nLab
split octonions

Contents

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,sr, s and 3-vectors by letters like x,yx, y. Let x,y\langle x, y \rangle denote the standard inner product

x 1y 1+x 2y 2+x 3y 3x_1 y_1 + x_2 y_2 + x_3 y_3

and let xyx \wedge y denote the standard cross-product, so that xy,z=det(x,y,z)\langle x \wedge y, z \rangle = det(x, y, z). Elements of VV are represented by 2×22 \times 2 arrays

(r x y s)\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:

(r x y s)(r x y s)=(rr+x,y rx+sx+yy ry+sy+xx y,x+ss)\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(r x y s)=rsx,yN\left( \begin{aligned} r & x\\ y & s \end{aligned} \right) = r s - \langle x, y \rangle

References

See also

Created on February 21, 2018 at 00:23:45. See the history of this page for a list of all contributions to it.