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Section A.3 Cayley–Dickson automorphisms

In Section 8.1, we discussed the automorphisms of the division algebras \(\RR\text{,}\) \(\CC\text{,}\) \(\HH\text{,}\) and \(\OO\text{;}\) the split case is similar. Here, we briefly discuss the automorphisms of more general Cayley–Dickson algebras, such as the sedenions, which were introduced in Section 2.9.

Consider the Cayley–Dickson algebra \(\KK\oplus\KK\,e\text{,}\) as constructed in Section 2.7, where \(e^2=-\epsilon=\mp1\text{.}\) If \(\phi\in\Aut(\KK)\text{,}\) then it is easy to see that the map

\begin{equation} \Phi(x_1+x_2\,e) = \phi(x_1)+\phi(x_2)\,e\tag{A.3.1} \end{equation}

is in fact an automorphism of \(\KK\oplus\KK\,e\text{.}\) Explicitly, we need to show that

\begin{equation} \Phi\bigl((x_1+x_2)(y_1+y_2)\bigr) = \Phi(x_1+x_2)\,\Phi(y_1+y_2)\tag{A.3.2} \end{equation}

which, when expanded, becomes

\begin{align} \phi(x_1 y_1 \amp+ \bar{y_2} x_2) + \phi(y_2 x_1 + x_2 \bar{y_1}) \,e\notag\\ \amp= (\phi(x_1)+\phi(x_2)\,e) (\phi(y_1)+\phi(y_2)\,e)\notag\\ \amp= (\phi(x_1) \phi(y_1) + \bar{\phi(y_2)} \phi(x_2)) + (\phi(y_2) \phi(x_1) + \phi(x_2) \bar{\phi(y_1)}) \,e\tag{A.3.3} \end{align}

which is clearly true, since corresponding terms are equal using the automorphism properties of \(\phi\text{.}\) An additional automorphism is obtained by reversing the sign of \(e\text{,}\) that is, the map

\begin{equation} \sigma(x_1+x_2\,e) = x_1-x_2\,e\tag{A.3.4} \end{equation}

is clearly an automorphism of \(\KK\oplus\KK\,e\text{.}\) Finally, direct computation verifies that the map

\begin{align} \Psi(x_1+x_2\,e) = \frac14\bigl(x_1\amp+3\bar{x_1}+\sqrt3(x_2-\bar{x_2})\bigr)\notag\\ \amp+ \frac14\bigl(x_2+3\bar{x_2}-\epsilon\sqrt3(x_1-\bar{x_1})\bigr) \,e\tag{A.3.5} \end{align}

is also an automorphism of \(\KK\oplus\KK\,e\text{.}\) Furthermore, the automorphisms \(\sigma\) and \(\Psi\) commute with \(\Phi\text{,}\) and together generate \(S_3\text{,}\) the symmetric group of permutations on three elements.

Although the construction above holds for any Cayley–Dickson algebra, in general it only shows that

\begin{equation} \Aut(\KK) \times S_3 \subset \Aut(\KK\oplus\KK\,e) .\tag{A.3.6} \end{equation}

In fact, it is easy to see that this construction does not yield all of the automorphisms of the complexes, quaternions, or octonions. Remarkably, however, it does give all of the automorphisms of the sedenions (and beyond). Explicitly, the automorphism group of the sedenions is \(G_2\times S_3\text{,}\) that is, the exceptional Lie group \(G_2\) together with the finite group \(S_3\text{.}\) This finite group fixes \(e\) (up to sign), and performs simultaneous rotations by \(\frac{2\pi}3\) in each plane spanned by \((q,q\,e)\text{,}\) with \(q\in\Im\OO\text{.}\)