Sedenions

Section 2.9 Sedenions

What happens if we add yet another, independent square root of \(-1\text{?}\) We digress briefly to discuss the sedenions \(\SS\text{,}\) the algebra obtained from the octonions using the Cayley–Dickson process in Section 2.7.

The sedenions can be thought of as a pair of octonions, so that

\begin{equation} \SS = \OO \oplus \OO\,e\tag{2.9.1} \end{equation}

where \(e^2=-1\text{.}\) The Cayley–Dickson process tells us that if \(x,y\in\SS\text{,}\) so that

\begin{equation} x=x_1+x_2 \,e, \qquad y=y_1+y_2 \,e\tag{2.9.2} \end{equation}

with \(x_m,y_m\in\OO\text{,}\) then

\begin{align} xy \amp= (x_1+x_2 \,e)(y_1+y_2 \,e)\notag\\ \amp= (x_1 y_1 - \bar{y_2} x_2) + (y_2 x_1 + x_2 \bar{y_1}) \,e .\tag{2.9.3} \end{align}

We also have

\begin{equation} \bar{x} = \bar{x_1}-x_2 \,e\tag{2.9.4} \end{equation}

so that

\begin{equation} |x|^2 = x\bar{x} = |x_1|^2 + |x_2|^2 .\tag{2.9.5} \end{equation}

Not surprisingly, the sedenions have some curious properties. In moving through the Cayley–Dickson process, we have given up commutativity and associativity. What have we lost now?

First of all, the sedenions contain zero divisors. For example,

\begin{align} (i\ell+j\,e)(j\ell+i\,e) \amp= \bigl( (i\ell)(j\ell) + ij \bigr) + \bigl(i(i\ell) - j(j\ell) \bigr) \,e\notag\\ \amp= (-k+k) + (-\ell+\ell) \,e = 0 .\tag{2.9.6} \end{align}

Thus, the sedenions are not a composition algebra, since \(|xy|\) can be zero even though both \(|x|\) and \(|y|\) are nonzero. Furthermore, the sedenions are not alternative, since, for example,

\begin{equation} \bigl((i\ell+j\,e)(i\ell+j\,e)\bigr)(j\ell+i\,e) = -2 (j\ell+i\,e)\tag{2.9.7} \end{equation}

whereas

\begin{equation} (i\ell+j\,e)\bigl((i\ell+j\,e)(j\ell+i\,e)\bigr) = (i\ell+j\,e) 0 = 0 .\tag{2.9.8} \end{equation}

Perhaps most remarkably, the sedenions no longer share the fundamental symmetry of the four division algebras, namely that the imaginary units are all equivalent. In the sedenions, \(e\) is fundamentally different from the rest, as we now show.

Recall that the alternativity of the octonions implies that

\begin{equation} x(xy) = x^2y\tag{2.9.9} \end{equation}

so long as \(x,y\in\OO\text{.}\) Over the sedenions, (2.9.9) still holds so long as \(x\in\OO\) or \(x\in\OO\,e\) (with now \(y\in\SS\)), but not in general. For instance,

\begin{equation} \bigl((i+j\,e)(i+j\,e)\bigr)y = (i+j\,e)\bigl((i+j\,e)y\bigr)\tag{2.9.10} \end{equation}

only holds if \(y\in\HH\oplus\HH\,e\text{,}\) where \(i,j\in\HH\text{.}\) If we want to find \(y\in\SS\) so that (2.9.9) is satisfied for all \(x\in\SS\text{,}\) all possible \(\HH\subset\OO\) arise when generalizing (2.9.10). Thus, in addition to the obvious choice \(y=1\text{,}\) the only other independent possibility is \(y=e\text{.}\) In other words, \(e\) has properties not shared by any of the other 14 imaginary units!