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For which k does R k admit a multiplicative structure?
Question 1.
Note that the cross-product in R 3 gives R 3 a multiplicative structure, but it has
zero divisors and there is no multiplicative identity and hence it makes no sense to
talk about multiplicative inverses. Therefore let us strengthen our question.
Question 2. For which k does R k admit a multiplicative structure that is a divi-
sion algebra over R ?
It turns out that there is a very precise answer to this.
20.2.1
Theorem.
A bilinear map
n
n
n
RR R
¥Æ
without zero divisors exists if and only if n = 1, 2, 4, or 8. (We do not require asso-
ciativity or a unit element here. If associativity is required, then n must be 1, 2, or 4.)
Proof. A proof of the fact that a division algebra over R has rank 2 k can be found in
[Shaf94]. The hard part of the theorem, namely, the only if part, is proved in [BotM58].
The product for the real numbers, the complex numbers, the quaternions, and the
octonions or Cayley numbers establishes the existence of the desired bilinear map for
n = 1, 2, 4, and 8, respectively. The quaternions will be described in this chapter. We do
not have time to describe the nonassociative algebra of Cayley numbers in this topic,
but the reader can find a definition in [Stee51] and a very extensive discussion of its
properties and connections with other areas of mathematics in [Baez02]. There is an
interesting discussion of the exceptional nature of some numbers in [Stil98].
The result of Theorem 20.2.1 is related to the question of how many linearly inde-
pendent vector fields there are on S n-1
(see Section 8.5 in [AgoM05]), but the latter
question is much harder however.
These introductory comments lead us to the subject of this chapter, quaternions,
and the fact that R 4 is a division algebra over the reals. The simplest way to show that
is to write down the formula that defines the product of two 4-vectors.
Notation. The symbols 1 , i , j , and k will denote the standard basis (1,0,0,0), (0,1,0,0),
(0,0,1,0), (0,0,0,1) of R 4 , respectively.
Definition.
The (bilinear) product
4
4
4
RR R
a, b
¥Æ
(
) Æ
ab
with unit 1 defined by the equations
2
2 2
== =-,
i
j
k 1
(20.1a)
ij
==-
k
ji jk
,
==-
i
kj
,
and
ki
==-
j
ik
(20.1b)
is called the quaternion product on R 4 .
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