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Re: Lie group F4 = Aut(OP2)
Posted:
Feb 14, 2006 9:28 AM


In article <7595736.1139632166640.JavaMail.jakarta@nitrogen.mathforum.org>, Marek Mitros <marekmit@neostrada.pl> wrote:
>Some poor uncited soul wrote:
>> Indeed (non)associativity spoils the conventional >> technique of using >> equivalence classes of scale invariant coordinates to >> construct OP^3 >> Instead one defines OP^3 using the exceptional Jordan >> algebra J(3,O), >> where points of OP^3 consist of those matrices of >> J(3,O) with vanishing >> Freudenthal product (A x A = 0).
You mean to say "nonzero matrices in J(3,O) with A x A = 0, modulo multiplication by nonzero real numbers".
>Thank you for this answer. However it is not enough for me. This >definition is not intuitive enough, not geometrical enough for me. I >would like to imagine how the OP2 looks.
I urge you to read this:
http://math.ucr.edu/home/baez/octonions/node8.html
I start from very geometrical considerations and sketch how points in projective spaces wind up being described as rank1 projections in formally real Jordan algebras.
Then here:
http://math.ucr.edu/home/baez/octonions/node12.html
I describe points in the octonionic projective plane OP^2 as rank1 projections in the exceptional Jordan algebra.
Then, I explain the Freudenthal cross product x in the exceptional Jordan algebra.
Then, I report Freudenthal's observation that the rank1 projections in the exceptional Jordan algebra are the same as nonzero elements with A x A = 0, modulo multiplication by nonzero real numbers.
You will have to do some calculations to verify this claim, and/or look at Freudenthal's paper.
But, if you're trying to work with points and lines in OP^2, the Freudenthal cross product is very useful.
>Do you know any analogy of the Jordan algebra and OP2 to quaternions?
The selfadjoint 3x3 real, complex or quaternionic matrices also form a Jordan algebra, and the rank1 projections in here are the points in the real, complex and quaternionic projective plane, respectively.
But an even simpler example is CP^1, and that's a good place to start:
http://math.ucr.edu/home/baez/octonions/node11.html
since it's the heavenly sphere!



