!by integration:

B_m~0!5D21

S

(

x,yÞ0

x1y50

@A~x!,B~_my!#

D

1lm_L

m, ~4.6!

where the last ~integration constant!term on the right hand side of~4.6!is of higher order than those in~4.5!. Its presence is allowed by the structure of~4.2!as long asL_mP Ker~adE).

Inserting~4.5!in~4.4!we find by collecting the coefficients oflk21

lB~_ml!~k21!52]B_m~l!

~k!2

(

x,yÞ0

x1y50

@A~l!,D21

@A~x!,B_m~y!~k!##2

(

x,yÞl x1y5l

@A~l!,B_m~y!~k!#; ~4.7!

for k5m we obtain

lB_m~l!~m21!52@A~l!,L_m#. ~4.8!

From ~4.7!we find that the general solution can be rewritten as

B_m~l!~k21!5

(

yÞ0

F_l,yBm~y!~k! ~4.9!

with

F_l,y5

1

l

S

Ddl,y2xPGrad

(

~M!

@ad_A~l!D21ad_A~x!d_x,2y1ad_A~x!d_l2x,y#

D

. ~4.10!

Using~4.9!repeatedly we are led to

B~_ml!~0!5

(

y₁, . . . ,ym21

F_l,y₁Fy₁,y₂• • •Fy_m22,y_m21Bm ~ym21!_~m

21!, ~4.11!

and after taking into account~4.8!

B_m~l!~0!5

(

y1, . . . ,ym21

F_l,y₁Fy₁,y₂• • •Fy_m22,y_m21 1

ym21

adL_mA~ym21!. ~4.12!

]mA~l!5]Bm ~l!

~0!2

(

x1y5l

@A~x!,B~_my!~0!#5l

(

y

F_l,yBm

~y!

~0!. ~4.13!

Substituting the solution ofB_m(l)(0) in terms of A(y) from~4.12!we arrive at

1

l ]mA

~l!

5

(

y1, . . . ,ym

F_l,y

1Fy1,y2• • •Fym21,ymadLm

S

A~ym!

ym

D

. ~4.14!

This expression leads to a recursion operator relating consecutive flows belonging the same family of flows generated by the specific elementL_m5(E˜l,l_M•_H(0)_{) from the center of Ker}~_{adE) as} explained in discussion around Eqs.~3.38!–~3.39!. Therefore these consecutive times have indices moduloK11 for the case ofsl(M1K11).

To get the closed expression for the recursion operator we compare Eq.~4.14! to the corre-sponding expression for]m2K21A(l). These flows are related through:

1

l ]mA

~l!

5

(

y1, . . . ,yK11

F_l,y₁• • •Fy_K,y_K11]m2K21

S

A~yK11!

y_K11

D

[R_l,K

11]m2K21A~yK11!/yK11.

~4.15!

This yields an expression for the recurrence operator R _as R₅FK11 _{for the} sl(M1K11)-matrix hierarchy.

V. THE SECOND BRACKET STRUCTURE

The potentialAintroduced in~3.3!is an element of the subalgebraGˆ_{0, and so we shall denote}

it as

A5hab_w~T

a!Tb, w~Ta![Tr~TaA!, ~5.1!

wherehab _{is the inverse of the trace form ofG}ˆ

0,hab[Tr(TaTb),a,b51,2, . . . ,dimGˆ0.

There is a natural Poisson bracket structure for the manifold spanned byw’s components of

A, induced by theG_{0-KM current algebra:}

$w~T!~x!,w~T

₈

!~y!%PB5w~@T,T

8

# !~x!d~x2y!1Tr~TT

8

!d

8

~x2y!, T,T

8

PGˆ0. ~5.2!

The model we are interested in is a constrained system where the components of A in Ker~adE) are set to zero@see~3.18!#. The bracket structure of such submodel is then given by the Dirac bracket associated to ~5.2!. We denote by K_{i, i51,2, . . . ,dim}K_{, and Mr,}

r51,2, . . . ,dimGˆ_02dimK_{, the generators of the subalgebra} K_{, defined in ~3.17!, and of its}

complementM _inGˆ_{0, respectively.}

The Dirac matrix is given by

D_{i j}~x,y![$w~K

i!~x!,w~Kj!~y!%'hi jd

8

~x2y!, ~5.3!

whereh_{i j}[Tr(K

iKj), and' means equality after the constraints are imposed. Therefore

D_{i j}21

~x,y!'hi j_]

x

21_d

~x2y!, i,j51,2, . . . ,dimK_{. ~5.4!}

Consequently the Dirac bracket is

$w~Mr!~x!,w~Ms!~y!%DB5w~@Mr,Ms# !~x!d~x2y!1Tr~MrMs!d

8

~x2y!

1hi jw~@K_{i,Mr# !~x!w~@}K_{j,Ms# !~y!]}

x

21_d

The subspaceM _{constitutes a representation of the subalgebra}K_,

@K

i,Mr#5Rrs~Ki!Ms. ~5.6!

Therefore, the second term on the r.h.s. of the bracket~5.5!can be calculated using representation theory. The relevant representation here is the tensor product of the representation,R^_{R, defined}

by the linear functionalsw(M_r)(x). We then write

X_rs~x,y![hi j_w~@K

i,Mr# !~x!w~@Kj,Ms# !~y!]x2

1_d

~x2y!

5hi jR_rt~K

i!Rsu~Kj!w~Mt!~x!w~Mu!~y!]x2

1_d

~x2y!

[Cu_M

r

&

x^uMs

&

y]x2

1_d

~x2y!, ~5.7!

where

C_[hi j_K

i^Kj ~5.8!

and where we have denoted states of the representation R^_{R as} w_(M

r)(x)w(Ms)(y)[ uM_r

&

x^uMs

&

y. So, the space variablesxandydefine the left and right entries, respectively, of the tensor product.

The operator~5.8!commutes with any generator

@C_{, 1}^K_i1K_i^_1#50, ~_5.9!

and, according to Schur’s lemma, it is proportional to the identity in each irreducible component of R^_{R. That fact simplifies substantially the evaluation of} ~_5.7!_{. Notice that}C_{is not the} qua-dratic Casimir operator in R^_{R. That operator is given by} C_R^R[hi j_(1^K

i1Ki^1)

3(1^K_j1K_j^_{1), and therefore we have}

C₅1

2~CR^R21^C2C^1!, ~5.10!

whereC5hi jK

iKj is the quadratic Casimir inR.

Decomposing the representationR^_{R in its irreducible components, one can evaluate~5.7!,}

using ~5.10!and the fact that the value of the quadratic Casimir operator in an irreducible repre-sentation isl(l12d), wherel is the highest weight, andd512(a.0 a5(a51

rank

l_a, witha being the positive roots, andl_a the fundamental weights ofK_.

A. The case ofsld(M1K11)-matrix integrable hierarchy

We now consider the example of the affine Kac–Moody algebrasl(M1K11) discussed in subsection III A. The subalgebra K _{and subspace} M _{are defined in ~3.40! and~3.41!,}

respec-tively. We shall denote byK˜_{i, i51,2, . . . ,M}2

21, the generators of the subalgebra sl(M) of

K_.

One can easily verify thatPj and P2j, j51,2, . . . ,M, transform under the representations

M andM¯ ofsl(M), respectively. The highest weights and highest weight states of these repre-sentations are (l1,P1) and (lM21,P2M), respectively. The remaining generators ofM ~3.41!, namely a_a•_H(0)_{, a5M11,M52, . . . ,M1K, are scalars under sl(M). The charges of the} U(1) factor of K _{~3.40!, generated by l}

M•H(0), are 1 for Pj, 21 for P2j, and 0 for

a_a•_H(0)_{. Therefore, the representation R of} K_5sl(M) % _{U(1), defined by} M _{~5.6!, is}

C₅h˜i j_K˜

i^K˜j1 1

Tr~ l_M•_H~0!_!2lM

•_H~0!^l_M•_H~0!

[C˜₁M1K11

M~K11!lM•H

~0!_^l

M•H~0!, ~5.11!

whereh˜i j_{is the inverse ofh˜}

i j5Tr(K˜iK˜j). Notice that Tr(• •), as introduced in~5.1!, is the trace form of the subalgebraGˆ_0.

Let us then analyze the various irreducible components of the representationR^_{R. The states}

uP_i

&

^u_P

j

&

decompose into the symmetric and antisymmetric parts, which are theM(M11)/2 and

M(M21)/2 irreducible representations of sl(M), respectively. The highest weights of these representations are 2l1 and 2l12a1, respectively. Therefore, using~5.11!, and then ~5.10!for

C

˜, one gets

Cu_Pi

_&

^u_Pj

&

₅

S

M1K11

M~K11!2l1~ l112d!

D

uPi

&

^uPj

&

1~ l1~ l11d!!~uPi

&

^uPj

&

1uPj

&

^uPi

&

)

11

4~~2l12a1!~2l12a112d!!~uPi

&

^uPj

&

2uPj

&

^uPi

&

). ~5.12!

Denoting the simple roots of sl(M) as aj5ej2ej11, j51,2, . . . ,M21, ej•ek5djk, one has

l_j5(k51jek2(j/M)(kM51 ek, and therefore d5(kM521 1 _l

k512(kM51 (M22k11)ek. Conse-quently

Cu_Pi

_&

^u_Pj

&

₅ 1

K11uPi

&

^uPj

&

1uPj

&

^uPi

&

. ~5.13!

By the same arguments, one gets

Cu_P

2i

&

^uP2j

&

5 1

K11uP2i

&

^uP2j

&

1uP2j

&

^uP2i

&

. ~5.14!

As for the states uPi

&

^uP2j

&

, we use the fact that the tensor product of the M and M¯ representations of sl(M) produces an adjoint and a singlet, i.e., M^_M¯_{5Adj11. The singlet is}

the state uS

&

[ (_j51MuPj

&

^uP2j

&

. The states of the adjoint are uPi

&

^uP2j

&

for i Þ j, and uP_j

&

^u_P

2j

&

2uPj11&^uP2(j11)&. The highest weight of the adjoint is the highest positive root c5a11a11. . .1aM215e12eM. Therefore, using~5.11!, and then~5.10!forC˜, one gets for

iÞj

Cu_Pi

_&

^u_P 2j

&

52

M1K11

M~K11!uPi

&

^uP2j

&

11

2~c~c12d!2l1~ l112d!2lM21~ lM2112d!!uPi

&

^uP2j

&

52 1

K11uPi

&

^uP2j

&

. ~5.15!

uPj

&

^uP2j

&

5 1

MuS

&

1uXj

&

, ~5.16!

where

uX_j

&

52 1 M

S

k

(

51

j21 kuv

k

&

2

(

k5j M21

~M2k!uv

k

&

D

~5.17!

and where we have denoteduv_k

&

[(uP_k

&

^u_P

2k

&

2uPk11&^uP2(k11)&).

Therefore

Cu_P

j

&

^uP2j

&

5 1

M

S

2

M1K11

M~K11!2

1

2~ l1~ l112d!1lM21~ lM2112d!!

D

uS

&

1

S

2M1K11

M~K11!1

1

2~c~c12d!2l1~ l112d!2lM21~ lM2112d!!

D

uXj

&

52 1

K11uPj

&

^uP2j

&

2uS

&

. ~5.18!

Consequently, from~5.15!and~5.18!,

Cu_Pi

_&

^u_P 2j

&

52

1

K11uPi

&

^uP2j

&

2di j

(

k51

M

uPk

&

^uP2k

&

. ~5.19!

In addition one has

Cu_P

6i

&

^uaa•H~0!

&

5Cuaa•H~0!

&

^uab•H~0!

&

50 ~5.20!

witha,b5M11,M12, . . . ,M1K,i51,2, . . . ,M.

From~3.42!,~5.1!Tr(EaE2a)51, and Tr(aa•H(0)ab•H(0))5aa•ab, one gets that

qi5w~P2i!, ri5w~Pi!, Ua5Kab2

1

w~a_b•_H~0!_{!, ~5.21!}

whereK_ab21 is the inverse ofKab5aa•ab, a,b5M11,M12, . . . ,M1K.

Therefore from~5.5!,~5.7!,~5.13!,~5.14!, ~5.19!, and~5.20!one gets the Dirac bracket for

sl(M1K11), which reproduces~after an appropriate Miura transformation! the second bracket of cKP(K11,M) hierarchy:

$ri~x!,qj~y!%5

S

]x2UM11~x!2

(

s51

M

rs~x!]x2

1

qs~x!

D

di jd~x2y!

2 1

K11ri~x!]x2

1

qj~x!d~x2y!, ~5.22!

$ri~x!,rj~y!%5 1

K11ri~x!]x

21

r_j~x!d~x2y!1r_j~x!]x2

1

r_i~x!d~x2y!, ~5.23!

$qi~x!,qj~y!%5 1

K11qi~x!]x

21

q_j~x!d~x2y!1q_j~x!]_x21

$qi~x!,Ub~y!%52 1

2qi~x!db M11d~x2y!;$ri~x!,Ub~y!%5 1

2ri~x!db M11d~x2y!,

~5.25!

$Ua~x!,Ub~y!%5 1

4Kab]xd~x2y!. ~5.26!

Note that forK50 ~andUa50) we recover from the above bracket structure the second bracket of the NLS-sl(M11) model.14,15This can also be checked directly by applying the same tech-nique as above to the model described in subsection III A 1. Calculation shows that the equation

~5.11!in this case is replaced by

C₅C˜₁M11

M lM•H

~0!_^l

M•H~0! ~5.27!

and Eqs.~5.13!,~5.14!and~5.19!hold withK50.

For the purpose of illustration let us consider the example of sl(3) with M5K51 corre-sponding tocKP2,1with the Lax operator as in~2.5!:

L15~D2v2!~D2v12rD2 1

q!5D21u1FD21_C

, ~5.28!

where u52v22v

8

2rq,F52r

8

2rv,C5q with v5v₂52v₁. The second bracket for u,F,C obtained from~5.22!–~5.26! indeed reproduces the standardcKP2,1second bracket ~see

e.g., Ref. 2!. One checks easily that the following equations:

]_t

2q52q

9

1

v

8

q₁rq2₁v2q,

]t₂r5r

9

1v

8

r1r2q1v2r, ~5.29!

]t₂v5~rq!

8

,

following from the Dirac bracket ~5.22!–~5.26! and the Hamiltonian H15*FC reproduce the

correct flows for u,F,C:

]F ]t25

]2_F

]x2 12u0F, ]C ]t252

]2_C

]x2 22u0C, ]u0 ]t25

]_x~ FC !. ~5.30!

Equation ~5.30! agrees with the second flow equation of the so-called Yajima–Oikawa hierarchy.1,2Note that in this calculation of thet2flow the consistency of the ZS problem required

use ofb(2)5l2

(l1•H) in~4.6!according to the discussion of Section III. Generally forsl(3) we

need to takeb(2k)5l2k_(l

1•H) andb(2k11)5l2k11E.

VI. DISCUSSION AND OUTLOOK

In the formalism based on the pseudo-differential Lax operator, thecKPhierarchy is obtained by constraining the complete KP hierarchy with the symmetry constraints expressed in terms of the eigenfunctions F_i,C_i from~1.1!and imposed on the isospectral flows.

In this paper we have obtained an alternative derivation of thecKPhierarchy as an integrable

sl

d(M1K11)-matrix hierarchy generalizing the Drinfeld–Sokolov hierarchy. The main ingredi-ents of this construction were the semisimple graded, nonregular elementE ofsl(M1K11) and the potentialA belonging to the grade zero subalgebraGˆ_{0. Both the Lax matrix operator as well}

symmetry the relevant phase space turned out to be the quotient spaceGˆ_{0/(Ker~adE)ù}Gˆ_{0). The}

structure of the flows of the hierarchy was shown to be related to the center of Ker~adE).9 The algebraic approach allowed us to write down in closed form a very simple expression for the second Hamiltonian structure with respect to which the flows are Hamiltonian. This bracket structure was explicitly calculated as a Dirac bracket emerging from reduction of Gˆ_{0 to the}

effective phase space of Gˆ_{0/(Ker~adE)ù}Gˆ_{0). Alternative approaches to calculate this bracket}

structure in various parametrizations of the constrained KP hierarchy have been proposed in the past and include e.g., applications of ther-matrix formulation developed in a discrete context23,24 and applications of the generalized Kupershmidt–Wilson theorem.3,25

One expects that several aspects of thecKPformalism will gain substantially by being treated within the algebraic formalism proposed in this paper. Work is in fact in progress regarding the following issues. Possible extensions of thecKPscalar Lax examples by going beyond the alge-braic construction based on thesl(M) algebra by employing different algebras. Calculation of the tau-function, following the dressing transformation method26 and Darboux–Backlund methods7 will help to further establish the connection between the pseudo-differential and matrix tech-niques. One also expects that the use of the matrix hierarchy will be essential for describing additional symmetries of thecKPmodels.

Note added:After this paper had been completed we received a copy of the paperExtensions of the matrix Gelfand–Dickey hierarchy from generalized Drinfeld–Sokolov reduction, by L. Fehe´r and I. Marshall~SWAT-95-61, hep-th/9503217!, which contains related results. We thank Dr. L. Fehe´r for sending us a hard copy of his paper.

ACKNOWLEDGMENTS

H. A. thanks Fapesp for financial support and IFT-Unesp for hospitality. H. A. was supported in part by U. S. Department of Energy, Contract No. DE-FG02-84ER40173. L. A. F., J. F. G, and A. H. Z. were supported in part by CNPq.

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