Math Forum Discussions

John Baez

Posts: 542
Registered: 12/6/04

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

Plain Text

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 rank-1
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
rank-1 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 rank-1
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 self-adjoint 3x3 real, complex or quaternionic matrices also form
a Jordan algebra, and the rank-1 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!