Re: Lie group F4 = Aut(OP2)
- From: Marek Mitros <marekmit@xxxxxxxxxxxx>
- Date: Wed, 22 Feb 2006 19:00:13 +0000 (UTC)
Hello,
Thank you for your answers on this interesting subject.
I have tried to check the "little calculations" mentioned in section 3.1 of your work.
I still don't understand why OP3 does not exist. We can continue projection Matrix definition also
for matrices M4O.
Consider matrix A from M2O: A=( (xx*, xy*), (yx*, yy*)), where v=(x,y) is unit vector from O2;
|x|^2 + |y|^2 = 1 (1)
The x* means conjugation of x.
Fact 1: Matrix A is projection on space {va: a belongs to O}. Indeed
Av = ( (xx*)x + (xy*)y, (yx*)x + (yy*)y ) = (x,y) = v.
Morover, if a belongs to O then A(va) = (( (xx*)(xa) + (xy*)(ya), (yx*)(xa) + (yy*)(ya) ) =
= (( x(x*x)a + x(y*y)a, y(x*x)a + y(y*y)a ).
I the last equality I use Moufang identity (I hope it is valid in this case also). Applying (1) we receive finally
A(va) = (xa, ya) = va.
Take perpendicular vector to v. It is w=(z,t) satisfying formula
x*z + y*t = 0 (2)
We would like to prove that Aw = 0. Let's try:
Aw = ( (xx*)z + (xy*)t, (yx*)z + (yy*)t ). I don't know yet how to prove that it is zero using formula (2) !?
Let us recall some facts regarding projective spaces.
I skip RPn.
-----
CP1 = S2 = S3/S1 = SU2/S1 = SU2/SU1 = Spin3/Spin2
CP2 = S5/S1 = SU3/(SU2*S1) = U3/(U2*S1)
CP3 = S7/S1 = SU4/(SU3*S1)
....
CPn = S(2n+1)/S1 = SU(n+1)/(SUn*S1)
----
HP1 = S4 = S7/S3 = Sp2/(S3*S3) = Spin5/Spin4
HP2 = S11/S3 = Sp3/(Sp2*S3)
HP3 = S15/S3 = Sp4/(Sp3*S3)
....
HPn = S(4n+3)/S3 = Sp(n+1)/(Spn*S3)
-----
OP1 = S8 = S15/S7 = Spin9/Spin8
OP2 = S23/S7 (?) = F4/Spin9
OP3 = S31/S7 ? I dont know whether it exists ?
...
-----
Let e1,...e7 be the base of ectonions such us e1e2=e3. Let v= (1,e4) belongs to O2.
Let Tv = <v, e1v,...,e7v> be 8-dimensional plane in R16. I have checked that
vector w = e2(e1v) does not belong to Tv. In other words rank of the matrix {v, e1,...,e7v, e2(e1v)} is 9.
I have checked this using Mathematica.
So what is the geometrical definition of OP1 ? How the Hopf transformation S15 -> S8 is defined ?
I propose following definition (x,y) -> (xx*, xy*).
I have checked that this is Hopf transformation S^3->S^2 for complex numbers (x,y).
----
Do you know any free tool for multiplying octonions, it can be under Linux ? I have downloaded CLICAL, but it does not work good under Windows XP.
Regards,
Marek Mitros
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