ELSEVIER

5 April 1999

Physics Letters A 254 ( 1999) 37-46

PHYSICS LETTERS A

Hirota bilinear forms with 2-toroidal symmetry

Kenj i Iohara a, ’ , Yoshihisa Saito b, Minoru Wakimoto ’ a ~e~artrnent of ~at~~nat~cs, Fuculty of Science, Kyoto ~niversi~ Kyoio 606-8502, gamut

h Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 73943526. Japan C Graduate School of Mathematics, Kyushu University. Fukuoka 812-8581, Japan

Received 9 November 1998; revised manuscript received 21 January 1999; accepted for publication 25 January 1999 Communicated by A.P. Fordy

Abstract

We compute Hirota bilinear forms arising from both homogeneous and principal realization of vertex representations of 2-toroidal Lie algebras of type At, Dt, Et. @ 1999 Elsevier Science B.V.

1. Introduction

Soliton equations are known as nonlinear partial differential equations with infinite dimensional symmetry.

For example, KP (Kadomtzev Petviashvili) hierarchy has GL;,- y s mmetry and KdV (Korteweg-de Vries)

hierarchy has &z-symmetry. Their solutions can be realized by means of representation theory. In particular, it was shown in Ref. [ 71 that KdV and NLS (Nonlinear Schriidinger) hierarchies can be obtained from the principal and the homogeneous realization of basic representations of &2, respectively. Here they derived the

~o~esponding Hirota bilinear form and by construction they showed that it possesses “soliton-type”’ solutions. There is a 2-dimensional generalization of affine Lie algebras known as 2-toroidal Lie algebras. This Lie

algebra is the universal central extension of Lie algebras g @I C[ s *’ , t*’ 1, where g denotes a simple finite

dimensional Lie algebra. Few results for representation theory of these algebras are known. In physics, it arises as current algebra of the 4-dimensional Kghler WZW (Wess Zumino Witten) model [ 51. It seems that Z-toroidal Lie algebras are nice candidates to describe higher-dimensional integrable systems.

In this Letter, we apply the method of Ref. [7] to obtain Hirota bilinear forms with 2-toroidal symmetry which include that with affine Lie algebra symmetry as sub-hierarchy. We calculate the Hirota bilinear form for g = A,, II!, El. Its special class of solutions, which might be called “soliton-type” solutions, are presented for g = 512. Some examples for g = 512 are also given.

’ JSPS Research Fellow.

03759601/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved. PfISO375-9601(99)00093-6

38 K. Iohara et al. /Physics Letters A 254 (1999) 37-46

2. P~e~i~na~es on Lie algebra

Here we collect some facts concerning our Lie algebra g& and its vertex representation.

2.1. The definition of Lie algebras

Let g be a simple finite dimensional Lie algebra over @ of rank 1 with a nondegenerate symmetric invariant bilinear form (a, a) and A be the set of all (nonzero) roots of g with respect to its Cartan subalgebra h. Set

A=C[s*‘,t*‘], f2; = Ads$Adt,

and let

-: Q; - &$/dA

be the canonical projection. We define the Lie algebra structure on

gtor := g @ A CD fi;/dA

by

U@3,Y@gl = [X,Yl @fg+(X,Y)tdf)g, [c, everything] = 0,

whereX@f,Y@gEg@AandcfKIA/dA. We remark that the Lie algebra gel which is defined as

with the standard commutation relation exactly coincides with the one considered by Saito and Yoshii [9]. Next, we define a much bigger Lie algebra g& as

with the commutation relations

E&,,*X@fl =X8 (40,J) 9 18&g f9 X@fl =X@Pg&gtf)*

[4,,,fdlogsl = (410gsf)dlogs, Wlo,,,fdlog~l = (g&,,fMw~

[4,,,fdhztl = (&,,fMbitt, [g~~~~,~dlogtl = (g~t~~~~)dlogt + f(W 9

[4cJgsrg4ogrl = (&s&)4osr 9

[.fh,,,,g4o,,l = {f(&,tg) - tG’~o~rf))~~ogt - h%g,g>{Wwf)~~

Note that the affine Lie algebra g is embedded into g& via the variable s, whose image is denoted by g,T. We also remark that the space of vector fields 2) := AC+,, 6~ AC&, naturally acts on gtor via

[“%.@t;tX@gl =X@ (~~l*~bg) t tj=.S,t,

lf40gs,gdlogsl = -ft&,tg)dht, U4o,,~gdlogtl = f(4,,g)dbt,

forXEg and f,gEA. (1)

But one cannot in~~uce a Lie algebra structure on gt,, 63 ?3 which has vertex representations and contains gtor as a Lie subalgebra.

K. Iohara et al. /Physics Letters A 254 (1999) 37-46 39

2.2. Vertex representalions

Let g be a Lie algebra of type AI, t>/ or El. Here we construct so-called vertex representations of g& from a more general point of view.

Let ‘8 be the Lie algebra generated by $%, qL ( k f Z \ (0)) and the central element c with the following relations,

kp:, (P/l = kak+l,OC 3 [pk*$‘11 =o, r&J,t1 =Ot Vk,IEZ\{O}.

Let 71+ be the subalgebra of 3-t generated by qk, 40: (k > 0) and c. We define the one-dimensional %+-module @ vat := C(0) by

cPk.lO) = 0, q&0)=0, ‘JkEZ,,, c.10) = IO).

and set

where C[ ZS,] = &,,=&emh 1s the group algebra of ZS,. We introduce a “I-I-module structure on F, in an obvious way.

For each X E g and 1 E Z, we set

Xl(Z) := CX@s”t’z-“-1 ) K;(z) := CsPt’dlog*i-“-1 ) D;(z) := ~SPt~&g*Z-~‘-‘,

PEZ PEZ PEZ

for * = s, r. These are the generating series of gtor @ V. We shall express the action of these generating series in terms of that of the affine Lie algebra 4 and the Virasoro algebra, which is denoted by

X(z) :=~XQsPz-p-1, T(z) := cL$-“-?

PEZ P@=

Next we introduce the elements dl, es1 E End(F’,) by

d,. (u @ en”,) := m( u @ e”*6’) , #,yq, @ en16,) := u @ e(nL+l)Sr for u @ e”“’ E .Fq

and some of the generating series in IEnd by

P(Z) := Cprz-k-‘, P+(Z) := (d, + CL&-~)~-‘, k+O k#O

b(z) :=exp(~~zk)exp(-~~z-k)e~F.

Lemma 2.1.

(i) There exists a functor .F from the category 0 of i,-module to the category g&-mod.;

(VT) - (V@.F,,?r) such that

*(Xl(z)) = r(X(z))@: A(z)’ :,

?r(l;rf(z)) = I@ : &)A(z)’ :,

40 K. Iohara et al. /Physics Letters A 254 (1999) 37-46

ii(K;‘(z)) = l@: A(Z)’ : ?;-I,

?rW;(z)> = 163 : ~+(z)A(z)” :,

+(&,g.y) = -Res,=o{z(r(T(z)) @ 1 + 18 : p(z)qt(z) :)} ,

(ii) ch3( V) = (chV) ]II( 1 - e-“‘*) -*S( 8,) , ll>O

where SC z > = CnEZ z” is the delta-function.

(iii) Let L(A) be the irreducible highest weight &-module with highest weight h E h*. Then _F( L( A)) is an irreducible ~~~-module.

We remark that one can define the action of Df( Z) on V @ 3P, which is compatible with ( 1 ), by

*(D;(z)) = -z{‘IT(T(z))@: A(z)’ : +l@ : ~(z)~+(z)A(z)’ :}.

Thus in particular, if we choose the homogeneous or the principal realization of the basic representation of g,s (see e.g. Ref. [ 61)) we recover the one which is obtained by Refs. [ 3,11, respectively. We also note that .FP

has the following realization,

via

‘Pk +--+

d

{-

k>O, ‘p: I-----+ auk

,rts, _ &i=t”

-ku_k k<O,

In the next section, we use this realization to obtain Hirota bilinear differential equations.

2.3. Generalized Casimir elements

Let us regard gtor@D as gtor- module. We introduce the symmetric bilinear form (. / .) tor on gta,. @ 2) as follows,

(i) (x @ dsk/y @ tYs’)tOr := (x, Y)~~~~~,O~k+~,O,

(ii> (tpskd ~og~~t~~~~,~~~)~~~ := ~~+~,o~k+~.o Q = s, t,

:i:‘; (d~W#t&+ftor := afi:i #t b = s, t 7 other pairs give zero.

We remark that this bilinear form is gm,-invariant, i.e,

([x3gll~hor= (4[s9rlhr forgE gtory X,Y E gtor@D. (2)

Moreover, this form is degenerate. But if we restrict ( ./.)tOr on (g @ A) x (g @ A), this form becomes

nondegenerate. Next we define “canonical” elements of gt, @ Z, with respect to ( ./.)torr what we call generalized Casimir

elements and denoted by a( z ) = CkEz &?-k-2, as follows (see also Ref. [2]). Let {I’) be an orthonormal

basis of g with respect to (‘, -). Set

dim $3

n(z) := CCI,a(z)~zftk(z)+~C{~~(z)~Drk(z)+D1(z)~K.Ck(z)}.

u=l kEZ *=s,I k&Z

K. lohara et al. /Physics Letters A 254 (1999) 37-46 41

Because of gtO,-invariance of ( .I.)iOr (i.e, (2) ), we have

[fxZ),Btorl =o. (3)

3. Hirota bilinear forms

In this section, we present the Hirota bilinear form associated to Lie algebras g& for g = A/, D/ and El

3.1. Hirota bilinear forms 1

Here we construct a bilinear form associated with the homogeneous realization of vertex representations of V

&or.

LetQ=@f=,Z~;betherootlatticeofgande:Q~Q---+{~l}beth f e unction that satisfies bimultiplicativity,

and the conditions

{

C-1) (&a,) if i < j ,

&(fX;,Cij) = (-l)t(ap,an) ifi=j,

1 if i > j.

Then, as is well known, the affine Lie algebra i,Y acts on

V:=@[x-:,“I1 <j<l, k E z>ol @@e(Q) 1

where Cc(Q) is the twisted group algebra of Q twisted by the cocycle E. Namely, Cc(Q) is a @-algebra spanned by {e”}nea and it satisfies

eneP = E( a, P)e”+P.

Hence by Lemma 2.1, g,“,, acts on .F( V) = V @ .Fv. Let G,,, be the group of linear transformations on F(V)

generated by the exponential action of locally finite elements in g @ A. Choose any orthonormal base {u(‘)} of h and set

I

c &(x);” := exp Cniz.j , CP,,‘“‘(n)z” :=exp CC(cU,U”‘)X,j”Z’

/I>0 { 1 ,I>0 il>O { j>O i=l I

Applying the method developed in Refs. [7,2], we obtain the following.

Theorem 3.1. If r = CPEe rpe@ E G,,,.{( 1 @e”) @ I}, th e completion of G,,,.{ ( 1 @ e”) @ 1) with respect

to the gradation defined by degxj’) = j, then it satisfies the following hierarchy of Hirota bilinear differential

equations,

42 K. Iohara et al. /Physics Letters A 254 (1999) 37-46

x exp (

r f: y(‘)D n>O ;=, n .$I) ..,(,fi.,,),., orp”+a

+ [@‘- @‘I2 + c {i ((@ - @‘,u(‘))D!;~ + $ 1 D$;‘D;’ + 2~lr):‘)D;5)

?I>0 i=l ,+k=,I, k>O j,k>O

+ 2 x(k - It)&-,& &((K - D,>ii> k>O J

x exp c&y(‘)D n,O i=, n ,,)exp(,,.D~,,),o,=O, forP’,P”EQ,

where D$:,), D,,, , D, and b,!b’ = iD$:: stand for Hirota bilinear derivative and K, y = { yi’)} , ii = {ii,,} , i7 = {fi,} are regarded as independent variables.

We remark that this theorem can be obtained by rewriting

Lb(rc3.) =o,

because of (3). Furthermore, we have set

u,=o, D,, = 0 , for it E Z,O ,

since, by definition, r is constant with respect to u,, (n > 0).

3.2. Hirota bilinear forms I1

Here we construct bilinear forms associated with the principal realization of vertex representations of g&

Let us set

E := {(i,r)/l < i < Z,r E Z} , E+ := {(i, r) E Elr E Z&O} .

Then, as is well known, the affine Lie algebra 6, acts on

VP’ := @[xi;,I(i, r) E E+] .

Hence by Lemma 2.1, g& acts on F( Vpr) = VP’ @ F,+,. Let h be the Coxeter number of the Lie algebra g.

Introduce a Z/h&gradation on g,

g = c g’.” ) g’.” := C&A,& ifjzO, j,-~,/,z

.iGZ/M? tl if j=O,

where we set Aj := {(Y E Alhta E jmodh}. The element e := cf=r e,, + e-0, where ep stands for a root vector of the root p and B is the highest root, is a regular semisimple element so its centralizer a in g is

a Cartan subalgebra. Let A P’ denote the set of all roots of g with respect to 5. The linear transformation

w := exp( VP”), where p” 1s the element in h such that (p”, cq) = 1 for any simple root (Y;, satisfies

K. Iohara et al./Physics Letten A 254 (1999) 37-46 4.7

WI~W = exp( 2?r’y ) idgc,, . One can decompose A P’ into Z-orbits via the (wj-action. Let (~1, ~2, . . . , ye} be a

set of its representatives. Since e E 9”) is a homogeneous element, one has 5 = CIEZIhZ5 n g(j). Choose its

homogeneous basis Slil E 5 n g(nri) ( 1 < i 6 i> such that (.!?‘I, Si.il) = h&+i,,, where

1 = rnf < r?E’ < . . ’ <FE&l <vq=h-I

is the set of exponents of g. For each a! E A P’, let er be a nonzero root VeCtOJ of the root (Y. We decompose

this element as

epr := 0 c ep~S.i) a ’ where e, v.(.i) E gCi) .

jEZ//lZ

Let GEr be the group of linear transformations on F( VP’) generated by the exponential actions of locally finite elements in g @ A. Set

E P,E(x)z” := exp C “j;J rr,, irh

IQ0 I

(.jJ)EE, }.

The following theorem is a generalization of that in Ref. [2].

Theorem 3.2. If 7 E G&( 1 @ l), the completion of GFi,..( 1 @ 1) with respect to the gradation defined by degx,:, = m; + rh, then it satisfies the following hierarchy of Hirota bilinear differential equations,

- hx { & (; c D,:,&,,,_,:, +2~(kk+m;)i.:~D~.:.~+,) It>0 i=1 ,,!G--1. k>O

,i.k>O

+ 2h x(k - It)&-,,L&,, S,,((K - &Iii) exp

(

c Y;;~D,:~

k>O (i;r)EE,

) exp(~WL,)~o7=0.

where Dx_ , D,,, , D, stand for Hirota bilinear derivative and K, .Y = { y,$‘) > , ii = {ii,,) , fi = {fi,,) are regarded as independent variables.

3.3. Special solutions (5&-case)

Here we describe a special classes of solutions, which might be called “soliton type” solutions, for the g = s12 case arising from the homogeneous realization. (See Ref. [ 21 for the principal realization.)

For simplicity, let us set

x, .= L,!‘) ” A’ ’

7 := c 7,&P * .sEZ

Under this setting, bilinear equations in Theorem 3.1 can be written only in terms of Schur polynomials as

follows,

44 K. Iohara et al. /Physics Letters A 254 (1999) 37-46

[(m-n)‘+C{2(m-n)D,+CD,D,,+2~ky,D~,~, i-20 , / k=r. k>O

.j.k>O

+2x(k - t-)&-r& U(K-&,)~ k>O

+ (-l),‘-, cc Fj((K - D,)ii)Sk(2Y)s,-2+2,,-*~(-2Bx) rgo /tk=r.

j&O

+ (-1)“‘-“C C sj((K-D~>1”1>Sk(-2Y)Sr-*-2rn+2n(2Bx) r>o ,+il=r.

,i,kW

x exp cy,D (,Y,, ( .,y)exp(,P,D.s)~n,+l oTn--l =O form,nEZ. (4)

Let V be the homogeneous realization of the basic representation of gK2. We define the operators z*“” on F(V)

as follows,

z*+(f&?nla) := z**nl(f@en’o ), for f E C[x,i,u,j,u,jlj E Go] , Pa E G(Q).

This system of equations have the following “soliton type” solutions in common: For N E I?&.~, E; E {*}, ai E C, k; E Z and Z; E @* (1 6 i < N),

where we set

In particular, for &I = . . . = EN = -f (=: CT), we have

‘~;:‘~:;,...,(,,N.kN);ZI....rZN (x,u) = c (_l)[r/21e’CI=,k’“)W, n (Zj, - Z.j,.)*

06, <N v=l I <v<pL&r

I<.jl <.j2<...<.jr<N

3.4. Examples (slz-case)

Here we show an example for g = 5t2. The coefficient of Sty1 in Eqs. (4) for m = n look as follows,

(2D,,D,,, - Dx2Dw)r, or,, - 2Dwrn-1 or,,+1 =O.

Let us introduce new variables pk, q* by

p+ := v%, , p_ := &x, , q+ := w , q- := x2 .

(5)

K. Iohara et al. /Physics Letters A 254 (1999) 37-46 45

In this new variables, the above equations are expressed as follows,

{ (&J+a,,_ - &J,_) .logr,} < = {&, . (b5%-1 - lqw,+,)} 7,-17,,+1

Set @,, := log 7, and put

Xl := p++p-, x2 := q+ - q- ,

TI := p+ - p- , T2 := q+ + q- ,

0 := a;, + a;, - a;, - a$ .

Eqs. (6) have the following form in this coordinate,

q .@,, = {(a,, + &).(@_, - C++,)} e@“~‘-2@“+rr~~+~

The special solutions corresponding to (5) are given as follows. For 0 6 r 6 N and g E {+}, set

PcTr - = fi a;,. <.j,(N v=l

(6)

(7)

Then the functions {ly,,}~, satisfy the system of Eqs. (7) for 0 < m < N and q .Pc = q .P<:,, = 0.

4. Discussion

Higher dimensional integrable systems, such as SDYM (self-dual Yang Mills), are known to possess soliton

solutions. Its Lie theoretic interpretation is still far from understood. In this Letter, we obtain Hirota bilinear

forms which possess 2-toroidal Lie algebraic symmetry for type g = A,, D/, El (see Theorems 3.1, 3.2.) In particular, for $I*, we gave a special class of solutions and some examples. But since we do not know how

to construct Lax pairs for our bilinear forms, we cannot call the above mentioned special solutions as soliton

solutions. Nevertheless, for KdV hierarchy, which can be obtained from the principal realizations of basic representations of st2, it was shown in Ref. [4] that one can construct Lax pair by purely representation

theoretical method. We hope that such treatment can be also generalized to our situation, at least for g = aI2. Let us make a comment on the above equation (6). Let {&},,Ez be a subset of Z, and suppose the above

equations (6) have the solutions of the form

{

eA.Yi r’ 0 n even,

7, := &4+T' I n odd,

for some rb and $. Then Eqs. (7) degenerate into the following form,

q .@, = (A,,_, - A,+1 ) /+-2@,,+@,t+l (8)

Note that the above equations have an S0(2,2)-symmetry. Moreover, Eqs. (8) are similar to the SDYM equation on a Lorentzian space with signature (2,2), and can be regarded as a deformation of the aftine Toda equation of type A, . (I) In particular, if we have either 7; = 1 or 7; = 1, Eqs. (8) essentially give the Liouville equation.

” Finally, let us remark on our Lie algebra gtor. As a first step, it is desirable to have a character of irreducible modules as ch( V) x vv2, where 77 stands for the Dedekind eta-function, from the view point of the flat invariants [ lo]. Lemma 2.1 says that this is exactly the case up to delta-function.

Thus, we believe that this approach will be useful in the future.

46 K. loham et al. /Physics Letters A 254 (1999) 37-46

Acknowledgement

The authors would like to thank T. Inami, H. Kanno, M. Kashiwara, N. Suzuki and Hiroshi Yamada for their

interest and discussion. They also would like to thank the referee for his careful reading.

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