Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The octonions, usually denoted , form the largest of the four normed division algebras over the real numbers.
The octonions is the nonassociative algebra over the real numbers which is generated from seven generators subject to the relations
for an edge or circle in the following diagram (a labeled version of the Fano plane) the relations
(graphics grabbed from Baez 02)
The octonions are not an associative algebra. The non-zero octonions and the unit octonions form Moufang loops.
The automorphism group of the octonions is G2.
A special triple or basic triple is a triple of three octonions such that
(Whitehead 71, p. 691)
(e.g. Baez 02, 4.1)
|Lorentzian spacetime dimension||spin group||normed division algebra||brane scan entry|
| the real numbers|
| the complex numbers|
| the quaternions||little string|
| the octonions||heterotic/type II string|
A survey is in
- John Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. (web)
The concept of “special triples” or (“basic triples”) used above seems to go back to
Revised on December 14, 2016 14:00:32
by Urs Schreiber