Hello Pat and all,
I'm getting the impression that the Octonions do not form a Clifford Algebra... is this correct?
Recall in my post below how I formed both the complex product and the quaternion product from the geometric product... in terms of multiplicative operation tables, all that can be put this way:
Note: I'm not sure I'm using the Cl<sub>p,q</sub> notation correctly... could someone check if I'm using it correctly?
The quaternion product also appears to be a geometric product:
However, I was having allot of trouble with the octonion product...
I just couldn't make it work... and then I consulted Tony Smith's, John Baez's webpages (which is where I got some of the diagrams above), and I believe I've found that I'm not just missing it... but, the Octonions are not a Clifford Algebra. Do the Octonions form what is called an 'exceptional' Lie Algebra?
It appears the octonion product can be put in terms of the geometric product, but only with some rather involved manipulations... see Lounesto's <u>Clifford Algebras and Spinors</u>, Chapter 23.
However, if I'm following Tony Smith correctly, it seems perhaps the easiest way to approach the octonion product through Clifford Algebra is through the formula below, which also applies to quaternions (and complex numbers if the wedge product is substituted for the cross product):
where we want to substitute the 7D cross product to form the octonion product. If I understand correctly, Tony says we're supposed to do this in Cl<sub>0,8</sub>... However, unless I'm misunderstanding the notation, that would seem to imply that we'd be dealing with 8 basis vectors squaring to -1. Don't we only want 7 basis vectors squaring to -1...?
This is where much of my confusion about the notation lies... p+q are supposed to equal n, the dimension of the space which the Clifford Algebra is embedded in... we do want a 8D space here, but we want a space which is a direct sum of a scalar real space and a 7D imaginary space... correct?
Can someone clarify all this? And, show how we form the octonion product from the 7D cross product as indicated above? Which 7D cross product is Tony referring to anyway?