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)
一65 (21)