1550251515-Classical_Complex_Analysis__Gonzalez_

70 Chapter 1

Now we come to the definition of the product xy of two elements of X.

If we wish multiplication to obey the distributive law, we must have

xy = (x1e1 + · · · + Xnen)(y1e1 + · · · + Ynen)

n

= L x;yke;ek

j,k=l

(1.19-2)

To obtain closure, that is, to have xy E X, it is necessary that the prod-
ucts e;ek of any two units be elements of X. Hence they must be linear

combinations of the units e 1 , ... , en, so we must have n^2 equations

n

e;ek = L c;k1e1 (j, k, = 1, ... , n) (1.19-3)

l=l

with coefficients Cjkl E F. What coefficients we choose determines the type
of hypercomplex numbers that we obtain. These n^3 numbers are called
the structure coefficients or multiplication constants of the corresponding
hypercomplex system.
Substituting (1.19-3) in (1.19-2), we have
n n
xy = L L CjklXjYkel
l=l j,k=l

(1.19-4)

The vector system X over F with multiplication defined as in (1.19-4) is
said to form a linear algebra of order n. Multiplication in a linear algebra
is in general neither commutative nor associative.
Multiplication is commutative iff
Cjkl = Ckjl

and associative iff it holds for the unit vectors, i.e.,

e;(e;ek) = (e;e;)ek

or
n n
e; L Cjk1e1 = L c;;1e1ek
1=1 l=l
or

tCjkl (t Ci/mem)

l=l m=l

and equating corresponding coefficients, we obtain
n n
L Cjk/Ci/m = L CijlC/km
l=l b l=l

(1.19-5)

(1.19-6)