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FROM GENERALIZED CAUCHY-RIEMANNEQUATIONS
TO LINEAR ALGEBRAS
JAMES A. WARD
In a previous paper [l] the author gave a definition of analytic
function in linear associative algebras with an identity. With each
such algebra there was associated a set of partial differential equations called generalized Cauchy-Riemann
differential equations which
serve as a criterion to determine whether a given function is analytic
in that algebra. A simplification for the commutative
case has been
given by Wagner
[2].
The purpose of this paper is to give sufficient conditions
2) that a set of equations
(1)
Ê dkij^
,-,i=i
dxj
= 0
(Theorem
(* = 1, 2, • • • , (n2 - «)),
where the du) are constants in a field F, determine a linear associative
commutative algebra A over F for which equations (1) are the generalized Cauchy-Riemann
differential equations. This will enable us
to find solutions of such a set by means of power series in the algebra.
Let €1, e2, • • • , e„ be a proper basis for a linear associative commutative algebra A with an identity over the field F. Multiplication
will be defined by
n
(2)
e.ey = X) Cijktk
*=i
(i, j = 1, 2, • • • , n)
where the && are in F. Denote by P< the matrix (Ci8T) where r
is the row index and î the column index. If a = aiei+a2e2+
• • • +a„e„
is any
element
of A, then
a<r->aiRi-)-a2R2-\-
• ■ • +anPn
is an iso-
morphism known as the first regular representation
of A by matrices.
Let U denote a system of functions y,(xi, x2, ■ ■ ■ , xn) of n variables Xi, Xt, • • • , «■ of F, and let the y¿ be analytic in a simply-connected region R of M-space. Then r¡ = ^"=1 yiU will be called a function over A of the variable £ = ^"=1 *<e<.
Since the algebra is commutative we define with Wagner [2, p. 456]
that r] = 23?=i y,e< be an analytic function of £ if the y¿ are in U and
the Jacobian matrix (dyT/dx,) is in A, that is, if there exist functions
Zi, z2, • • • , z„ such that
Received by the editors April 15, 1952.
456
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FROM CAUCHY-RIEMANNEQUATIONSTO LINEAR ALGEBRAS
(3)
457
( — J = ziRi + z2R2 + • • • + znRn.
\dx,/
This implies a set of homogeneous
the generalized Cauchy-Riemann
relations among the dyr/dx, called
differential equations of the alge-
bra with basis ei, e2, ■ ■ • , („.
Lemma. // A is a linear associative commutative algebra of order n
with an identity over F, then a necessary condition that rj = E"=i y,etbe analytic is that the components of r\ satisfy a linearly independent
system of differential equations
(4)
È im ^ = o
i,j=i
(k = i, 2, • • •, (y - »)),
dx¡
where the dkii are constants in F depending only on the multiplication
constants Cm of the algebra, and such that
(5)
¿ im = 0
<-i
By the definition
of derivative
(k = 1, 2, • • • , (n2 - n)).
the matrix equation
dyA
(6)
WiiCi + u2R2 + • • • + unRn =
gives w2 equations
set Cijh. From any
is not zero, we can
equations leads to
dxj
in the n unknowns, with coefficients among the
n of the equations (6) for which the determinant
solve for the w¿, and substitution in the remaining
n2 —n equations of the form (4). Under a change
of basis
n
(7)
i,-= E UA
;-i
tijEF
(i=l,2,---,n),
we have x¡ = X)"=i *«*< and y'j — E"=i U¡yi- Therefore
/ ;
i,)=l
dkij -
— Z-,
OXj
«Auv —~
u, v=l
•
C %v
Let Dh - (dkrs) and D't = (d'trs); then
(8)
Du = T~*DkT
where T= (t„). Since the trace of a matrix is invariant under similarity transformation,
we see that E?=i ^*» is invariant under change
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458
J. A. WARD
[June
of basis of the algebra A.
Wagner [2, equations 3] has shown that if ei is the identity, the
generalized Cauchy-Riemann
differential equations may be written
(9)
oyr
"
dyi
dx,
j_i
dxi
- — + Z Cur ~
Since ei is the identity,
= 0
(r, s = 1, 2, • • • , n).
Cur = S,r (Kronecker
5). For 5 = 1 we get n
equations
-\-
dxi
which are identically
2-, cnr ■— = 0
,«.1
(r = 1, 2, • • ■, n)
dxi
zero. The n2 —n remaining
form (4). If, in (9), r = s^l
equations
are in the
we get
_ dy.
dyi
dx,
» r
dxi
j_2
dJl
âaci
because G„ = l, hence ^?=i ¿*,-,= —1+ 1 =0. If r^s, s^l,
Cnr = 0
and the equations reduce to
-^+¿C,r^
dx.
<=2
dzi
= 0
so that dkii = 0 for every i. Hence in every case we have
(10)
¿ dkii = 0
and the lemma is proved.
Theorem 1. If Ai = (aitr) (i = 1, 2, ■ • • , n) is a set of n by n matrices
over F such that
(11)
AiA^AjAi
(*,i-
1,2, •••,*)
ana if there is a p such that
(12)
aipr = 5ir
(i, r = 1, 2, • • • , »),
¿Äew
n
(13)
j4*4j = X «iyt^t
«=i
(*■/ = 1. 2, • • • , n)
and the Ai form a basis for a linear, commutative, associative algebra of
order n over F with an identity element.
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1953]
FROM CAUCHY-RIEMANNEQUATIONSTO LINEAR ALGEBRAS
459
By (12) the element of uth row and pth column of AiA; is
n
n
2-1 aitvßjvt
= ¿_, auuSjt
t-1
— anu-
<=-l
By change of notation we obtain the element of the wth row and the
pth column of A ¡A, to be ajiu. Thus we have
(14)
aiju = aiiu
(i, j, u = 1, 2, • • • , «).
Another form of (11) is
- n
(15)
n
¿j anua]Vt = ¿_i Qjtußivt.
t—\
<=i
Thus we have
AiA¡= Í E aurait) = í E «y<rö,-.(J (by (15))
= ( 2~2aJtra.itJ
(by (14))
= I Z ßtir«y.i
J
(by (15))
= ( E durant J
(by (14))
n
= 2^antAt.
From (12) we see that the At are linearly independent with respect
to F. Equations (12) and (14) together show that A, = I. Hence the
Ai form a basis for an algebra A whose constants of multiplication
are Cijk= a</*,so that Ri = A,-.
Let
(16)
/» - L im ~
•,3=i
dXj
(k=l,2,---,(n2-
»))
such that dun are in a field F and the /* are linearly independent with
respect to F. If there is a p such that it is possible to solve for each
dyi/dxj in the system
(17)
E im-— - 0 (k = 1, 2, • • • , (y - »))
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460
[June
J. A. WARD
as a linear function of dyi/dxp, dy2/dxp, ■ • ■ , dyn/dxp, then (17) may
be put in a matrix form
n*
idyr\
/v
dy<\
X^f
\dx,/
\ 1=1
d^j,/
(=i
by adjunction
ï dyt
dxp
V a dyi
t=i
dxp
of the n identities
dyt
dyi
/O OCp= /O Xp
(¿=1,2,...,«).
Theorem 2. Suppose the system of differential equations (17) has
the property that for some fixed integer p, it implies the set (18). Suppose further that the matrices A,• = (a18r), i—\,2,
■ • ■ , n, are commutative and
n
(19)
E iku = 0
>'=i
(k = 1, 2, • • • , (n2 - n)).
Then there is a uniquely determined linear, commutative associative
algebra A, over F, for which (17) is a set of generalized Cauchy-Riemann differential equations (in the sense of the lemma).
Since each equation of (18) which is not an identity is a linear
combination
of the set (17), it follows from (19) that alpr = ôrt.
Hence by Theorem 1 the Ai (i=\,
2, • • ■ , n) form a basis for an
algebra A with AP = I as the identity element. It is seen that (17)
and hence (16) is a set of generalized Cauchy-Riemann
differential
equations for the algebra.
For n = 2 we have the
Corollary.
A necessary and sufficient condition that the linearly independent equations
(20)
E im—
,-,3=i
ax,
=0
(k= 1,2)
determine an algebra A for which (20) is a set of generalized CauchyRiemann differential equations is that
(21)
itn + im = 0
(k = 1, 2).
The necessity of (21) follows from the lemma.
Since (21) holds and the matrix of the coefficients of (20) is of
rank 2, (20) may be put into form (18). Therefore either Ai = I and
therefore is commutative
with A2 or A2 = I and is commutative
with ^4i. Therefore the corollary follows from Theorem 2.
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19531
FROM CAUCHY-RIEMANNEQUATIONSTO LINEAR ALGEBRAS
Ketchum
[3, pp.
646 and
653] proved
that
yi. y»> ' ' ' » y» of the generalized Cauchy-Riemann
tions for a commutative
algebra may be expressed
n
every
461
solution
differential equain the form
co
v = X)y#i= H aA'
i=l
,=o
where £= /"=i *»€i. Therefore if equations (1) determine an algebra,
the solutions of the equations may be obtained by use of power series
in the algebra.
References
1. James A. Ward, A theory of analytic functions in linear associative algebras,
Duke Math. J. vol. 7 (1940)pp. 233-248.
2. Raphael D. Wagner, The generalized Laplace equations in a function theory for
commutative algebras, Duke Math. J. vol. 15 (1948) pp. 455-461.
3. P. W. Ketchum,
Analytic functions
of hypercomplex variables. Trans.
Math. Soc. vol. 30 (1928) pp. 641-667.
University
of Kentucky
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Amer.