Clifford algebras

The Clifford algebra $C_k$ is the algebra over $\mathbb{R}$ with $k$ generators $e_1, \dots, e_k$ and relations

$$e_i^2 = -1 i=1,\dots, k, \cr e_ie_j = - e_je_i i \ne j.$$

The first few Clifford algebras are easy to describe

$$C_0 = \mathbb{R},\quad C_1 = \mathbb{C},\quad C_2 = \mathbb{H}=\{$$   quaternions$$\}.$$

If $\mathbb{F}$ is a field, denote by $\mathbb{F}(n)$ the algebra of all $n\times n$ matrices with entries in $\mathbb{F}$ . We call $\mathbb{F}(n)$ 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:

$$\begin{array}{ccc\vert ccc\vert cc} k & C_k && k & C_k && k ... ... && 15 & \mathbb{R}(128)\oplus \mathbb{R}(128) &&& \end{array}$$

This table exhibits clearly a periodic pattern of period $8$ , except for the dimension increase after each period. The $8$ -fold periodicity of the Clifford algebras, long known to algebraists, is reminiscent of the $8$ -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 $k$ real matrices $e_1, \dots, e_k$ of size $n\times n$ satisfying

$$e_i^2 = -1, \quad e_ie_j=-e_je_i$$   for $$i \ne j.$$

This corresponds to a real representation of the Clifford algebra $C_k$ . The associated Dirac operator $D=D_{k,n}$ is the linear first-order differential operator

$$D= I \frac{\partial}{\partial{x_0}} + e_1 \frac{\partial}{\partial{x_1}} + \dots + e_k \frac{\partial}{\partial{x_k}},$$

where $I$ is the $n\times n$ identity matrix. Such a differential operator on $\mathbb{R}^{k+1}$ has a symbol $\sigma_D(\xi)$ obtained by replacing $\partial/\partial{x_i}$ by a variable $\xi_i$ :

$$\sigma_D(\xi) = I \xi_0 + e_1 \xi_1 + \dots + e_k \xi_k.$$

The Dirac operator $D$ is readily shown to be elliptic; this means its symbol $\sigma_D(\xi)$ is nonsingular for all $\xi \ne 0$ in $\mathbb{R}^{k+1}$ . Therefore, when restricted to the unit sphere in $\mathbb{R}^{k+1}$ , the symbol of the Dirac operator gives a map

$$\sigma_D(\xi): S^k \to \operatorname{GL}(n, \mathbb{R}).$$

Since $\operatorname{GL}(n,\mathbb{R})$ has the homotopy type of $O(n)$ , this map given by the symbol of the Dirac operator defines an element of the homotopy group $\pi_k(\operatorname{GL}(n,\mathbb{R})) = \pi_k (O(n))$ .

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 $8$ -fold periodicity of the Clifford algebras reappears as the $8$ -fold periodicity of the stable homotopy groups of the orthogonal group.