And here I add some supporting information for what I wrote previously from Tony Smith that he e-mailed to me about the above transformations and ideas: (his webspace is quoted a lot, by John Baez etc) http://www.tony5m17h.net/1TSphysics.html#clifvodou
In terms of Lie Groups:
And about automorphism groups of projective spaces:
not that I know it all, but suddenly it becomes very easy since we have tetration/selfroot ( and more) and their geometrical interpretation which EXACTLY fits the things the above groups, algebras, spinor spaces, covers describe.
From here, its a mere question of technique to develop both hyperops and understand and VISUALIZE complicated symmetries in space.
Tell me what You think about it, please.
Ivars
Quote: Spheres and projective spaces can be thought of like this (I hope I have got this right,
but could be making some errors in notation or thought):
n-dim sphere Sn = n-dim real space Rn plus {point at infinity}
n-dim real projective space RPn = n-dim real space Rn plus { RP(n-1) at infinity } =
= Rn plus R(n-1) plus { RP(n-2) at infinity } =
= Rn plus R(n-1) plus R(n-2) plus R(n-3) plus ... plus { RP0 = two points = +1 , -1 at infinity }
n-dim complex projective space CPn = n-dim complex space Cn plus { CP(n-1) at infinity } =
= Cn plus C(n-1) plus { CP(n-2) at infinity } =
= Cn plus C(n-1) plus C(n-2) plus C(n-3) plus ... plus { CP0 = S1 = circle at infinity }
n-dim quaternion projective space QPn = n-dim quaternion space Qn plus { QP(n-1) at infinity } =
= Qn plus Q(n-1) plus { QP(n-2) at infinity } =
= Qn plus Q(n-1) plus Q(n-2) plus Q(n-3) plus ... plus { QP0 = S3 = 3-sphere at infinity }
My point is that the
real projective spaces have a two-point { +1 , -1 } structure at infinity
so that does look like spinor + - structure or a discrete Z2 2-element group at infinity and
complex projective spaces have a circle S1 1-dim-sphere at infinity
and since S1 = U(1) you can get a U(1) continuous symmetry group at infinity
(like electromagnetic U(1) gauge group)
and
quaternion projective spaces have a 3-sphere S3 at infinity
and since S3 = SU(2) you can get a SU(2)continuous symmetry group at infinity
(like weak force SU(2) gauge group)
To get the color force SU(3), you have to go to octonion projective spaces,
which (due to non-associativity) stop at OP2,
but you do have
OP2 = O2 plus { OP1 at infinity } =
= O2 plus O1 plus { OP0= S7 = 7-sphere at infinity }
so
octonions give you a 7-sphere S7 at infinity
Although S7 is not a Lie group, it naturally expands to Spin(= S7 + G2 + S7
and G2 contains SU(3)
so that you can get a SU(3) continuous symmetry group at infinity
(like color force SU(3) gauge group)
and
furthermore if you look at Spin(as a gauge group,
you can get (as I do in my E8 physics model) both Gravity and the Standard Model.
Tony
In terms of Lie Groups:
Quote:As to what I wrote in terms of Lie groups,
I hope this helps and does not have too many mistakes:
spheres Sn = SO(n+1) / SO(n)
real projective RPn = SO(n+1) / O(n) = SO(n+1) / SO(n) x {+1,-1}
note that O(n) is a double cover of SO(n)
which double cover comes from the {+1,-1} structure at infinity of real projective spaces and is geometrically related to the antipodal map between two hemispheres of a sphere.
complex projective CPn = SU(n+1) / U(n) = SU(n+1) / SU(n)xU(1)
and CP1 = SU(2) / U(1) = S3 / S1 = S2 by the Hopf fibration S1 -> S3 -> S2
quaternion projective QPn = Sp(n+1) / Sp(n)xSp(1)
and QP1 = Sp(2) / Sp(1)xSp(1) = Spin(5) / Spin(3)xSpin(3) = Spin(5) / Spin(4) = S4
where Spin(n) has the same Lie algebra as SO(n)
For octonions:
OP1 = Spin(9) / Spin(= S8 which is also S15 / S7 by the Hopf fibration S7 -> S15 -> S8
and
OP2 = F4 / Spin(9)
If you enlarge F4 by
complexifying F4 you get E6
quaternifying F4 you get E7
octonifying F4 you get E8
Tony
And about automorphism groups of projective spaces:
Quote:Here are some remarks about automorphisms of structures at infinity:
spheres - only one point at infinity - the only automorphism is the identity 1
real projective spaces - {+1,-1} at infinity - automorphism is Z2, the 2-element group
complex projective spaces - 1-sphere S1 at infinity - automorphism group U(1) = S1
quaternion projective spaces - 3-sphere S3 at infinity - automorphism group SU(2) = S3
so for those, the structure at infinity can be given group structure so that
it looks like its own automorphism group
For octonions,
with an S7 at infinity, the automorphism group is not S7 itself because S7 is not a group,
but the S7 expands naturally to Spin(= S7 + S7 + G2
where G2 is the octonion automorphism group
You might say that the expansion of S7 occurs because
the automorphism group of the octonions G2 is 14-dim and so bigger than 7-dim S7,
so that S7 has to expand by G2 to get group structure,
and then that process brings in a second S7 to produce S7 + G2 + S7 = 28-dim Spin(.
Tony
not that I know it all, but suddenly it becomes very easy since we have tetration/selfroot ( and more) and their geometrical interpretation which EXACTLY fits the things the above groups, algebras, spinor spaces, covers describe.
From here, its a mere question of technique to develop both hyperops and understand and VISUALIZE complicated symmetries in space.
Tell me what You think about it, please.
Ivars


= S7 + G2 + S7