Next: Chirality Up: Clifford algebras and spinor Previous: The trace Contents

Periodicity

Write $\Cl_{pq} := \Cl(\bR^{p+q}, g)$ , where has signature , and the orthonormal basis is written as $\{e_1,\dots, e_p, \eps_1, \dots, \eps_q\}$ , where $e_1^2 =\cdots= e_p^2 = 1$ and $\eps_1^2 =\cdots= \eps_q^2 = -1$ . For example,
$\begin{align*} \Cl_{10} &= \bR \oplus \bR ; \\ \Cl_{01} &= \bC, \words{with} \e... ...2} &= \bH, \words{with} \eps_1 = i, \eps_2 = j, \eps_1\eps_2 = k. \end{align*}$

$\begin{lem}[\lq\lq (1,1)-periodicity''] $\Cl_{p+1, q+1} \isom \Cl_{pq} \ox M_2(\bR)$. \end{lem}$

$\begin{proof} Take $V = \bR^{p+q+2}$, $A = \Cl_{pq}\ox M_2(\bR)$. Define $f\: V... ... It is an isomorphism, because the dimensions over $\bR$ are equal. \end{proof}$

$\begin{lem} $\Cl^0_{p+1, q} \isom \Cl_{qp}$. \end{lem}$

$\begin{proof} Define $f\: \bR^{q+p} \to \Cl^0_{p+1, q}$ on basic vectors by \b... ...icommute. The rest of the proof is like that of the previous Lemma. \end{proof}$

$\begin{lem} $\Cl_{p+4, q} \isom \Cl_{pq} \ox M_2(\bH) \isom \Cl_{p, q+4}$. \end{lem}$

$\begin{proof} We will prove the first isomorphism. Take $A = \Cl_{pq} \ox M_2(\b... ...atrix}1 & 0 \ 0 & -1 \end{pmatrix}. \tag*{\qed} \end{align}\hideqed \end{proof}$

$\begin{cor}[\lq\lq (+8)-periodicity''] $\Cl_{p+8, q} \isom \Cl_{pq} \ox M_{16}(\bR) \isom \Cl_{p, q+8}$. \end{cor}$

$\begin{proof} This reduces to $M_2(\bH) \ox_\bR M_2(\bH) \isom M_{16}(\bR)$, tha... ...s to $\bH \ox_\bR \bH \isom M_4(\bR)$, which is left as an exercise. \end{proof}$

All $\Cl_{pq}$ are given, up to $M_N(\bR)$ tensor factors, by $\Cl_{p0}$ for $p = 1,\dots,8$ :
$\begin{align} \Cl_{10} &= \bR \oplus \bR \notag \\ \Cl_{20} &= M_2(\bR) \notag ... ... \notag \\ \Cl_{70} &= M_8(\bC) \notag \\ \Cl_{80} &= M_{16}(\bR) \end{align}$
Two algebras $\Cl_{10}$ and $\Cl_{50}$ are direct sums of simple algebras, and the others are simple. We could also define $\Cl_{00} = \bR$ (the base field), so that Corollary holds even when .

Those eight algebras $\Cl_{p0}$ can be arranged on a ``spinorial clock'', which is taken from Budinich and Trautman's book [BT].

$\begin{diagram}[small] && && \bC && && \\ && &\ldTo^{6}& &\luTo^{7}& && \\ && ... ...&& && \bR && \\ && &\rdTo_{3}& &\ruTo_{2}& && \\ && && \bC && && \end{diagram}$

If $p - q \equiv m \bmod 8$ , then $\Cl_{pq}$ is of the form $A \ox M_N(\bR)$ , where is the diagram entry at the head of the arrow labelled . Moreover, Lemma says that the even subalgebra $\Cl^0_{pq}$ is of the same kind, where is now the diagram entry at the tail of the arrow labelled . The matrix size is easily determined from the real dimension, in each case. In this way, the spinorial clock displays the full classification of real Clifford algebras.

Next: Chirality Up: Clifford algebras and spinor Previous: The trace Contents

Pawel Witkowski 2006-03-14