2.3 Clifford Algebras

William Clifford invented his algebras in 1876 as an attempt to
generalize the quaternions to higher dimensions, and he published a
paper about them two years later [20]. Given a real inner
product space , the **Clifford algebra** is the
associative algebra freely generated by modulo the relations

for all . Equivalently, it is the associative algebra freely generated by modulo the relations

for all . If with its usual inner product, we call this Clifford algebra . Concretely, this is the associative algebra freely generated by anticommuting square roots of . From this we easily see that

So far this sequence resembles the iterated Cayley-Dickson construction -- but the octonions are

To see this relation, first suppose is a normed division algebra.
Left multiplication by any element gives an operator

If , the operator is norm-preserving, so it maps the unit sphere of to itself. Since is a division algebra, we can find an operator of this form mapping any point on the unit sphere to any other point. The only way the unit sphere in can have this much symmetry is if the norm on comes from an inner product. Even better, this inner product is unique, since we can use the polarization identity

to recover it from the norm.

Using this inner product, we say an element is **imaginary**
if it is orthogonal to the element , and we let be the space
of imaginary elements of . We can also think of as the
tangent space of the unit sphere in at the point . This has a
nice consequence: since maps the unit sphere in to
the Lie group of orthogonal transformations of , it must send
to the Lie algebra of skew-adjoint transformations of .
In short, is skew-adjoint whenever is imaginary.

The relation to Clifford algebras shows up when we compute the square of
for
. We can do this most easily when has norm
. Then is both orthogonal and skew-adjoint. For any
orthogonal transformation, we can find some orthonormal basis in which
its matrix is block diagonal, built from blocks that look
like this:

and possibly a block like this: . Such a transformation can only be skew-adjoint if it consists solely of blocks of this form:

In this case, its square is . We thus have when has norm 1. It follows that

for all . We thus obtain a representation of the Clifford algebra on . Any -dimensional normed division algebra thus gives an -dimensional representation of . As we shall see, this is very constraining.

We have already described the Clifford algebras up to . Further calculations [50,73] give the following table, where we use to stand for matrices with entries in the algebra :

Table 2 -- Clifford Algebras

Starting at dimension 8, something marvelous happens: the table continues
in the following fashion:

In other words, consists of matrices with entries in . This `period-8' behavior was discovered by Cartan in 1908 [13], but we will take the liberty of calling it

Since Clifford algebras are built from matrix algebras over and
, it is easy to determine their representations. Every
representation is a direct sum of irreducible ones, or **irreps**. The
only irrep of is its obvious one via matrix multiplication on
. Similarly, the only irrep of is the obvious
representation on , and the only irrep of is the obvious
one on .

Glancing at the above table, we see that unless equals or
modulo , is a real, complex or quaternionic matrix
algebra, so it has a unique irrep. For reasons to be explained later,
this irrep is known as the space of **pinors** and denoted .
When is or modulo , the algebra is a direct
sum of two real or quaternionic matrix algebras, so it has two irreps,
which we call the **positive pinors** and **negative pinors**
. We summarize these results in the following table:

irreps of | ||

Table 3 -- Pinor Representations

Examining this table, we see that in the range of dimensions listed there is an -dimensional representation of only for and . What about higher dimensions? By Bott periodicity, the irreducible representations of are obtained by tensoring those of by . This multiplies the dimension by 16, so one can easily check that for , the irreducible representations of always have dimension greater than .

It follows that normed division algebras are only possible in dimensions
and . Having constructed and , we also know
that normed division algebras *exist* in these dimensions. The only
remaining question is whether they are *unique*. For this it helps
to investigate more deeply the relation between normed division algebras
and the Cayley-Dickson construction. In what follows, we outline an
approach based on ideas in the book by Springer and Veldkamp [83].

First, suppose is a normed division algebra. Then there is a unique linear operator such that and for . With some calculation one can prove this makes into a nicely normed -algebra.

Next, suppose that is any subalgebra of the normed division algebra
. It is easy to check that is a nicely normed -algebra in
its own right. If is not all of , we can find an element that is orthogonal to every element of . Without loss of
generality we shall assume this element has norm 1. Since this element
is orthogonal to , it is imaginary. From the definition
of the operator it follows that , and from results
earlier in this section we have . With further calculation
one can show that for all we have

A glance at equation (1) reveals that these are exactly the relations defining the Cayley-Dickson construction! With a little thought, it follows that the subalgebra of generated by and is isomorphic as a -algebra to , the -algebra obtained from by the Cayley-Dickson construction.

Thus, whenever we have a normed division algebra we can find a chain of subalgebras such that . To construct , we simply need to choose a norm-one element of that is orthogonal to every element of . It follows that the only normed division algebras of dimension 1, 2, 4 and 8 are and . This also gives an alternate proof that there are no normed division algebras of other dimensions: if there were any, there would have to be a 16-dimensional one, namely -- the sedenions. But as mentioned in Section 2.2, one can check explicitly that the sedenions are not a division algebra.

© 2001 John Baez