Division Algebras, Lie Algebras, Lie Groups: 9

Division Algebras, Lie Algebras, Lie Groups and Spinors

9. Octonion Introduction

This gif animation illustrates the most popular of the cyclic octonion multiplication tables
(the dual of the one employed in the book, but used by John Conway, Martin Cederwall
and Pertti Lounesto, among others, so I have changed my preference). As this is covered
in detail elsewhere on this site, I won't go into too much detail here, except to say that
e₀=1 is the octonion identity, and e_a, a=1,...,7, will be refered to as the imaginary units.

Like we did for the quaternions, if A is an imaginary octonion (no real part), then
A = uθ, where |u|=1, and |A| = θ, so u is an element of the imaginary octonion
unit 6-sphere and is the direction of A, and θ is the magnitude of A. Again, u² = -1,
so u behaves like i when exponentiated, and exp(A) = exp(uθ) = cos(θ) + u sin(θ).
The set of unit octonions = {exp(A): A imaginary} = S⁷ = 7-sphere. Again, since
|XY| = |X||Y|, X and Y arbitrary octonions, this 7-sphere is closed under multiplication,
and it is easily shown to be parallelizable. But it is NOT a Lie group, because it is not
associative, and hence it can NOT be represented by a matrix algebra.

But there are real matrices associated with the octonions, and lots of Lie groups. But for
that we need a new page. (Note: I said these matrices are associated with the octonions,
not that they represent the octonions. I've seen lots of papers claiming to represent
the octonions by real matrices - or complex. Can't be done. There are associations
that can be made, but real and complex matrix algebras are necessarily associative,
hence whatever algebra results is not the octonion algebra - it's something else.
Don't be fooled by substitutes - look for the real octonion label. Of course, if you
adopt a multiplication other than conventional matrix multiplication, anything can be done,
and I do this on page 17 of this section, but my motivation is pure, and the result beautiful.)