Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. Plural form of octonion.
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Examples

I hope Eva and Clifford are right, but in case SUSY is essential there must be a deeper way to understand it, perhaps afforded by octonions.

But there are close relationships between the representation theory of SO (8)  the transverse directions in 9+1 dim spacetime and the octonions that allow a supersymmetric theory to be defined.

There is no SL (2, O), because the octonions are nonassociative.

If I may, one last beautiful fact: These three representations organize themselves into components of a very beautiful algebrawhich is the algebra of 3 by 3 hermitian matrices of octonions.

This has to do with the division algebras: A = R (real), C (complex), Q (quaternions) and O (octonions).

Moshe, yes, thanks, the connection between strings and octonions is recognized but seems a possible clue to a deeper underestanding of string theory that is not yet well explored.

We also know that the octonions are not only three dimensional (four if you count the zero dimensional scalars), but that they are very curious too (we read also read Baez, you know).

Doug, being a reader of Baez, mentioned octonions.

We know that increasing the dimension of the integers leads to complex numbers, and then to quaternions and then to octonions.

Not even otherwise natural structure constants of the octonions seem to satisfy the required mutated Jacobi identity.
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