The *Clifford algebra*
is the algebra over
with
generators
and relations

for | for all |

The first few Clifford algebras are easy to describe

quaternions

If is a field, denote by the algebra of all matrices with entries in . We call a full matrix algebra. It turns out that the Clifford algebras are all full matrix algebras or the direct sums of two full matrix algebras:

This table exhibits clearly a periodic pattern of period , except for the dimension increase after each period. The -fold periodicity of the Clifford algebras, long known to algebraists, is reminiscent of the -fold periodicity of the stable homotopy groups of the orthogonal group.

In the early Sixties Michael Atiyah, Raoul Bott, and Arnold Shapiro
found an explanation
for this tantalizing connection. The link is provided by
a class of linear differential
operators called the *Dirac operators*. The link between differential
equations and homotopy groups first came about as a result of the realization
that ellipticity of a differential operator
can be defined in terms of the symbol of the differential operator.

Suppose we can find real matrices of size satisfying

for

This corresponds to a real representation of the Clifford algebra . The associated Dirac operator is the linear first-order differential operator

where is the identity matrix. Such a differential operator on has a

The Dirac operator is readily shown to be

Since has the homotopy type of , this map given by the symbol of the Dirac operator defines an element of the homotopy group .

The paper [33] shows that the minimal-dimensional representations of the Clifford algebras give rise to Dirac operators whose symbols generate the stable homotopy groups of the orthogonal group. In this way, the -fold periodicity of the Clifford algebras reappears as the -fold periodicity of the stable homotopy groups of the orthogonal group.