Once you have a multiplication table for octonions, the
algebra formed by the transpose of this table meets
all requirements expected of octonions, yet is not
isomorphic to the original.
I call these two forms Right Octonion Algebra and Left
Octonion Algebra, for reasons I explain in the PDF
available on my website:
www.octospace.com
I would like to pose the following question to all the
smart math-heads that view the posts here:
What is the impact of having two non-isomorphic
representations for octonions on their application
in mathematics?
I can narrowly answer the question for the application
of octonion algebra to the physics of electrodynamics
and a possible unification with non-metric gravitation.
The short answer is that it does not matter, since all
differential forms for observables like work, force,
energy transport, conservation of energy and
momentum do not care if Left Octonion Algebra or
Right Octonion Algebra is used, since they all are
Algebraic Invariants. Any of 8 twists on each of the
two types of octonion algebra yield identical results.
I call this the "Law of Octonion Algebraic Invariance".
I look forward to your informed opinions.
Rick Lockyer
|
