The Inst. of Natural Scielrces Nihon Univ. Proc. of The Inst. of Natural Sciences Vol. 25 (1990) pp. 49-65

HURWITZ PAIRS, SUPERCOMPLEX STRUCTURES, SOLITON EQUATIONS,

AND QUASICOMFORMAL MAPPlNGS

Osamu Suzuki, Julian Lawrynowicz and Jerzy Kalina

(Received October 31. 1989)

Hurwitz pairs are discussed in connection with algebra, complex analysis, and the field

theory. The following results are obtained :

( i ) A Hurwitz pair determines a special Clifford algebra (Theorem I).

(ii) A field operator of the Dirac type, which is called a Hurwitz operator, is introduced by

use of a Hurwitz pair and its characterization is given (Theorem II). A field equation

of the Neveu-Schwarz model of the superstring theory is obtained by the Hurwitz pair

(E4, E3).

(iil) Isospectral deformations of the Hurwitz operator of the Hurwitz pair (E2, E2) induce

various soliton equations (Theorem 111).

(iv) A special complex structure, which is called supercomplex structure, is introduced

(Definition (2. 3)) and there exists a correspondence between supercomplex structures

of a Hilbert space and reduction solutions 0L the Sato's K.-P. system (Theorem IV).

( v ) A quasicomformal mapping is obtained from a generalized Hurwitz pair (Theorem V).

By these results we may conclude that Hurwitz pairs give rise to an interesting field not

only in mathematics but also in physics.

Introduction

This paper is the second part of our study on Hurwitz pairs. In the first part [1], we

have given an outline of a field theory defined by Hurwitz pairs. We have introduced a

field equation of the Dirac type and soliton equations by Hurwitz pairs. In this paper we

shall give a systematic treatment on Hurwitz pairs in connection with algebra, complex

analysis and field theory. Our main concern is not to perform detailed studies but only to

suggest relationships between Hurwitz pairs and other fields. Hence we shall restrict our

considerations mainly to the simplest Hurwitz pair (E2, E2) and state our results in a simple

manner. Our main results are stated in Theorems I - V. The hearts of this paper are

sections 4, 5, 6 and 7. The first three sections supply preliminaries to the latter part.

Although the preliminary part is overlapped with the results of the previous paper [1l], we

do not hesitate to repeat them. During the preparation of this paper, generalizations

* The third narned author was supported by the Polish Academy of Sciences and the University of L6dz during the preparation of this work.

o. Suzuki, J. Lalvrvnowicz J. Kahna

of some results of this paper are given in [18] and [5]. In [18] Hurwitz pairs are

defined for pairs of pseudo-euclidean spaces and Theorem I is generalized to the case of

indefinite Hurwitz pairs. In [5] a concept of Hermitian Hurwitz pairs is introduced and a

generalization of Theorem 11 is given. Further generalizations of other results will be given

in the forthcoming papers. Here we make some comments on references of recent works

on Hurwitz pairs. The second named author's study on the supercomplex structure provides

a quite detailed guide to Hurwitz pairs ([16], [17]). Here we treat Hurwitz pairs only in

the case of Euclidean spaces. As for Hurwitz pairs for pseudoeuclidean spaces, we refer to

[18] and ~9]. In [9], the Minkowski space is discussed in connection with Hurwitz pairs.

In [8], Iow dimensional Hurwitz pairs are discussed by use of the multiplication structure

of the octonions.

We describe the contents of this paper in more details. In 1898 and 1923. A. Hurwitz

discoverd very misterious pairs (En, EP) of Euclidean spaces En and EP, which we call

Hurwitz pairs ([6], [7]). W~e may understand that he had introduced such pairs in con-

nection with the composition problem of quadratic forms in number theory. Fifty years

later A. Andreotti had interest in Hur~vitz pairs in connection to field theory, because a

special class of Clifford algebras appear from Hurwitz pairs. He had proposed to the second

named author a characterization problem of Hurwitz pairs in terms of fleld theory. This

had given a strong motivation of further reseraches on Hurwitz pairs. In fact, the second

named author discussed Hurwitz pairs in the mordern frame work of the theory of Clifford

algebras and discoverd a concept of supercomplex structures ([16], [17]). In [1l], we have

introduced a field operator of the Dirac type by use of the special properties of the Clifford

algebras which are deflned by Hurwitz pairs. These operators have been called Hurwitz

operators and the corresponding equations have been called Hurwitz equations. We have

given a characterization of Hurwitz pairs in terms of Hurwitz operators. This may be re-

garded as an answer to the question ~vhich lvas arised by A. Andreotti.

In this paper we set Hurwitz equations as a back born of our researches and wander in

various directions (S 3). If we want to go back to Clifford algebras, we have to treat them

by considering the spin structure of the fermion fleld which is defined by the equation

(Theorem I, S 1). If we want to discuss complex analysis by use of Hurwitz pairs, we treat

our equations as defining equations of (generalized) holomorphic functions in the sense of

Clifford analysis (S 3). In fact, our equation is equivalent to the Cauchy-Riemann equation

in the case of the Hurwitz pair (E2, E2) and it is equivalent to the Feuter equation in the

case of Hurwitz pair (E4, E4) (Proposition (3. 8), S 3). Morover, we can obtain a special

class of quasicomformal mappings when we consider generalized Hurwitz pairs (Theorem V,

S 7). If we want to discuss on Hurwitz pairs in connection with fleld theory, we suffice to

discuss on the Hurwitz equations themselves. The Hurwitz equation is nothing but the non-

relativistic Dirac equation in the case of (E4, E4) and it gives rise to the field equations

which appear in the Neveu-Schwarz model of the superstring theory in the case of (E4, E3)

respectively (Theorem II, S 3 and S 4). If we want to discuss on soliton equations in

HURWITZ PAIRS, SUPERCOMPLEX STRUCTURES. SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPINGS

connection with Hurwitz pairs, we have to consider isospectral deformations of Hurwitz

operators. We want to underline that the Hurwitz operator of the Hurwitz pair (E2, E2)

is equivalent to that of Abrowit2: et al. ([1]). Hence we can see easily that various solition

equations are obtained by use of our isospectral deformations (Theorem 111). Moreover, we

can discuss about the K.-P, system for the Hurwitz operator and obtain its linearization

equation (Theorem 111). Next we state a result which suggests a relationship between

complex analysis and solition equations. We generalize a concept of a pre-Hurwitz pair to

a pair (H, Ep) of a separable Hilbert space H and the Euclidean space Ep and define

supercomplex structures on H. Then we can show that there exists a correspondencce

between supercomplex structures of a Hilbert space and reduction solutions of the o:ririnal

K.-P. system (Theorem IV).

By these discussions of this paper, we can conclude that Hurwitz pairs have many con-

nections with various fields in mathematics and physics. Hence we may expect for further

researches on Hurwitz pairs.

One of the authors would like to express his hearty thanks to Profs. S. Sasaki, J.

Yamashita and T. Mukasa and Dre. M. Jimbo and S. Kanemaki for their variable discus-

sions and suggestions.

1. Hurwitz pair

In this section we recall some basic facts on Hurwitz pairs. Hurwitz considered the

following problem : Find a pair (En, EP) of the n-dimensional and p-dimensional Euclidean

spaces En and EP which admits a bilinear mapping f : E"XEP --~ E" with the following

condition :

(1. 1) l:f(x,y)ll=1lx I l!yl for any xeEn and y~EP,

where j I [ denotes the usual norms of Euclidean spaces. This condition is called the

Hurwitz condition and the mapping f is called the Hurwitz mapping. Hurwitz discovered

that such pairs are strongly restricted. We examine some simple cases. The pair (El, Ei)

satisfies the condition (1. 1) with respect to f(x, y)=xy. For any pair of complex numbers

z, weC, we see that

(1. 2) zwl = Izl lw .

Hence the pair (E2, E2) satisfies the condition (1. 1). In a similar manner, we see that

(1. 2) holds for any pair of quatanion numbers or octonion numbers. Hence (E4, E4) and

(E8, E8) satisfy the condition (1. 1). If we choose a pair (E~, Eq) with the condition (1. 1),

then we see that a pair (E"(~E", EP) also satisfles the condition (1. 1). Hence we have to

introduce the concept of the irreducibility : f is called irreduciale if f does not preserve

any non-trivial proper subspace V of E", i.e., there never exists any non=trivial subspace V

of E" which satisfies

fvxE" : VXEp--~ V. (1. 3)

O. Suzuki, J. La¥vryno~vicz J. Kalina

Then we introduce the follo~ving deflnition :

Definition (1. 4) A pair of Euclidean spaces En and EP is called a pre-Hurwitz pair

if there exists a bilinear mapping f: E1~XEP-~,E7e 1~e'hich satisfies (1. 1). Moreover, if f

is irreducible, it is calZed a Hurwitz pair.

Hurwitz has proved the folknving theorem :

Theorem (1. 6)([7]). If (En EP) Is a Hul wltz palr (n>p) then the followlng (n p)

are possible :

( i ) n=2r and p=2r, r=4s+1 or 4s+3, (ii) n=2r and p=2r, r =4s +2,

(iii) n=2r-1 and p=2r, r =4s +4,

(iv) n=2r+1 and p=2r+1, r=4s+1 or 4s+2,

(v ) n=2r and p=21-, r=4s+3 or 4s+4.

The first possible pairs ('z, p) are listed as follo~vs :

(1, 1) (8, 5) (16, 9) (2, 2) (8, 6) (32, 9) (4, 3) (8, 7) (64, 11) (4, 4) (8, 8) (64, 12)

The table (i)-(v) in Theorem (1. 6) suggests that En is the representation space of

the Clifford algebra defined by EP, what has been discovered by Hurwitz (~7]). In order

to get the relationship to the Clifford algebra, we write down the condition (1. l) using the

orthonormal basis el' e2, "' , en of En and el, e2, "', eq of EP. We define nXn-matrices

C (c, ")(a 1 2 p) by f(e ,e.)=~ c~.ek. k=1

Then the Hurwitz mapping can be written in the form of

(1. 7) f(x,y) = f(y)x, f(y) =ylCl+ "' +y~)Cp.

We can prove the following proposition :

Proposition (1.8). The mapping f in (1. 7) statisfies the Hurwitz condition (1. l) if

and only if

(1. 9) tC.Cp+tC,9C.=26.fi In (a, P= 1, 2, ..., p),

where In is the identity 1?latrix and a.,3 is the Kronecke7~ sysl'tbol.

Corollary (1. 10). C.(a =1,2, ...,p) is an o'-thogonal Inatl~ix.

As for proofs, see [16].

By this proposition we can prove the following lemma :

(8) - 52 -

HURWITZ PAIRS, SUSERCOMPLEX STRUCTURES, SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPINGS

Lemma (1. 11) ([16]). Let r^(a=1,2, ･･･,p-1) be a complex matrix defined by C~=i

Cp T~･ Then

(1. 12) r* rp+rp r*=2a~p In (a, p=1, 2, ･･･, p-1), tr*= -r* and Re r~=0. (1. 13)

Hence we see that a pre-Hurwi~~ pair determines a Clifford algebra wlth the aditional

condition (1. 13). Conversely, given a Clifford algebra with the codition (1.13), we can

obtain a pre-Hurwitz pair in the following manner : Choosing an arbitrary orthogonal matrix

Cp, we define C (of 1,2, ･･･,p-1) by C.=iCpr.･ Then we see that {C } satrsfy the condrtlon

(1. 9). By these matrices we define f by (1. 7). Then we obtain a desired Hurwitz mapping.

Moreover, if we choose an irreducible Clifford algebra with (1. 13), we obtain a Hurwitz

pair. Therefore we can prove the following theorem :

Theorem I ([16], [17]). The determination of a Hurwitz pair is equivalent to that

of an irreducibJe Cltfford algebra which admits a representation with the condition (1. 13).

Hence (n, p) can be characterized by

2[,'p~~] p~O, 1,7 (mod 8) {

(1. 14) - 2[~p+~] pE2 3 4 5 6 (mod 8) n

,,'' Finally we make a remark on a Hurwitz mapping. If f is a Hurwitz mapping, then

Cf is also a Hurwitz mapping, where C is an orthogonal matrix. Hence a Hurwitz mapping

can be written in terms of r* :

(1. 15) f = -iyl ri -iy2 r2 - "' -iyp-1 rp-1 +yp I?e'

where I~ is the identity matrix.

2. Supercomplex structure

In this section we recall some basic facts on the supercomplex structure. As for details

we refer to [16] and [17]. Let (En, EP) be a Hurwitz pair. Let r.(oi=1,2, ･･･,p-1) be a

system of matrices in (1. 11). We introduce S^((x=1, 2, ･･･,p-1) by

S.= -iT~ ((~=1, 2, ..., p-1). (2. l)

-1,e' Hence we know that S. determines a complex structure of En. Then we see that S~=

Next we consider the (p-2)-dimensional sphere SP-2. For a point i~~SP-2, ~=(nl' n2, "',

np_1) with n~+n~+･･･+n~_1=1, we set p- l

(2. 2) J~= ~ n. S..

We have also a complex structure.

Definition (2. 3). The complex strucutre J~ is called the supercomplex strucuture for

a direction i~. The set of supercomplex structures is denoted by Supcom (E?e, EP) , namely

(2. 4) Supcom (En, EP)= {J-" : i~~SP-2}.

- 53 - (9)

O. Suzuki, J. La~vrynowicz J. Kalina

Here we state some basic facts on the supercomplex structure.

Proposition (2. 5) ([17]). Let J be a supercomplex strucuture of a Hurwitz pair

(En, EP). 'rhen ~)e can faid an eZement ReO(n) such that

O J R J R-1, where J = o 1le/2 O

Propositron (2 7) (1) Supcom (E E ) {Jo}' (ii) Supcom (E4. E4)~:P1(C).

Moreover, a paranretrization. of Supcom (E4, E4) gives rise to the folZowing holomorphic

fibre space : There exists a holomorphic fibre space lr : P3(C)-P1(C)--~P(C) such that

lr~1(h) is biholomorphic to (B4, J~).

The assertion of (i) is trivial. As for the proof of (ii), see [2] (p. 39).

Remark. The above fibre space is connected to the Penrose transform of S4. Let t :

P3(C) --~ S4 be the Penrose transform of S4. Then we see that every fibre l~~1(p) is

biholomorphic to P1(C). Choosing a point ~o~~S4, we make the restriction f/ of T to

P3(C)-t-1(00). Then we can obtain r!: p3(C)-P1(C)-->R4. We see that T/ maps C~

to its realization R4. As for details, we refer to E2] (Chap 111).

3. Hurwitz equation

In this section we introduce a concept of quantization and obtain the Cauchy-Riemann

equation, the Feuter equation and the Dirac equation as the images of quantizations of the

Hurwitz mappings.

At first we define quantization. By k[yl'y2"',yp] we denote the algebra of polynomials

over k(=R or C). By k[a/axl'a/ax2, -',alaxq] we denote the algebra of differential

operators with constant coeflicients on Rq, where xl'x2"',xq are the cartesian coordinates

0L Rq.

Definition (3.1). An aZgebraic homomorphism

(3. 2) X : k[yl'y2, "',yp] - k[a/ax a/ax2, "', alaxq]

is called qtiantization.

Let Ind;~(k) denote the algebra of n-sized matrrces over k. Then x can be extended to

(3. 3) - ~4; (k)ek[o/axl' "" alaxq] X : ind;n(k)._.~x)kryl' ~" yp] -

in a natural manner. This is also called quantization and we use this extended quantization

in the following discussions.

Here we consider a Hurwitz pair (E", EP) with the Hurwitz mapping given by (1, 15).

By (2. 1) and (1. 15) we see that f can be written as

(3. 4) f =ylSl +y2S2 + " yp-1Sp_1 +ypl~

HURWITZ PAIRS, SUSERCOMPLEX STRUCTURES, SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPINGS

We shall show that we can obtain well known field equations by quantizations of Hurwitz

ma ppings.

( I ) Generalized Feuter equations ([18])

We obtain the so called generalized Feuter equations.

Definition (3. 5). The quantization XF : M~(B)RB[yl' y2, "', yp]--~'M~(B)~B[a/axl'

a/ax2, "', a/axp] defined by

(3. 6) (p,n) : yl ~~> a/axl' ""yp-~ alaxp XF

is called the quantization of Feuter type. The operator D(1~'"'P) = X(Fn,p)(f), where f is the

Hurwitz mapping (3. 4) is called the generalized Feuter operator of a Hurwitz pair (En

EP) and the corresponding equation D(F"'P) ~ = O is called the generalized Feuter equation.

The generalized Feuter equation can be written

p-* (3. 7) , D(F'e,p)= ~ Ska/axk+1~a/axp. D(Fn,p)~=0 k=*

We see the proposition

Proposition (3. 8)([17], [18]). (i) D(F2,2)c=0 rs the Cauchy Rremann equatron

(ii) D(F4,4)ip=0 is the Feuter equation ([4]).

( H ) Hurwitz equation ([1l])

We obtain the equation of the Dirac type.

Definition (3. 9). The quantiation x : M,e(C)OC[yl' ""yp]--->M (C)OC[a/axl' (n, p)

D a/axp_1' alat] defined by

X (?~,p) . (3. lO) . yl~~a/axl' "" yp-i-=>a/axp_1' yp-~ -ialat D

is called the quantization of Dirac type (or time-separation quantization). The operator

D(n,p) = x(Dn, p)(f) is called the time-dependent Hurwitz operator of the Hurwitz pair (En

Ep). The equation D(n,p) c = O is called the time-dependent Hurwitz equaiion.

This equation can be written

p-1 (3. 11) ia~'flat=~ Sja~laxj

,*l

Proposition (3. 12) ([1l]). D(4,4)~)f =0 is the non-relativistic massless Dirac equation.

Remark. Operating D(?e,p) with

p-l (3. 13) D(n,p)* = ialat+ ~ Ska/axk,

k*l

we obtain the so called Klein-Gordon operator :

(3. 14) D(fe,p)D(n,p)* = (a2/ at2 - Ap_1)In'

O Suzuki. J. Lall-rynowicz J. Kalina

where Ap_1 is the usual Laplacian on RP-1

(m) The 1?lass produci,lg quantization*

In order to get mass in physics, we consider the following quantization :

Definition (3. 15). The quantization X(,n'e,p) : Mn(C)OC[yl' "" yp]--~M,e(C)OC[a/axl'

a/ax2, "', a/axp_l] defined by

(3. 16) v(1~.P) . . yl =~> a/axl' "" yp-1 =~~ a/axq-1' yp --> m'l, 'L'n

where m is a non-negative constant and I is the identity operator, is called a mass pro-

ducing quantization, je;nn,p)= 7(?7~n,p)(f) is called the Hul-witz operator of the Hurwitz

pair (En, EP) ~)ith mass m. Especially whell ~)e choose m=0, we call it the Hurwitz

operator and is denoted by ye(n,p). The equation j~e(n,p)u! =0 is called the Hurwitz equation.

The Hurwitz equation is written as

ye(?e,p)c =0. X(?e,p) = ~ iTkalax~ k*1

Here we notice the selL-adjointness of the Hurwitz operator. We choose the linear space

Fo(RP-1. O~) of n-component functions of C=-class with compact supports. Then we see

that

(3. 18) ye(n,p) : ro(RP-1, cn) _=~ Fo(RP-1, cn).

We introduce the inner product by

(3. 19) J (~,c)= RP_1 ~*~)! dv for p,9)f~Fo(RP-1, cn).

Then the formal abjoint operator je(n,p)* of X(n,p) is given by

p-1 (3. 20) X(?e,p) = ~ ir~alaxk.

~=1

Hence the operator ~e(1e,p) is self-adjoint if and only if r~=rk(k=1,2, ...,p-1)

In terms of Hurwitz equations, we can give a characterization of pre-Hurwitz pairs :

Theorem 11 (L1l]). The foZlo~)ing statelnents af~e equivale'rt:

( i ) (En, EP) is a p7-e-Hul~-z()itz pair.

( ii ) There e,xist pure imaginal~y n-sized Inatl~ices rl'T2, "',rp-1 which satisfy the con-

dition (1. 12) and (1. 13).

(ili) ;e is a self-adjoilrt opel~ator of the form (3. 17) with pure imaginary matrices

rl' r2, "', rp-1 satisfying

(3. 21) = Ap_1 In' ;e2 -

* The introduction of mass by use of the spontanous symmetry breaking of the gauge structure 0L Hurwitz

operator is suggested by Dr. S. Kanemaki [also see [5~).

HURWITZ PAIRS, SUSERCOMPLEX STRUCTURES, SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPl~~:'GS

Proof. (i) ~> (ii) is a direct consequence of the discussions in S 1. (ii) ~~ (iii). By

rl'T2, "',rp-1' we consider the operator ~e of the L0rm (3. 17). Then we see that its self-

adjointness follows from the conditon (1. 13). Also we see that (3. 21) follows from (1. 12).

(iii) ~> (ii). Since ye is a self-adjoint operator, we see that T~=rk(k=1,2, ･･･,p-1).

By the condition that rk is pure imaginary, we see that (1. 12) holds. Also we see that

(3. 21) implies (1. 12).

Remark. The Hurwitz condition (1. 1) can be stated in terms of the Hurwitz operator

X(n,p) as follows :

(-Ap_1 Inc, c) = I I je(?e,p)c1 12 for c~Fo(Rp-1, C"), (3. 22)

where I I I is defined by the inner product (3. 19).

4. The Neveu-Schwarz model

In this section we shall be concerned with the complex version of the Hurwitz pairs

(E4, E4) and (E4, E3) and obtain the equations of Neveu-Schwarz model of the superstring

thoery ( [24]).

At flrst we treat the Hurwitz condition in the complex form. We identify the flrst

element E4 of (E4, E4) with the hermitian space E~ m a natural manner and consider

the Hurwitz mapping as fc : E~XE4-->E~･ This mapping is called the complex form of the Hurwitz mapping fr. We write down its explicit form. At first we notice that the

Hurwitz mapping of (E4, E4) can be witten as follows :

f (x, y) = ( - iylTl ~ iy2r2 - iy3T3 +y414) x, (4. 1)

where

(4. 2) = i O and r3=t O (13 ( O~ a a2 ( . rl - i O ' r2=~0 (T2/ (F3 O Here ol'(T2,(T3 are the well known Pauli matrices

O -i ~l O)' 2=(i O) and (;3= (Fl (;

Then we see that the complex form fc : E~XE4---~E2 becomes of the form c

(4. 4) fc= (iylerl + iy2a2 + iy3(73 +y412 ) x.

By Theorem I we see that (E4, E3) is also a Hurwitz pair. The Hurwitz mapping and its

complex form are

(4. 5) f (x, y) = ( - iylrl ~ iy2r2 +y314)x

and

(4. 6) fc(x, y) = (iyl(TI + iy2(T2 +y312)x.

* The Hurwitz condition in the complex form introduces the concept of Hermitian Hurwitz pairs. Detaile

studies are given in [9] and [5].

O. Suzuki, J. Lawryno~vicz J. Kalina

Here we consider the following mass producing quantization :

(4. 7) Xo : M2(C)OC[yl'y2,y3]'->M2(C)OC[ala(T, a/ a T]

defined by

(4. 8) Xo : yl~=~-ialaa, y2--~'a/a T, y3-~0 '1.

This may be regarded as the complex version of the mass producing quantization of S 3 (see

(3. 16)). Then the X(c4,3) =xo(fc) becomes

(4. 9) X(c4, 3) = (Tla / ac + i,T2 a / a lr ,

The corresponding Klein-Gordon operator is

(4. 10) a2/ aa2 - a2/ a -, 2.

Hence- the correponding equations are

(4. 11) (Tla~laa+ia2a~)r/aT=0 and a2p/aa2-a2~/a-,2=0.

We shall show that these equations are nothing but the equations which appear in the

Neveu-Schwarz model ([24]). We recall the Lagrangian of the Neveu-Schwar~ model

L = - (1/2a')JdadT {a.Xka"Xk+i~-kp"a.~k} (4. 12)

where

(O -i~ O ' 2 = ( i f) (4 13) , pl = ~ i O) p

and ~k(k 1 2) rs a two component splnor (see (2 32) n p 240 m E24]). As for the

notations we refer to [24]. The Euler equations of the Lagrangian are written as follows :

(4. 14) allklaa+a~lk/aT =0, al2kla(T- a~2k/af =0, a2Xk/acF2-a2Xk/ a ~2=0

We see easily that these equations are equivalent to the ones in (4. 11). In fact, these two

equations are transformed each o.ther by the transformation '

O l , ( , = ) ~ =A~'/ A I O

where we set ),, one of ;,k(k=1,2) by

~1 2=/'~ * ~),2)

5. Isospectral deformations of the Hurwitz operator of (E2, E2)

In [1l], we have given isospectral deformations of the Hurwitz operator in the case of

n =p=2 and optain the K-dV and the modifled K~lV equations. Unfortunately other

important equations are out 0L our donsiderations. In this section we shall give a full

description of isospectral deformations of the Hurwitz operator ;e(=X(z,2)). Isospectral

deformations of Hurwitz operators of more general cases are important in connection with

Gauge fields and will be discussed in the forthcoming paper. Firstly we show that isospectral

deformations of Je(= ije) are equivalent to those which have been treated by Abrowitz et

(14) - 58 =

HURWITZ PAIRS, SUSERCOMPLEX STRUCTURES, SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPINGS

al. [1]. Hence we may say that known soliton equations are obtained by use of isospectral

deformations of our Hurwitz operator. Secondly we apply the 2-component K.-P, theory

due to M. and Y. Sato [23] to je and discuss soliton equations for our case.

We set

(O -i~ (5. 1) ^ and D=d/dx ;e=(TD, where (T= ¥ i O)

and consider an isospectral deformation on je with a deL0rmation parameter t whose generator

is given by a differential operator B :

(5. 2) Lc = ~c, L=;e + U {

ct =Bc

where ~ is a spectral parameter and U(=U(x, t)) is a matrix valued function. We show

that (5. 2) can be transformed to the isospectral deformation which has been introduced by

Abrowitz et al. ([14]) : Put

(5 3) _(- i -1~ ~l i )

and make A-lye A. Then we we see that this is equal to

1 O ( , = ) KD, where K o _1

which is nothing but the operator which is introduced by Abrowitz. Hence we can trans-

forrn the results to our case.

Proposition (5. 4). By isospectral deformations (5. 2), we can obtain the K.-dV, mo-

dlfied K.-dV. , sine-Gordon, ard nonlinear Schrddinger equations. More exactly, if we set

U=1(~(r+q)i -r+q) 2 ~_r +q (r+q) i

where r and q are functions of x and t, then we obtain

( i ) the modified K.-dV equation : rt+6r2rx+r*xx=0, when we choose r=q, real and

B as a differentiaj operatot~ of the third order,

( ii ) the nonliner Schredinger equation : irt+rxx+2jrl2r=0, when we choose r=q*, the

complex conjucate and B as a differential operator of the second order,

(ili) the sine-Gordon equation : rtx=sin r if we choose r=q, real and B as a pseudo-

differential operator of the order - 1.

As for proofs, see [14].

Next we deflne the K.-P. system for the operator je. Following [23], we introduce a

pseudo-differential operator P :

(5. 5) P= Po + plD-1 + . .. + p~D-n + . . .,

where P~=P~(x) is a matrix valued function and Po is a constant invertible matrix. We

set

O. Suzuki, J. LaT~;~rynowicz J. Kalina

(5. 6) A L=Pj~P-1 and L,e=P(~Dnp-1.

Then we have the following decomposition :

Ln = (L~) + + (L*) _,

(L~)+=B(n)Dn+B(n"-)1Dn-1+...+B(n) and (L ) B(")D 1+B(")D +

We notice that B(I~) = PoaD?ep~l. Here we introduce an infinite number of time parameters

t = (tl' t2, "') and consider P(=P(t)) depending on t. We introduce

Definition (5. 7). The equation

(5. 8) aL/at~= [(L~)+' L] n =1, 2, ...

is called the K.-P. system of the Hurwitz pair (E2, E2).

Following the discussions in [19], we can show that every solution of (5. 9) can be

obtained from that of the linear equation

(5. 10) aU/at~=aD7eU, n=1, 2, ....

We get special solutions of (5. 9) :

Proposition (5. 11). The K.-P. system can be reduced to the K.-P, system for the

KD when Po is chosen as A-1, ~e'hel'e A is given in (5. 3), Hence nonlinear Schredinger

equation and the 1'rodlfied K.-dV, equation can be obtained in the cases of n=2 and n=3

respectively.

Proof We choose P A-1. For a solution P of (5. 8), we set

(5. Il) P =P'A-1, P'=12 +p~D-1 +p2"D-2+ ...

L'=P'KDP'-1 and L~=P'KDnp'-1.

Then we see that L' satisfies

(5. 12) aL'/atn=[(L~)+' L'] (n=1 2 ...) ,, ,

which is nothing but the K.-P. system for KD. The latter part of the assertions are well

known*.

By summarizing the discussions we can obtain the foll07ving theorem :

Theorem 111. Isospectral deformations of the Hurwitz operator ~e oj (E2, E2) give

rise to the K.-dV, the modtfied K.-dV. , sine-Gordon and nonlinear Schrbdinger equations.

The K.-P. system for ;e can be defined by (5. 8) and linearization is given by (5. 10). In

the cases of n = 2, 3, this equation gives the modtfied K.-dV. and nonlinear Schrbdinger

equations respectively.

* Calculating the integrability conditions in the case of n=1, 2, we can obtain the so called nonlinear Schr6dinger equation ...etc (this is a private communication by Dr. Jimbo).

HURWITZ PAIRS, SUSERCOMPLEX STRUCTURES, SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPlNGS

6. Supercomplex structure of a Hilbert space and reduction solutions of the K.-P.

system

In this section we generalize a concept of supercomplex structure to a Hilbert space

and state a correspondence between supercomplex structures and reduction solutions of the

original K.-P. system. We know that reduction solutions of the K.-P, system give those

of the K.-dV and other soliton equations ([3]). On the other hand, we know that the

Virassoro algebra is used to describe soliton equations ([22]). Taking into account the

fact that the Virassoro algebra can be obtained from the group of biholomorphic mappings

of C* we may say that our result will give one of the understandings of a reason why

the Virassoro algebra is used in the theory of soliton equations.

We begin with generalizing the concept of pre-Hurwitz pair to a Hilbert space. By H

we denote the separable Hilbert space over R. We make the following definition :

Definition (6. 1). A pair (H, EP) is called a pre-Hurwitz pair, if there exists a

bilinear mapping f: HX EP --~ H satisfying

(6. 2) Ilf(x,y)ll=1lxll llyll for any xeH and y~EP,

where I x I , I I yll denote the norms of H and EP respectively. A pre-Hurwitz pair is called

a decomposable pre-Hurwitz pair, if H has the decolmposition

(6. 3) H= (T"~ Ek where (E~) is a Hurwitz pair. k=1

In the following we treat only decomposable pre-Hurwitz pairs. Choosing supercomplex

structures' Jk on E~, we define a supercomplex structure J on H by

(6. 4) J= ~) Jk. k=1

The complex structure J is called a decomposabZe supercowplex structure on H, or simply,

a supercomplex structure on H. Then we can prove the following theorem :

Tneorem IV. There exists a correspondence between a set of decomposable supercom-

plex structures and a set of ~30 (p-1)-reduction solutions of the original Sato's K.-P.

system .

Before going to the proof of Theorem IV, we recall basic facts on reduction solutions

of the K.-P. system ([3]). The original K.-P. system is given in the following manner :

We set

(6. 5) L= WDW-1 where D=d/dx and W=1+ulD-1+u2D-2+ ... Here ul'u2, "' are functions of x and t, t=(tl' t2, "')' We call the equation

(6. 6) aL/at~= [(L")+' L] (n= 1, 2, 3, ･･･) the (original) K.-P, system ([23]). A solution of the K.-P, system is called an l-reduction

solution if

O. Suz*ki. J. La+ry~o~vic' J. Kalina

(6. 7) (Lt) + = Lz

with some integer l. We know that 2-reduction (or 5-reduction) solutions give rise to

those of the K.~lV. (resp. Boussinesq) equation. It is well known that l-reduction solutions

can be characterized in terms of the so called Kac-Moody Lie algebra, more exactly the

Lie algebra A~1)1 ([3]). An element ~~A(1) can be written as l-1

(6. 8) ~= ~ X(k);,k k=-* '

where X(k)e~~iC(1, R) and ), is a parameter. In a similar manner we can difine ~r-reduction

solutions. We choose a Lie subalgebra ~r of ~~~C(1, R) and set ~ by (6. 8) with X(k)e:~r

(keZ). The corresponding solutions of the K.-P. system are called ~r-reduction solutions.

Proof of Theorem IV. Let ~ be an ~~O (p-1)-reduction solution. Then ~ has the

form of (6. 8). Hence we have a sequence of infinite elements {X(k)}ke2; X(k)~J~50(p-1,R).

We set gk=exp(X(k))(e~SO(p-1,R)). By use of the identification (see (2. 4))

(6. 9) -, : Supcom (E", E") - SO(p-1,R)/SO(p-2,R),

we have elements Jk=-, (gk)(k~~Z), hence we have a supercomplex structure of H. Con-

versely, we choose a supercomplex structure J of H. Then we have a sequence of infinite

supercomplex structures Jk of the form (6. 4). By (6. 9) we have elements gk(~SO(p-1, R))

satisfying Jk= T (gk). Then we have elements X(k) (~~~O(p-1, R)) (kEZ). Hence we have

an ~~0(p-1, B)-reduction solution of the form (6. 8).

7. Generalized Hurwitz pair and quasicomformal mapping

In this section ¥ve generalize a concept of Hurwitz pair to vector spaces wlth more

general metrics and show that such a pair induces a quasicomformal mapping.

We choose a function a on R" with (T(x) ~0 for any x~R and consider the metnc

I i I l.=(T2jl I I, where I I is the usual Euclidean metric on R1?. The Rn endowed with metric

l I I l. is denoted by E~. We choos" functions a' and a on R" and RP and consider a pair

(E~,, E"P). We consider the Hurwitz condition for this pair : If a mapping f: E~. X E~-->E~,

satisfles the condition

(7. 1) Ilf(x, y)ll.,= Ilxll.,llyjl. for any x~~E~, and any yeE~,

we say that f satisfies the generalized Hurwitz condition. A pair (E~,, E~) satisfying this

condition is called a generalized pre-Hurl~)itz pair. Moreover, the pair with the f which

satisfies the irreducibility condition (see (1. 3)), it is called a generatized Hurwitz pair.

Remark. The condition (7. 1) can be written as

(7. 1) ' llf(x, y)ll =1 x] I Ilyl ..

Hence the essential contribution to the condition of (7. 1) is (i and not (7'. Therefore, in

the following we restrict our considerations to a ger,eralized Hurwitz pair (E", E~).

( 1 8) - 62 -

HURWITZ PAIRS, SUSERCOMPLEX STRUCTURES, SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPlNGS

We derive a similar condition to the one (1. 9) for a generalized Hurwitz pair. We set

(7. 2) f.(x, y) = a~2f(x, y).

Then we see easily that f.(x, y) satisfies (1. 1). Next we show that f. becomes a bilinear

mapping. In fact, we set ~);~)(x) = I !yoll-If(x, yo) for yo(~0) eEP. Then this is isometry,

hence linear with respect to x. Therefore f.(x, y) is a linear mapping with respect to x.

In a similar manner, we see that f.(x, y) is also a linear mapping with respect to y. Hence

by Proposition (1. 8) we can obtain matrices C.((~=1,2, ...,p) which satisfy the condition

(1. 7) and (1. 8) with respect to f.. We set

(7. 3) C(')=0C (a 1,2, ...,p-1) and C(')=Cp.

Then we see that

(7. 4) tC(')Cfi(") + tCfi(")C(') =2026~pln((r, ~ = 1, 2, p 1)

tC(')C(f) +tC(p")C(') =0 and tC(p")C(p") =1,e

Here we define r(')(a=1,2, ...,p-1) by

(7. 5) C(') = i C(') r (')

Then as in Section 1, we see

r (') Y (') (7.6) " 'p + r~")r(') 2(F2a pln(a P 1 2 p l)

tr(') = -r() Re r(')=0

We deflne a generalized Feuter equation by

p-l (7. 7) ~e~~~c O ~e~,~=-~ irka/axk+1na/axp. k=1

Hereafter we restrict ourselves to the case n=p=2. We choose a measurable function

cr(~0) on R2 satisfying

(7.8) l[(1-(r)/(1+a) *<1 (see [15], [20] and [2l]) and consider a generalized Hurwitz pair (E2, E~). The cor-

responding Feuter equation is

(7. 9) (12a/ ax2 - i r"a/ axl)c = O,

O -' lc ( ' O) where r"= . Then we can prove the following theorem t(1

Theorem V. The generalized Feuter equation (7. 9) of the generalized Hurwitz pair

(E2, E~) gives rise to the so called Beltrami equation in the theory of quasicomformal

mapping : Namely, for a solution c, tc=(~1' ip2) of (7. 12), we set f=ipl+i~2' Then we

see

(7. 10) arf=pazf, where p= (a -1)/(a +1)

and z=x2+ixl'

Proof is only a simple calculation and may be omitted.

0．Suzuki，J．Lawrynowicz　J．Kalina

　　Remarks．　（1）Theorem　V　tells　that　deformations　of　metrics　give　rise　to　those　of

complex　stmctures　of　special　type．　（2）Further　developments　conceming　Hurwitz　pairs

and　quasicomformal　mappings，we　refer　to［12｝

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Department　of　Mathematics

College　of　Humanities　and　Sciences

Nihon　University，Tokyo156，Japan

（0．Suzuki）

Institute　of　Mathematics　of　the

Polish　Academy　of　Sciences，L6dz　Branch

Narutowicza56，PL－90－136，L6dz，Poland

（J，Lawrynowicz　and　J．Kalina）

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