It would be nice to have a construction of the normed division algebras that explained why each one fits neatly inside the next. It would be nice if this construction made it clear why is noncommutative and is nonassociative. It would be even better if this construction gave an infinite sequence of algebras, doubling in dimension each time, with the normed division algebras as the first four. In fact, there is such a construction: it's called the Cayley-Dickson construction.
As Hamilton noted, the complex number can be thought of as a pair
of real numbers. Addition is done component-wise, and
multiplication goes like this:
Now that we have the complex numbers, we can define the
quaternions in a similar way. A quaternion can be thought of
as a pair of complex numbers. Addition is done component-wise,
and multiplication goes like this:
The game continues! Now we can define an octonion to be a pair of quaternions. We add and multiply them using the same formulas that worked for the quaternions. This trick for getting new algebras from old is called the Cayley-Dickson construction.
Why do the real numbers, complex numbers, quaternions
and octonions have multiplicative inverses? I take it as
obvious for the real numbers. For the complex numbers,
one can check that
But this, of course, raises the question: why isn't there an infinite sequence of division algebras, each one obtained from the preceding one by the Cayley-Dickson construction? The answer is that each time we apply the construction, our algebra gets a bit worse. First we lose the fact that every element is its own conjugate, then we lose commutativity, then we lose associativity, and finally we lose the division algebra property.
To see this clearly, it helps to be a bit more formal. Define a -algebra to be an algebra equipped with a conjugation,
that is, a real-linear map
Starting from any -algebra , the Cayley-Dickson construction gives a new -algebra . Elements of are pairs , multiplcation is defined by
All of these follow from straightforward calculations; to prove them here would merely deprive the reader of the pleasure of doing so. It follows from these propositions that:
and therefore that and are normed division algebras. It also follows that the octonions are neither real, nor commutative, nor associative.
If we keep applying the Cayley-Dickson process to the octonions we get a sequence of -algebras of dimension 16, 32, 64, and so on. The first of these is called the sedenions, presumably alluding to the fact that it is 16-dimensional . It follows from the above results that all the -algebras in this sequence are nicely normed but neither real, nor commutative, nor alternative. They all have multiplicative inverses, since they are nicely normed. But they are not division algebras, since an explicit calculation demonstrates that the sedenions, and thus all the rest, have zero divisors. In fact [21,68], the zero divisors of norm one in the sedenions form a subspace that is homeomorphic to the exceptional Lie group .
The Cayley-Dickson construction provides a nice way to obtain the
sequence and the basic properties of these algebras. But
what is the meaning of this construction? To answer this, it is better
to define as the algebra formed by adjoining to an element
satisfying together with the following relations:
What is the significance of the relations in (1)? Simply
this: they express conjugation in terms of conjugation! This is a pun
on the double meaning of the word 'conjugation'. What I really mean is
that they express the operation in as conjugation by . In
particular, we have
This interpretation of the Cayley-Dickson construction makes it easier to see what happens as we repeatedly apply the construction starting with . In the operation does nothing, so when we do the Cayley-Dickson construction, conjugation by must have no effect on elements of . Since is commutative, this means that is commutative. But is no longer real, since .
Next let us apply the Cayley-Dickson construction to . Since
is commutative, the operation in is an automorphism. Whenever
we have an associative algebra equipped with an automorphism ,
we can always extend to a larger associative algebra by adjoining an
invertible element with
Finally, let us apply the Cayley-Dickson construction to . Since is noncommutative, the operation in is not an automorphism; it is merely an antiautomorphism. This means we cannot express it as conjugation by some element of a larger associative algebra. Thus must be nonassociative.
© 2001 John Baez