Why is any of this interesting?
With g in SO(8) the product
g-1[g[A]g[B]] = AoB
defines a simple change of basis for the underlying octonion algebra
(the simplicity of the thing was pointed out to me by Shahn Majid).
In fact, given any n-dimensional algebra, and g in SO(n), this will
result in a change of basis.
What's interesting in the octonion case is that the X,Y-product also
gives a change of basis, and if we insist on restricting the change
in such a way that
ea o eb = ± ec,
then really interesting lattices start popping up, in particular the
laminated lattices of dimension 8 and 16. So what?
Well, my interest in in the division algebras arose from what seemed
obvious mathematical connections to physics
at the Standard Model
level, and the fact that the division algebras were examples of
mathematical specialness, or, as I like to think of it, resonance.
When I finished my book I decided that mathematical resonance was
a good thing, and the more one knew about it, the better prepared one
would be to see how it connected to our physical reality. So, right
off the bat, yes, I have a strong faith or intuition that mathematical
resonance is intimately tied to the design of our physical reality.
To me this seems very pragmatic and down to earth. For others
pragmatism lies more with letting our physical reality itself tell
us the nature of its design. (Well, sure. That's reasonable. What it
has told me is that mathematical exceptionalness is of key importance
in that design; it requires rich mathematical structures to give rise
to a structure as rich as this universe.)
Through browsing Conway and Sloane's addictive "Sphere Packings..."
book I became aware of the Leech lattice, which is the laminated lattice
of dimension 24. It is intimately related to the octonion algebra,
as are the laminated lattices of dimensions 8 and 16, but the Leech
lattice seems to be about the most exceptional, resonant lattice
there is, and I wish I could tell you why with some authority, but
it's been a while since I last looked at the book. (Indeed, I shouldn't
even be writing this; I should be working on a real life, but things
keep cropping up that get me thinking, and that together with an
Anyway, as mentioned, and as published in various places, I've used the
X,Y-product to generate really nice copies of the 8 and 16-dimensional
laminated lattices, and I hungered for the Leech, the 24-dimensional
laminated lattice. I felt and feel convinced that once understood,
it will provide some stunning insights into the exceptional nature
of octonion mathematics (see
John Baez's mathematical weeklies, #104 I believe
for what provoked this new page).
Well, avoiding all coyness, my most recent idea was to see if this
algebraic change of basis,
g-1[g[A]g[B]] = AoB
could somehow be exploited to go beyond the 16-dimensional laminated
lattice up to the 24-dimensional case. The two thoughts I'd had
revolved around the octonion triple product (see Okubo's book), and
the exceptional Jordan algebra of 3x3 Hermitian matrices over the
octonions. I have lots of ideas, and very little time. I throw this into
the wind for what it's worth, and what better wind than the world wide wind.
(The start of all this work lies in physical mathematics with my demonstration
that the mathematics of the division algebras comforms in many remarkable
ways with the mathematics of the standard model of quarks and leptons.
An introduction to that work can be found on my
Note: that there is no page 13 is an accident.
Octoshop III page.