affine Group E9 and symmetries of its Coxeterelement
by Mag.rer.nat. Kronberger Reinhard
Why do we consider the E9 group (more specifically
the Coxeter element of this group)?
1) E9 is an affine group and thus has something to do with
2) The extension is flat as the universe.
3) The action of the Coxeter elements of the group produces
symmetries involving our current standard model.
Symmetries which arise from the Coxeter element of
= SU(5) x SU(3) x SU(2) x U(1) x U(1) (SU(n) =
Special unitary group, U(1) unitary group)
evidently SU(5) x SU(3) x SU(2) x U(1)
SU(3)c x SU(2)L
x U(1)Y(Color charge, isospin, Hyper charge)
Write the symmetry in order to:
E9CS = SU(5)S
x SU(2)L x
actual Standard Model)
Dynkin Diagram E9 (affine one point extension of group E8):
Derivative of the symmetries of E9CS from the
invariants of the Coxeter elements E9:
A Coxeter element is a product of
the generating reflections of E9.
For example: Coxeterelement = e1.e2.e3.e4.e5.e6.e7.e8.e9
The Coxeterpolynom is the
characteristic polynomial of Coxeter elements and
has the form:
E9(x) is a polynom with terms of
cyclotomic factors for n>1 and
(x-1) for n= 1.
The cyclotomic factors are the characteristical polynom of the An-1
(which is the Dynkin diagram for the SU(n) Liegroup.See more here:
So finally the symmetry space by the actions of the Coxeterelements
is SU(5) x SU(3) x SU(2) x U(1) x U(1)
What bring us the additional
(1) These have the
potential to describe new particles.
(2) These have the
potential to describe the space and time.
These have the potential to describe gravity.
Wish to analogously represent Graviton to the
photon as a blend (Weinberg angle see <8>).
Light and gravitation just like
photon and graviton have something in common.
Both are massless and propagate
with the speed of light.
We know that light by the Symmetry breaking 1: SU(2)xU(1)--> U(1)e is described
as a mixture.
So light is a part of the electro-weak interactions.
we consider analog gravity as a
result of a further symmetry breaking
Symmetry breaking 2: SU(5) x U(1) x U(1) --> U(1)g
Our extended standard model allows
We will now like to
assign our relevant SU(n)'s to division algebras
(real numbers, complex numbers, ...).
This 4 divison algebras (real
numbers, complex numbers, quaternions and octonions) develop
through the doubling process
Considering the rank of the SU(2)
= 1,SU(3)= 2, SU(5) = 4 then this is double as well.
There appears to be a connection
between the division algebras and the SU(n)’s (n = 2,3,5)..
I think the trivial connections are the rank (maximal torus) of the
SU(n) and the orthogonal complex subspaces of
the divison algebra.
For example the quaternions have two orthogonal complex
subspaces a+b.i1 and c.i2+d.i3 (a,b,c,d real).
The SU(3) has also 2 neutral elements (toris).
With this assignment we can create backgroundfields to the SU(3)
and SU(5) like it is the higgsfield for SU(2).
Therefore, we rely
analogously on the Higgsfield (2
x complex = doublet)
<2>the Octoquintenfield (5 x
This provides 40 degrees of
24 of which will be "spent" for
our SU(5) tensor bosons for the 5th longitudinal spin degree of
freedom (24 Goldstone bosons swallowed over gauge
transformation) thus remain 16 left.
The S, F, R, G and H charges are
the 5 charges of the SU (5) analogous to the 3 color charges of
SU (3) and the 2 charges (+ .-) of SU (2).
The letters stand for S = See, F =
feeling, R=smelling G = Taste and H = Hear
Calling therefore the charges of
the SU (5) sense charges.
Note: These charges
have (such as the color charges of quarks with color) nothing to
do with the senses, but to give a name to the child for reference
We now want to look at the 16 (40-24 = 16) remaining degrees of
Make the following
division for the 40 field components of the Octoquinten field as a
Take care that the division is not unique because for the left half
4 gray fields we can use 220.127.116.11= 24 Permutations of them in the
And for the left 4 charges we have five over 4 = 5 Permutations.
So at all we have 5 x 24 = 120
On the Higgsfield we have 2 x 1= 2 permutations.
Analogeous to the Higgspotential we declare a Potential on the
<3>Potential over the Octoquintenfield
<3.2> Getting a new
particle by the Octoquintenfield of m ~ 28 GeV mass.
With this assumption we can calculate the higgsmass.See Assumption 1
at the end.
A second particle can be maybe found by the first term of the
Hint:The Higgsmass as calculated above comes from the second term of
By using Planckunits we get from the Einstein-Form
of the potential the
16-Cell (Coxeter group B4,D4):
But the 16-Cell is not the only 4 polytope which fits for the
See more APPENDIX III at the end of the document (600-Cell).
of the Octoquintenfield/Octoquintenpotential
I do the same steps as shown in this cooking recipe for the
STEP 1:Lorentzinvariant Lagrangedensity for the
by the Octoquintenfield
The construction comes from multiplications (symmetric to the
diagonal) by 2 degrees of freedome (complex subspaces).
With this construction the tensor is symmetric in the diagonal.
10 independent fields.
Second CURVATURE TENSOR from the Octoquintenfield
(generates a spinpotential)
The construction comes from multiplications by 4 degrees of
freedome (quaternionic subspaces).
With this construction the tensor is symmetric in both diagonals.
5 independent fields A,B,C and two in the diagonal (blue and
So finally we get three derivation- or curvaturetensors of
the Octoquintenpotential for twisted spacetime excitation
the ART by the second curvaturetensor
The Octoquintenpotential has two symmetric
curvaturetensors.This motivates us to extend the Einstein Equation.
I think this shows that the GR (General
Relativity) has to be extended by an imaginäry part (spinpart) to be
a consistent quantumtheorie.
So finally we expect something like GR+i.GR°
where GR° is the spinpart.
more detailed with the two curvaturetensors of the Octoquintenfield:
The second curvaturetensor Cspin is
determinded by the first curevaturetensor Cem
because its components are a mix of the components of Cem.
vacuumpart of the extended Einstein equation then is:
<6.2> Scalefactor for the accelerated expanding Universe by
Getting a closed form for the Extended
For that we define the operator for 4 x 4 matrices or tensors:
and possible proofing of the Extended
<8>Some important points of the Octoquintenpotential
interpretation of the roots (zeropoints) in cubic
equations with 3 real zeropoints
Hint:On our special OQP (Octoquintenpotential) the zeropoints
(spheres) comes from a pentagon.
<9>Candidates for the dark matter in the universe.
Dark Energy comes by
definition from the Octoquintenpotential (the
second term in the potential). Dark Matter could be the W , Z Bosons and
the particles by the SU(5) Symmetry (adjoint
and fundamental presentation).
1) The 16-Cell
On the 16-Cell each vertizes is connected by an edge to all other
vertizes except the opposite one!
We have 4 disjunct such pairs which are not connected by an edge.
Now exchanging this points which are not connected is an
automorphism (for example p1 with p1-line) because
p1 has the same connections as p1-line.
So at all we generate 2^4=16 automorphisms by this actions because
we can say for all 4 pairs 0 means pair IS NOT exchanged and 1 for
pair IS exchanged.
So every binary code like (0,1,0,0) is an automorphism.
But this are not all automophism.Independend from that we can
permutate p1,p2,p3,p4 when we simultaneously permutate their
This give us 4!=24 automorphisms independend from the 16
So at all we get 16.24=384 automorphisms.
With this we can divide the Aut(C16) into
AutOpposite(C16) and AutFakt(C16)
so that Aut(C16) = AutOpposite(C16) x AutFakt(C16) = 16 x
AutOpposite(C16) are the Automorphisms of the
16-Cell C16 which comes from exchanging the
opposite vertizes (points).
AutFakt(C16) are the Automorphisms of the
16-Cell C16 which comes from permutating
p1,p2,p3,p4 vertizes (points) as described above.
AutFakt(C16) is simply the permutation-group
Sym(4) = S4.
Embedding (projection) the 16-Cell into
the quartenionic subgroups of the Octoquintenfield:
See also Quaterniongroup Q8
We have seen in <3.4> that the 16-Cell fits for the
combinatorial form of the Oktoquintenpotential.
But also the 600-Cell fits in the same way.
Together with the symmetries which comes from the pentagon of the
zeropoints (see above) we have the symmetries of the D5 demipenteract.
The order of the group = 10.192=1920 =16.5!
See also 10 regular Clebsch Graph.
Calculating backward from the Combinatorial- to the Planckform of
the Oktoquintenpotential (see <3.5>) we get the formular:
But i have to say that the assumption in
<3.2> is not absolute sure!
possible combinatorial massrelation of electron,myon and tauon.
Hint: there are other subgroups of S8 which
have also order 12 or 192!