Jackson Seligman, & Dr. Ivona Grzegorczyk
In this presentation, I am investigating Octonions (O) as rank three tensors, their multiplication, and their various constructions Octonions are an Algebra that have the following properties: The are non-commutative (ab ̸ = ba) nor are they anti-commutative (ab ̸ = -ba). They also are non-associative as (a(bc)̸ =(ab)c). They do however abide by the basis square multiplication of: e2_1 = e2_2 = e2_3 = e2_4 = e2_5 = e2_6 = e2_7 = −1 = −(e0)2 where e0 is the real basis and e1, e2, e3, e4, e5, e6, e7 are the imaginary basis, also known as the 7 vector portion of the octonion.
Octonions are an extension of quaternions with additional imaginary basis e_4 with the associated multiplication giving 8 basis. They pose an important role at the current edge of particle physics research being the mathematical system proposed to describe supersymmetry.
A common issue with octonions is that they are not associative. They are commonly attempted to be transformed into matrices which in themselves are associative. This is a contradiction but is considered necessary by many physicists. This will also be investigated as it is currently under great debate.