nLab split octonions

Contents

Context

Algebra

higher algebra

universal algebra

Contents

Idea

Up to isomorphism there is only one split composition algebra of dimension 8; this is called the algebra of split octonions.

Definition

The multiplication may be deduced from the fundamental result on doubling multiplication or may be expressed as follows. Denote scalars by letters like $r, s$ and 3-vectors by letters like $x, y$. Let $\langle x, y \rangle$ denote the standard inner product

$x_1 y_1 + x_2 y_2 + x_3 y_3$

and let $x \wedge y$ denote the standard cross-product, so that $\langle x \wedge y, z \rangle = det(x, y, z)$. Elements of $V$ are represented by $2 \times 2$ arrays

\left( \begin{aligned} r & x\\ y & s \end{aligned} \right)

and multiplication is given by the following formula, highly reminiscent of matrix multiplication but with some cross-product cross terms:

\left( \begin{aligned} r & x\\ y & s \end{aligned} \right) \cdot \left( \begin{aligned} r' & x'\\ y' & s' \end{aligned} \right) = \left( \begin{aligned} r r' + \langle x, y' \rangle & r x' + s' x + y \wedge y'\\ r' y + s y' + x \wedge x' & \langle y, x' \rangle + s s' \end{aligned} \right)

The norm is given by a kind of determinant formula

N\left( \begin{aligned} r & x\\ y & s \end{aligned} \right) = r s - \langle x, y \rangle