nLab
octonion

Octonions

Idea

The octonions form the largest of the four normed division algebras, denoted 𝕆\mathbb{O}.

Properties

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.

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})\mathbb{R} the real numbers
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})\mathbb{C} the complex numbers
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})\mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) somesenseSL(2,𝕆)Spin(9,1) \simeq_{some\,sense} SL(2,\mathbb{O})𝕆\mathbb{O} the octonionsheterotic/type II string

References

A survey is

Revised on September 2, 2013 23:26:15 by Urs Schreiber (89.204.130.58)