1.1 Preliminaries

Before our tour begins, let us settle on some definitions. For us a
**vector space** will always be a finite-dimensional module over the
field of real numbers. An **algebra** will be a vector space that
is equipped with a bilinear map
called
'multiplication' and a nonzero element called the 'unit' such that
. As usual, we abbreviate as . We do
not assume our algebras are associative! Given an algebra, we will
freely think of real numbers as elements of this algebra via the map
.

An algebra is a **division algebra** if given with , then either or . Equivalently, is a division
algebra if the operations of left and right multiplication by any
nonzero element are invertible. A **normed division algebra** is an
algebra that is also a normed vector space with
. This implies that is a division algebra and that .

We should warn the reader of some subtleties. We say an algebra has
**multiplicative inverses** if for any nonzero there is an
element with
. An associative
algebra has multiplicative inverses iff it is a division
algebra. However, this fails for nonassociative algebras! In Section
2.2 we shall construct algebras that have
multiplicative inverses, but are not division algebras. On the other
hand, we can construct a division algebra without multiplicative
inverses by taking the quaternions and modifying the product slightly,
setting
for some small nonzero real number
while leaving the rest of the multiplication table unchanged.
The element then has both right and left inverses, but they are not
equal. (We thank David Rusin for this example.)

There are three levels of associativity. An algebra is **power-associative** if the subalgebra generated by any one element is
associative. It is **alternative** if the subalgebra generated by any
two elements is associative. Finally, if the subalgebra generated by any
three elements is associative, the algebra is associative.

As we shall see, the octonions are not associative, but they are alternative.
How does one check a thing like this? By a theorem of Emil Artin
[77], an algebra is alternative iff for all we have

In fact, any two of these equations implies the remaining one, so people usually take the first and last as the definition of 'alternative'. To see this fact, note that any algebra has a trilinear map

called the

The associator measures the failure of associativity just as the commutator measures the failure of commutativity. Now, the commutator is an alternating bilinear map, meaning that it switches sign whenever the two arguments are exchanged:

or equivalently, that it vanishes when they are equal:

This raises the question of whether the associator is alternating too. In fact, this holds precisely when is alternative! The reason is that each equation in (1.1) says that the associator vanishes when a certain pair of arguments are equal, or equivalently, that it changes sign when that pair of arguments is switched. Note, however, that if the associator changes sign when we switch the th and th arguments, and also when we switch the th and th arguments, it must change sign when we switch the th and th. Thus any two of equations (1.1) imply the third.

Now we can say what is so great about and :

The first theorem goes back to an 1898 paper by Hurwitz [51]. It was subsequently generalized in many directions, for example, to algebras over other fields. A version of the second theorem appears in an 1930 paper by Zorn [93] -- the guy with the lemma. For modern proofs of both these theorems, see Schafer's excellent book on nonassociative algebras [77]. We sketch a couple proofs of Hurwitz's theorem in Section 2.3.

Note that we did *not* state that and are the only
division algebras. This is not true. For example, we have already
described a way to get 4-dimensional division algebras that do not have
multiplicative inverses. However, we do have this fact:

This was independently proved by Kervaire [58] and Bott-Milnor [9] in 1958. We will say a bit about the proof in Section 3.1. However, in what follows our main focus will not be on general results about division algebras. Instead, we concentrate on special features of the octonions. Let us begin by constructing them.

© 2001 John Baez