IN CAYLEY ALGEBRA BASED ON A GENERALIZED AXIALLY SYMMETRIC
POTENTIAL THEORY OPERATOR
HIROMI KORIYAMA and KIYOHARU N ˆONO
In this paper, we give a characterization of octonionic H-regular functions defined by continuously differentiable solutions to a linearization of generalized axially symmetric potential theory operator equations.
AMS 2010 Subject Classification: 30G35, 32A30, 32D05.
Key words: Cayley algebra, Octonionic H-regular functions, generalized axially symmetric potential theory operator.
1. INTRODUCTION
The Cayley algebra (the algebra of octonions) O was constructed by A. Cayley in 1845 as a non-associative extension of quaternion field. O is a composition algebra
|zw|=|z| |w|, z, w∈O, where | · |is the absolute value of octonions.
In 1934, R. Fueter ([3]) has given a definition of regular quaternionic functions by means of extended Cauchy-Riemann equations. Many authors have developed a quaternionic regular function theory ([7], [19]).
In 1973, P. Dentoni and M. Sce ([2]) gave a definition of octonionic regular functions as continuously differentiable solutions to the octonionic dif- ferentiable equation
(1.1)
7
X
i=0
ei
∂f
∂xi
= 0,
wheree0, e1, . . . , e7 is the basis ofOandf =
7
P
i=0
eifi is an octonionic function.
They and many other authors gave several properties of octonionic regular
REV. ROUMAINE MATH. PURES APPL.,56(2011),2, 157–167
functions. Also, the theory of octonionic regular functions has been applied in the theoretical physics ([9], [10]).
A. Weinstein ([20]) and R.P. Gilbert ([4], [5], [6]) studied the following second order partial differential equation which is called the generalized axially symmetric potential equation
(1.2)
n
X
i=0
∂2u
∂x2i − 2v xn
∂u
∂xn = 0, −∞< v <∞.
H. Leutwiler ([13]) considered the following system as a generalization of the Cauchy-Riemann system
(1.3) xn
n
X
i=0
ei∂f
∂xi
+ (n−1)fn= 0, f =
n
X
i=0
eifi,
where {ei} is the basis of the Clifford algebraCln and studied the properties of solutions of this system ([8], [12]). They gave several properties of regular functions defined by (1.3), using the Clifford algebra as an algebraic system.
In [17], we gave a regularity of functions with values in the Clifford algebra defined by continuously differentiable solutions of a linearization
(1.4) xk
n
X
i=0
ei
∂f
∂xi + (n−1)fk= 0, k= 1,2, . . . , n,
of generalized axially symmetric potential theory operator equations.
In the case of n = 7, if we adopt the Cayley algebra as an algebraic system, the properties given in [17] are not trivial, because the Clifford algebra is a non-commutative and associative algebra, but the Cayley algebra is a non- commutative algebra and non-associative algebra.
The aim of this paper is to give an H-regularity of octonionic functions defined by a linearization (2.4) of the following system of the partial differen- tial equations
xk
7
X
i=0
∂2u
∂x2i −6 ∂u
∂xk
= 0, k= 1,2, . . . ,7, (1.5)
x2k∆fk = 6
xk
∂fk
∂xk −fk
, k= 1,2, . . . ,7, (1.6)
and to give several properties of the octonionic H-regular functions.
2. PRELIMINARIES AND DEFINITIONS
LetObe the Cayley algebra (the algebra of octonions) with the canonical basis {e0, e1, . . . , e7} over the field R of real numbers. Its basis satisfy the
following rules
(2.1) eiej+ejei=−2δije0, e0= 1, e2i =−1, i, j= 1,2, . . . ,7, (2.2) e1e2 =e3, e1e4=e5, e1e6 =e7, e2e4=−e6,
e3e4 =e7, e3e5=e6, e2e5 =e7. Then any octonionz inOis represented as
z=
7
X
i=0
eixi.
Let z = P7 i=0
eixi be an octonion in O. The conjugate number z∗, norm N(z), absolute |z|and inversez−1 ofz are defined by the following formulas:
z∗ =x0−
7
X
i=1
xiei, N(z) =
7
X
i=0
x2i,
|z|= q
x20+x21+· · ·+x27, z−1 = z∗ N(z). By using the mapping
(x0, . . . , x7)∈R8 →
7
X
i=0
eixi ∈O, we can identify Owith R8.
Let Ω be a domain in R8 and f be a function defined in Ω with values in O, then f is represented as
f :z=
7
X
i=0
eixi ∈Ω→w=f(z) =
7
X
i=0
eigi(z)∈O, where wj =gj(z),j = 0,1, . . . ,7, are real valued functions.
In this paper, we consider the following differential operators (2.3) D=
7
X
i=0
ei ∂
∂xi, D∗ = ∂
∂x0 −
7
X
i=1
ei ∂
∂xi, ∆ =
7
X
i=0
ei ∂2
∂x2i.
Also, we consider the following system of first order partial differential equa- tions
(2.4) xjDf(z) =−6fj(z), j= 1,2, . . . ,7, where z=
7
P
i=0
eixi.
Definition 1. f(z) =
7
P
j=0
ejfj(z) is said to be H-regular in Ω iff (1)fj ∈C1(Ω),j= 0,1, . . . ,7,
(2)xjDf(z) =−6fj(z) in Ω, j= 1,2, . . . ,7.
Remark. The condition (2) in Definition 1 is equivalent to the system of the following differential equations:
(1)xj
∂f0
∂x0 −P7
i=1
∂fi
∂xi
=−6fj,j= 1,2, . . . ,7;
(2) ∂x∂fi
0 =−∂f∂x0
i,i= 1,2, . . . ,7;
(3) ∂f∂xj
i = ∂x∂fi
j,i, j= 1,2, . . . ,7.
From Definition 1 and the direct calculation, we obtain that following proposition.
Proposition 1. Let Ωbe a domain in R8 andf =
7
P
i=0
eifi be H-regular in Ω. Then it follows that:
(1)xifj =xjfi,i, j= 1,2, . . . ,7;
(2)xj∂fi
∂xk =xi∂fj
∂xk, k6=i,k6=j,i, j= 1,2, . . . ,7,k= 0,1, . . . ,7;
(3)xj∂f0
∂xi =xi∂f0
∂xj, i, j= 1,2, . . . ,7;
(4)fj∂x∂fi
k =fi∂x∂fj
k, k6=i,k6=j,i, j= 1,2, . . . ,7,k= 0,1, . . . ,7;
(5)fj∂f∂x0
i =fi∂f∂x0
j, i, j= 1,2, . . . ,7;
(6)fj+xi∂fj
∂xi =xj∂fi
∂xi, j6=i,i, j= 1,2, . . . ,7;
(7)fifj∂fj
∂xj −fifj∂fi
∂xi −fj2∂x∂fi
j +fi2∂f∂xj
i = 0,i, j= 1,2, . . . ,7;
(8) xj∂f∂xi
i −fj
fj−xjfi∂f∂xj
i = 0, j 6=i,i, j = 1,2, . . . ,7.
Definition 2. Let Ω be a domain in R8 and u = u(z) be a real valued function defined in Ω. We say thatu(z) satisfies the condition GASPTE in Ω iff
(1)u∈C2(Ω);
(2)xj
7
P
i=0
∂2u
∂x2i = 6∂x∂u
j,j= 1,2, . . . ,7 in Ω.
From Proposition 1, by a direct calculation, we obtain that following proposition.
Proposition 2. Let Ω be a domain in R8 and f(z) =
7
P
i=0
eifi(z) be a twice continuously differentiable function defined in Ω. Iff(z) is H-regular in
Ω, then
(1) xj∆f0 = 6∂f0
∂xj, j= 1,2, . . . ,7, (2.5)
(2) xj∆fi = 6∂fi
∂xj
, i6=j, i, j= 1,2, . . . ,7, (2.6)
(3) x2j∆fj = 6
xj
∂fj
∂xj −fj
, j= 1,2, . . . ,7 (2.7)
in Ω, where ∆ is Laplacian of 8-real variables. Therefore, f0(x) satisfies GASPTE in Ω.
Proof. By z∗(zw) = (z∗z)w, it follows that D∗(Df) = (D∗D)f = ∆f
in Ω. SinceD∗(xkDf) =−6D∗fk,k= 1,2, . . . ,7,we have that (2.5), (2.6), (2.7).
3. REGULAR FUNCTIONS
In this section, we give several properties of H-regular functions. The following two propositions are obtained by direct calculation.
Proposition 3. Let Ω be a domain in R8 and a, b be real numbers. If f and g are H-regular in Ω, then af+bg, 12(f g+gf) and N(z)f∗ are H-regular in Ω.
Proposition 4. Let Ω1 and Ω2 be two domains in R8 and octonionic functions f and g be H-regular in Ω1 andΩ2, respectively. If g(Ω2)⊂Ω1, the composite function f◦g is H-regular in Ω1.
Theorem 1.Let Ωbe a domain inR8. Iff(z) =
7
P
i=0
eifi(z) is H-regular in Ω, then
7
X
i=0
∂f0
∂xi
∂fj
∂xi
= 0, j= 1,2, . . . ,7in Ω.
Proof. Since f is H-regular, by Remark and Proposition 1, we have that
7
X
i=0
∂f0
∂xi
∂fj
∂xi = ∂f0
∂x0
∂fj
∂x0 + ∂f0
∂xj
∂fj
∂xj +
7
X
i6=j i=1
∂f0
∂xi
∂fj
∂xi =
=
7
X
i6=j i=1
∂fi
∂xi − 6 xkfk
!∂fj
∂x0 + ∂fj
∂xj
∂fj
∂x0 − ∂fj
∂x0
∂fj
∂xj +
7
X
i6=j i=1
∂f0
∂xi
∂fj
∂xi =
=
7
X
i6=j i=1
∂fj
∂xi
∂f0
∂xi
+ ∂fi
∂xi
+fi xi
∂fj
∂x0
=
7
X
i6=j i=1
∂fj
∂xi
∂fj
∂x0
+ xi xj
∂f0
∂xi
= 0.
Theorem 2. Let Ωbe a domain inR8. Iff(z) =
7
P
i=0
eifi(z) is H-regular in Ω, then there exists a real valued function K(z) in Ωsuch that
J2 =
∂(f0, f1, . . . , f7)
∂(x0, x1, . . . , x7) 2
=K(z)
∂f(z)
∂x0
2
in Ω.
Proof. Since f(z) is H-regular in Ω, by Theorem 1, we have that (3.1)
7
X
i=0
∂f0
∂xi
∂fj
∂xi
= 0, j= 1,2, . . . ,7 in Ω. From (3.1) , we have that
J2 =
7
X
i=0
∂f0
∂xi
2 7
X
i=0
∂f0
∂xi
∂f1
∂xi · · ·
7
X
i=0
∂f0
∂xi
∂f7
∂xi
7
X
i=0
∂f1
∂xi
∂f0
∂xi 7
X
i=0
∂f1
∂xi
2
· · ·
7
X
i=0
∂f1
∂xi
∂f7
∂xi
... ...
7
X
i=0
∂f7
∂xi
∂f0
∂xi
7
X
i=0
∂f7
∂xi
∂f1
∂xi · · ·
7
X
i=0
∂f7
∂xi 2
=
=
7
X
i=0
∂f0
∂xi
2
7
X
i=0
∂f1
∂xi
2
· · ·
7
X
i=0
∂f1
∂xi
∂f7
∂xi
... ...
7
X
i=0
∂f7
∂xi
∂f1
∂xi · · ·
7
X
i=0
∂f7
∂xi 2
.
Hence, there exists a real valued function K(z) such that J2 =
7
X
i=0
∂f0
∂xi
2
K(z) =K(z)
7
X
i=0
∂fi
∂x0
2
=K(z)
∂f(z)
∂x0
2
.
4. ANALYTIC REGULAR FUNCTIONS
In this section, we give four analytic properties of H-regular functions.
Theorem 3. Let f(z)be an H-regular homogeneous polynomial of degree m with respect to real variablesx0, x1, . . . , x7, then
f(z) = 1 m!
∂mf(z)
∂xm0 zm, z=
7
X
i=0
eixi ∈O.
Proof.Since f(z) =
7
P
i=0
eifi(z) is an H-regular homogeneous polynomial,
7
X
i=0
xi∂f
∂xi =mf, ekxk
7
X
i=0
ei∂f
∂xi =−6ekfk, k= 1,2, . . . ,7.
Hence, we have mf = 6
7
X
k=1
ekfk+
7
X
i=0
eixi
∂f
∂x0 +
7
X
i=1
xi
∂f
∂xi +
7
X
i=1 7
X
j=1
(eixi)
ej
∂f
∂xj
. Since f is H-regular, we have the following
6
7
X
k=1
ekfk+
7
X
i=1
xi∂f
∂xi +
7
X
i=1 7
X
j=1
(eixi)
ej ∂f
∂xj
=
= 6
7
X
k=1
ekfk+
7
X
i<j 7
X
k=1
ei
ejxiek∂fk
∂xj
+ej
eixjek∂fk
∂xi
. By (2.2) and a direct calculation,
7
X
i<j 7
X
k=1
ei
ejxiek∂fk
∂xj
+ej
eixjek∂fk
∂xi
=
=X
i<j
ei(ejei)
xi
∂fi
∂xi
−xj
∂fi
∂xi
+X
i>j
ej(eiej)
xj
∂fj
∂xi
−xi
∂fj
∂xj
. Also, by (2.2) and Proposition 1, we have the following
6
7
X
k=1
ekfk+
7
X
i<j 7
X
k=1
ei
ejxiek
∂fk
∂xj
+ej
eixjek
∂fk
∂xi
=
= 6
7
X
k=1
ekfk+
7
X
j=1 j6=i
ej
xi∂fi
∂xj
−xj∂fi
∂xi
+
7
X
i<j
ei
xj∂f0
∂xj
−xj∂f0
∂xi
=
=
7
X
i=1
ei 7
X
k=1 k6=i
fi−xi
∂fk
∂xk
+xk
∂fk
∂xi
= 0.
Theorem 4. If f(z) is expanded in a power series with real cofficients
(4.1) f(z) =
∞
X
n=0
anzn
in a neighborhood U of the point 0∈R8, then f(z) is H-regular in U. Proof. Put f(z) =
7
P
i=0
eifi(z) andPn(z) =anzn, n= 0,1, . . .. Since the polynomialsPn(z) =
7
P
i=0
eiPni(z) are all H-regular,
(4.2) xjDPk(z) =−6Pkj(z), j= 1,2, . . . ,7,
in U. Therefore, by the termwise defferentiation of (4.2), we have that xjDf(z) =
∞
X
n=0
DPn(z) =
∞
X
n=0
(−6Pnj(z)) =−6fj(z), j = 1,2, . . . ,7, in U. Hence,f is H-regular in U.
Theorem 5.Iff(z) =
7
P
i=0
eifi(z)is an analytic function of real variables x0, x1, . . . , x7 in a neighborhood U of the point 0∈O and iff(z) is H-regular in U, then
f(z) =
∞
X
n=0
anzn, an= 1 n!
∂nf
∂xn0(0), z∈O.
Proof. Since each fi(z) is analytic in the variables x0, x1, . . . , x7, f(z) =
∞
X
k=0
Pk(z), z∈U,
where Pk(z) =
7
P
i=0
eiPki(z) is a homogeneous polynomial of degree k. By H-regularity and analyticity of f, we have that
∞
X
k=0
(xjDPk+ 6Pkj) =xj
∞
X
k=0
DPk+ 6
∞
X
k=0
Pkj
=xjDf + 6fj = 0, j = 1,2, . . . ,7.
Putting Qk(z) = xjDPk(z) + 6Pkj(z), Qk(z) is homogeneous polynomial of degree k. Since
∞
P
k=0
Qk(z) = 0, we have that 0 =
∞
X
k=0
Qk(tz) =
∞
X
k=0
Qk(z)tk, t∈R. Putting ψ(t) =
∞
P
k=0
Qk(z)tk= 0, −δ < t < δ, we have that Qk(z) = 1
k!ψ(k)(0) = 0 inU.
Therefore,
xjDPk(z) + 6Pkj(z) = 0 inU, j= 1,2, . . . ,7.
HencePk(z) is a H-regular homogeneous polynomial. From Theorem 3, we have Pk(z) = 1
k!
∂kPk(z)
∂xk0 zk. Since k!1 ∂k∂xPkk(z)
0
zk= k!1 ∂∂xkfk 0
(0)zk, the proof is complete.
Theorem 6.LetU be a star-sharped domain inR8, iff0(z)is real valued function defined in Usuch that
xk
7
X
j=0
∂2f0
∂x2j (z) = 6∂f0
∂xk(z), z∈U, then there exists a function f(z) =
7
P
j=0
ejfj(z) defined in U such that f(z) is H-regular in U.
Proof. Iff0(z) is real valued function defined inU, Re
Z 1 0
Df0(tz)z dt= Z 1
0
( x0
∂f0
∂x0(tz) +
7
X
j=1
xj
∂f0
∂xj(tz) )
dt=f0(z)−f0(0).
Therefore,
f0(z) = Re Z 1
0
Df0(tz)zdt+f0(0).
Putting f(z) = 12R1
0 (Df0(tz)z+zDf0(tz)) dt+f0(0), we have that f(z) =
Z 1 0
7
X
j=0
xj
∂f0
∂xj(tz) dt+f0(0) +
7
X
j=1
ej
Z 1 0
∂f0
∂x0(tz)xj −x0
∂f0
∂xj(tz)
dt.
Since f0(z) =
Z 1 0
7
X
j=0
xj∂f0
∂xj
(tz) dt+f0(0), fj(z) = Z 1
0
∂f0
∂x0
(tz)xj−x0∂f0
∂xj
(tz)
, j = 1,2, . . . ,7, we have that
∂f0
∂x0 −
7
X
j=1
∂fj
∂xj =−6 Z 1
0
∂f0
∂x0(tz) dt+ Z 1
0
tx0
7
X
j=0
∂2f0
∂x2j (tz) dt
=− 6 xk
Z 1 0
xk∂f0
∂x0
(tz)−x0∂f0
∂xk
(tz)
dt
=− 6
xkfk, k= 1,2, . . . ,7.
∂f0
∂xi + ∂fi
∂x0 =
= Z 1
0
( 7 X
j=0
txj ∂2f0
∂xj∂xi
(tz) + 6t∂f0
∂xi
(tz)−
7
X
j=1
txi∂2f0
∂x2j (tz)−tx0 ∂2f0
∂xi∂x0
(tz) )
dt
= Z 1
0
( 7 X
j=0
t xj ∂2f0
∂xj∂xi(tz)−xj∂2f0
∂x2j (tz)
!
+ 7t∂f0
∂xi(tz) )
dt
= Z 1
0
−7t∂f0
∂xi
(tz) + 7t∂f0
∂xi
(tz)
dt= 0.
By Proposition 1, we have that
∂fi
∂xj
−∂fj
∂xi
= Z 1
0
t
xi
∂2f0
∂x0∂xj
−xj
∂2f0
∂x0∂xi
dt= 0.
Acknowledgements. The authors would like to thank the referee and the editorial board for their helpful remarks and suggestions.
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Received 1 December 2010 Fukuoka University of Education Department of Mathematics Munakata, Fukuoka, 811-4192, Japan
