symmetric monoidal (∞,1)-category of spectra
Up to isomorphism there is only one split composition algebra of dimension 8; this is called the algebra of split octonions.
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
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
and multiplication is given by the following formula, highly reminiscent of matrix multiplication but with some cross-product cross terms:
The norm is given by a kind of determinant formula
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.