# What IS a Jordan-like Algebra?

Jordan Algebras, Generalizations, and Algebraic Geometry

Exceptional Jordan and Lie Structures

```F4 + J3(O)o -> E6 + Cx( J3(O)=J4(Q)o ) + U(1) -> E7 + Qx( J4(Q) ) + SU(2) -> E8
|                                                         |
F4         <-          OP2 + B4          <-         OP1 + D4                   ```
Jordan, Hilbert, D4, and Cl(8)
```

To construct a Jordan Algebra:

with entries that are elements of one of
the division algebras over the reals:
the real numbers;
the complex numbers;
the quaternions; or
the octonions.

Consider the matrix product AB.

Note that the product AB can be decomposed
into antisymmetric and symmetric parts:

AB =  (1/2)(AB - BA)  +  (1/2)(AB + BA)

Recall that the Lie algebra product is

[A,B] =  (1/2)(AB - BA)

that is, the antisymmetric part of the matrix product AB.

What about the symmetric part of the matrix product AB?
It is
AoB =  (1/2)(AB + BA)
IT IS THE JORDAN ALGEBRA PRODUCT.

```

John Baez, in his Week 162, has an excellent introduction to Jordan Algebras. John Baez says:

"... Their classification is nice and succinct. An "ideal" in the Jordan algebra A is a subspace B such that if b is in B, a o b lies in B for all a in A. A Jordan algebra A is "simple" if its only ideals are {0} and A itself. Every formally real Jordan algebra is a direct sum of simple ones. The simple formally real Jordan algebras consist of 4 infinite families and one exception:

• The algebra of n x n self-adjoint real matrices with the product x o y = (xy + yx)/2.
• The algebra of n x n self-adjoint complex matrices with the product x o y = (xy + yx)/2.
• The algebra of n x n self-adjoint quaternionic matrices with the product x o y = (xy + yx)/2.
• ... Given an n-dimensional real inner product space V, the "spin factor" J(V) is the Jordan algebra generated by V with the relations v o w = <v,w> This should remind you of the definition of a Clifford algebra, and indeed, they're related - they have the same representations! This sets up a connection to spinors, which is why these Jordan algebras are called "spin factors". ... J(V) is isomorphic to the direct sum V + R equipped with the product (v,a) o (w,b) = (aw + bv, <v,w> + ab) ...
• ... we can think of J(V) = V + R as "Minkowski spacetime", with V as space and R as time. The reason is that J(V) is naturally equipped with a dot product: (v,a) . (w,b) = <v,w> - ab which is just the usual Minkowski metric in slight disguise. ... define the "lightcone" to consist of all nonzero x in J(V) with x . x = 0 A 1-dimensional subspace of J(V) spanned by an element of the lightcone is called a "light ray", and the space of all light rays is called the "heavenly sphere" S(V). We can identify the heavenly sphere with the sphere of unit vectors in V, since every light ray is spanned by an element of J(V) of the form (v,1) where v is a unit vector in V. ... the Lorentz group acts as conformal transformations of the heavenly sphere.
• The algebra of 3 x 3 self-adjoint octonionic matrices with the product x o y = (xy + yx)/2. This is called the "exceptional Jordan algebra".

... think about 2 x 2 hermitian matrices with entries in any n-dimensional normed division algebra, say K. ... The space h2(K) of hermitian 2 x 2 matrices with entries in K is a Jordan algebra with the product x o y = (xy + yx)/2 ... this Jordan algebra is ... a spin factor There is an isomorphism

f: h2(K) -> J(K + R) = K + R + R

which sends the hermitian matrix

```a+b     k
k*    a-b```

to the element (k,b,a) in K + R + R.

Furthermore, the determinant of matrices in h2(K) is just the Minkowski metric in disguise, since the determinant of

```a+b     k
k*    a-b```

is a2 - b2 - <k,k> ...

... since the Jordan algebras J(K + R) and h2(K) are isomorphic, so are their associated projective spaces. ... the former space is the heavenly sphere S(K + R); ... the latter is the projective line KP1. It follows that these are the same! This shows that:

• h2(R) is 3d Minkowski spacetime, and RP1 is the heavenly sphere S1;
• h2(C) is 4d Minkowski spacetime, and CP1 is the heavenly sphere S2;
• h2(H) is 6d Minkowski spacetime, and HP1 is the heavenly sphere S4;
• h2(O) is 10d Minkowski spacetime, and OP1 is the heavenly sphere S8. ...

... it follows ... that:

• SL(2,R) is the double cover of the Lorentz group SO0(2,1);
• SL(2,C) is the double cover of the Lorentz group SO0(3,1);
• SL(2,H) is the double cover of the Lorentz group SO0(5,1);
• SL(2,O) is the double cover of the Lorentz group SO0(9,1).
[ Note that in my signature convention SL(2,O) = Spin(1,9) and that the Clifford algebra Cl(1,9) = M(32,R) = Cl(2,8). ]

and thus:

• SL(2,R) acts as conformal transformations of the sphere S1 = RP1;
• SL(2,C) acts as conformal transformations of the sphere S2 = CP1;
• SL(2,H) acts as conformal transformations of the sphere S4 = HP1;
• SL(2,O) acts as conformal transformations of the sphere S8 = OP1. ...".

As John Baez says, "... if the connected component of the group of linear metric-preserving transformations of R^{p,q} is called SO_0(p,q), then the connected component of the group of conformal transformations of the conformal compactification of R^{p,q} is SO_0(p+1,q+1) ...[and Spin(p,q) is the double cover of SO_0(p,q)]...

For example:

• the conformal compactification of the Euclidean space R^p is the sphere S^p, so [the connected component of] the group of conformal transformations of this sphere is SO_0(p+1,1). ...
• [the connected component of] the conformal group of [the conformal compactification of] the Minkowski space R^{p,1} is SO_0(p+1,2). ...".

Here are three examples that are relevant to the D4-D5-E6-E7-E8 VoDou physics model ( Note that I write V(p,q) for a vector space V with p negative definite dimensions and q positive definite dimensions, while some others, including John Baez above, write V(q,p) ) :

• Let V = O + R (where O = Octonions) = 9-dim
• unit sphere in V = S(V) = OP1 = S8 = 8-dim
• J(V) = J(O+R) = (O+R) + R = h2(O) = 10-dim
• Spin(1,9) is
• the double cover of the Lorentz group over J(V) and
• the double cover of the Conformal group over S8 = S(V)
• Spin(2,10) is the double cover of the Conformal group over 10-dim J(V).

```
Roughly speaking:

Antisymmetric and Antihermitian matrices
form Lie algebras under the antisymmetric Lie product

(There are some subtleties about derivations and the
Jacobi identity  that I am ignoring here -
see, for example, Jordan Algebras and Their Applications,
by Kevin McCrimmon, Bull. Am. Math. Soc. 84 (1978) 612-627.);

and

Symmetric and Hermitian matrices
form Jordan algebras under the symmetric Jordan product.

Here is a NON-rigorous
GEOMETRIC way to compare Lie and Jordan algebras:

The antisymmetric Lie algebras generate generalized rotations.
the Lie algebras are infinitesimal generators of
generalized rotations or flows
mapping the generalized sphere into itself.

The symmetric Jordan algebras generate generalized radial distortions.
the Jordan algebras generate
mapping the generalized sphere into a generalized ellipsoid.

Here is a NON-rigorous
PHYSICAL way to compare Lie and Jordan algebras:

The Lie algebras are infinitesimal generators of
gauge groups.

The Jordan algebras correspond to the matrix
algebra of quantum mechanical states,
that is, from a particle physics point of view,
the configuration of particles in spacetime
upon which the gauge groups act.

For a given spacetime and configuration of particles on it,
a gauge group acts on the spacetime as
a separate independent LOCAL generalized rotation
at EACH POINT of the spacetime.
The generalized rotation is NOT
a generalized rotation of spacetime itself,
but
IS a generalized rotation in another space
(the space describing the identity of the particles
in the particle physics model)
a copy of which is attached to each point of spacetime,
sort of like tangent spaces.
For example, the color space on which
the SU(3) color force acts is a 3-dim space.

For the octonion non-associative division algebra,
the largest matrices that form a Jordan algebra are 3x3,
forming the 27-dimensional exceptional Jordan algebra J3(O),
which represents the 27-dimensional MacroSpace
of the D4-D5-E6-E7-E8 VoDou physics model.

The 26-dimensional traceless subalgebra J3(O)o can represent
the 26-dimensional Bosonic String Structure of MacroSpace. ```

Jordan Algebras can be Generalized by processes similar to:
• Complexification, producing Freudenthal Algebras; and
• Quaternification, producing ternary non-binary Brown Algebras.

To see details of how these processes work, see materials in the references:

The Freudenthal Algebra Fr3(O), the Brown Algebra Br3(O), and Graded Lie Algebras are useful in describing the Algebraic Structure of the MacroSpace of Many-Worlds in the D4-D5-E6-E7-E8 VoDou Physics model.

To see how useful they are, consider the Early History of the D4-D5-E6-E7-E8 VoDou Physics model.

### Exceptional Jordan and Lie Structures

```The following table (table 5.1, of the article
Jordan Algebras and their Applications
by Kevin McCrimmon (Bull. A.M.S. 84 (1978) 612-627)
with a reference citation to the book
Exceptional Lie Algebras,
by N. Jacobson, Dekker, New York, 1971)
(due to typgraphical limitations I have used + for direct sum
and * for overbar, and I will use x sometimes to mean tensor product):
```
```----------------------------------------------------------------------

Type   Lie Algebra       Lie (or Algebraic) Group      Dimension

G2   Derivations of O      Automorphisms of O              14

F4   Derivations of H3(O)  Automorphisms of H3(O)          52

E6   Reduced structure     Reduced structure group     52+(27-1)=78
algebra Strlo(J)=     Strl(J)/R  Id of H3(O)
= Der J + VJo

E7   Superstructure        Superstructure group        27+79+27=133
algebra Strlo(J)=         of H3(O)
J + Strl(J) + J*
of H3(O)

E8         ?                       ?                       248

----------------------------------------------------------------------```
```In my opinion, the Lie algebra D4 should be added to the list
as an exceptional Lie algebra. After the table from page 540 of
The Book of Involutions
```
```dim A   F     FxFxF    H3(F,a)  H3(K,a)  H3(Q,a)  H3(C,a)
1     0       0        A1       A2       C3       F4
2     0       U        A2      A2xA2     A5       E6
4     A1   A1xA1xA1    C3       A5       D6       E7
8     G2      D4       F4       E6       E7       E8   ```
```the authors say:
"... Here ... Q for Quaternion algebra and C for a Cayley algebra;
U is a 2-dimensional abelian Lie algebra. ```
```The fact that D4 appears in the last row is one more argument for
considering D4 as exceptional. ...".
-------------------------------------------------------------------```
`About E8, McCrimmon (1978) says: `
```"... when A = O [octonions], J = J3(O), the Lie algebra
Der A  +  ( Ao x Jo )  +  Der J
will have dimension 14 + (7x26) + 52 = 248.   ...".```
```Rosenfeld (1997) says:
"... If we replace the elliptic planes ... by lines
in these planes and use the interpretation of these lines in
real elliptic spaces ...
[such as the theorem ...
The Hermitian elliptic lines (QxO)S1 and (OxO)S1 admit
interpretations as the manifold of 3-planes in the space S11,
respectively as the manifold of 7-planes in the space S15. ..]
we obtain ... Theorem 7.24.
The groups of motions in lines in the planes whose groups of
motions are the compact groups in the Freudenthal magic square
are locally isomorphic to the groups of motions in the
following real elliptic spaces:

[Type]   [Real Elliptic Spaces]

[F4]          S8

[E6]          S9

[E7]          S11

[E8]          S15                            ...".

If you look at the Freudenthal-Tits magic square,
you see that the diagonal entries are related to Hopf fibrations:

[Real Elliptic Spaces]

S1     S2     S4     S8

S2     S3     S5     S9

S4     S5     S7     S11

S8     S9     S11    S15

John Baez says
"... the complex numbers have a distinct advantage
... Only in this case can we turn any
and vice versa, by multiplying by i.
I.e., only in this case
can we naturally identify the Jordan algebra of OBSERVABLES
with the Lie algebra of SYMMETRY GENERATORS.

... We don't just want a Jordan algebra
... we don't just want a Lie algebra
... we want something that's both ...".

In other words, for complex 3x3 matrices
(the number entries denote dimension,
and * denotes an entry that by symmetry is not independent
of other entries with numbers):

Jordan                          Lie
Hermitian                 Skew-Hermitian (Anti-Hermitian)

1    2    2                     1    2    2
J3(C)  =  *    1    2          L3(C) =    *    1    2  = 9-dim U(3)
*    *    1                     *    *    1

Here J3(C) is a nice 9-dim Jordan algebra,
and
L3(C) is 9-dim U(3) = SU(3) x U(1).

Since U(3) reduces, by cutting out 1-dim U(1), to 8-dim SU(3)
and
since J3(C) has a nice traceless 8-dim subalgebra J3(C)o,
you get a hint that a useful way to rewrite the relation is

Traceless Jordan                 Irreducible Lie
Hermitian                 Skew-Hermitian (Anti-Hermitian)

1    2    2                     1    2    2
J3(C)o =  *    -    2          L3(C) =    *    -    2  = 8-dim SU(3)
*    *    1                     *    *    1

The - marks the dimension lost due to the trace zero condition.

==================================================================

In the octonion case, the correspondence is not as simple.
for example (using the traceless version):

Jordan                          Lie
Hermitian                 Skew-Hermitian (Anti-Hermitian)

1    8    8                     7    8    8
J3(O) =  *    -    8          L3(O) =    *    -    8
*    *    1                     *    *    7

Here J3(O)o is the 26-dim subalgebra of the 27-dim Jordan algebra J3(O),
but
L3(O) is a 38-dim thing that is NOT an exceptional Lie algebra.
To get L3(O) to be a Lie algebra,
you have to add the 14-dim automorphism group G2 of the Octonions O,
thus getting the 38+14 = 52-dim exceptional Lie algebra F4.

In comparing the Complex and Octonionic cases,
you see that
the Jordan algebra can be made only of the 3x3 matrices,
but
the Lie algebra also needs the derivations of the Division Algebra
that is used in the 3x3 matrices.
Also,
you see that (as John Baez had noted) the Hermitian condition
and the skew(anti)Hermitian condition lead to different
dimensionalities on the diagonal of the 3x3 matrices,
because only in the Complex case
does the real dimension equal the imaginary dimension.

For the octonion case, Rosenfeld (1997),
gives on pages 79-80 a theorem of Vinberg
(I use Q instead of H, and I use * and ' for
conjugations in writing the theorem):

"... The ... Lie algebras ... F4, E6, E7, and E8
are direct sums of
the linear spaces of skew-Hermitian 3x3 matrices
whose entries are elements in the algebras O, CxO, QxO, and OxO,
respectively, with zer traces
and of
the ... Lie algebras of ... automorphisms in these algebras. ...
The condition of skew-Hermiticity is
a_ij = a_ij* for octonionic matrices
and
a_ij = a_ij*' for matrices with entries from tensor products ..."

Vinberg Lie Algebra Constructions in Rosenfeld:

7    8    8
52-dim F4 =    *    -    8      + 14
*    *    7

8   2x8  2x8
78-dim E6 =    *    -   2x8     + 14
*    *    8

10   4x8  4x8
133-dim E7 =    *    -   4x8     + 14 +  3
*    *   10

14   8x8  8x8
248-dim E8 =    *    -   8x8     + 14 + 14
*    *   14

Compare the Vinberg constructions with the
Freudenthal-Tits Lie Algebra Constructions in McCrimmon
in which the exceptional Lie algebras F4, E6, E7, and E8
also correspond to the pairs
(A,J) = (R,O), (C,O), (Q,O), and (O,O)
but according to the formula

Lie Algebra  =   Der A  +  ( Ao x J3(O)o )  +  Der J3(O)

Freudenthal-Tits Lie Algebra Constructions in McCrimmon:

52-dim F4 =    0  +  (0x26)      +  52

78-dim E6 =    0  +  (1x26)      +  52

133-dim E7 =    3  +  (3x26)      +  52

248-dim E8 =   14  +  (7x26)      +  52

which can be written as:

7    8    8
52-dim F4 =    0  +  (0x26)        +   *    -    8   + 14
*    *    7

1    8    8       7    8    8
78-dim E6 =    0  +  *    -    8   +   *    -    8   + 14
*    *    1       *    *    7

3   24   24       7    8    8
133-dim E7 =    3  +  *    -   24   +   *    -    8   + 14
*    *    3       *    *    7

7   56   56       7    8    8
248-dim E8 =   14  +  *    -   56   +   *    -    8   + 14
*    *    7       *    *    7

which in turn can be written so that the Vinberg relation is clear:

7    8    8
52-dim F4 =    0  +  *    -    8    + 14
*    *    7

8   16   16
78-dim E6 =    0  +  *    -   16    + 14
*    *    8

10   32   32
133-dim E7 =    3  +  *    -   32    + 14
*    *   10

14   64   64
248-dim E8 =   14  +  *    -   64    + 14
*    *   14

How do these Octonionic Lie algebras correspond to
the corresponding Jordan algebras, which I will designate
by J3(O), J3(CxO), J3(QxO), and J3(OxO) ?

It seems that you do NOT just take
the non-skew3x3 Hermitian KxO matrices.

For example,
it is known that the Freudenthal algebra whose automorphism
group is E6 is 56-dimensional.
Since the 3x3 matrices of CxO are 9x2x8 = 144-dimensional,
and since the Vinberg skew part of E6 is 48-dimensional,
it seems to me that the non-skew part is 144-48 = 96-dimensional,
which is bigger than the 56-dimensional Freudenthal algebra.

From Rosenfeld (1997), pages 91 and 56, it seems to me
that 56-dim Fr3(O) with automorphisms E6 should
be written as a 2x2 Zorn-type array:
```
```                 1 8 8
1             * 1 8
* * 1

1 8 8
* 1 8              1
* * 1                 ```
```
If you try to "think like a Vegan",
you might see that
E7 and E8 might be represented as higher-dim arrays,
such as 2x2x2 and 2x2x2x2.

When you go to a 3-dim 2x2x2 array
for the 112-dim Brown "algebra-like thing" corresponding to E7,
you get a picture like this:
```
```                          1 8 8
1 ------------- * 1 8
/  |              * * 1
/   |            /   |
/    |           /    |
/     |          /     |
1 8 8      |         /      |
* 1 8 ------------ 1        |
* * 1      |                |
|        |       |        |
|        |       |        |
|        |       |        |
|      1 8 8     |
|      * 1 8 ----|------- 1
|      * * 1     |       /
|    /           |      /
|   /            |     /
|  /             |    /
/            1 8 8
1 ------------ * 1 8
* * 1         ```
```Note that the J3(O) corners and the 1 corners correspond
to two tetrahedra within the cube.

When you go to a 4-dim 2x2x2x2 array for E8,
you get a 224-dim "thing" that looks like a tesseract:```
```
1 8 8
* 1 8 --------------------------------------------------- 1
* * 1                                                   //|
/  |  \                                                 // |
/   |  \                                                //  |
/    |   \                                              //   |
/     |   \                                             //    |
/      |    \                                           //     |
/       |     \                                         //      |
/        |     \                                        //       |
/         |      \                                      //        |
/          |       \                                 1 8 8         |
1 ---------------------------------------------------- * 1 8         |
| \          |         \                               * * 1         |
|  \         |          \                            /  /|           |
|   \        |          \                          /   / |           |
|    \       |           \                        /1 8 8 |           |
|      \     |           1 ----------------------/ * 1 8 |           |
|       \    |         /  |                     /  * * 1 |           |
|        \   |        /   |                    / /   |   |           |
|         \  |       /    |                   / /    |   |           |
|           \|      /     |                  / /     |   |           |
|            \ 1 8 8      |                 / /      |   |           |
|             \* 1 8 ---------------------- 1        |   |           |
|            | * * 1      |                          |   |           |
|            |   |        |                 |        |   |           |
|            |   |        |                 |        |   |           |
|            |   |        |                 |        |   |           |
|            |   |      1 8 8               |        |   |           |
|            |   |      * 1 8 --------------|------- 1   |           |
|            |   |      * * 1               |       /  \ |           |
|            |   |    /   /                 |      /    \|           |
|            |   |   /   /                  |     /      |           |
|            |   |  /  /                    |    /       |\          |
|            |   | /  /                   1 8 8 /        | \         |
|            |   1 ---------------------- * 1 8          |   \       |
|            | /    /                     * * 1          |    \      |
|            |/   /                         \            |     \     |
|            /   /                           \           |       \   |
|           /| /                              \          |        \1 8 8
|          / 1 --------------------------------\---------|-------- * 1 8
|         / /                                   \        |         * * 1
|        / /                                     \       |        /
|       / /                                       \      |       /
|      / /                                         \     |      /
|     / /                                           \    |     /
|    / /                                             \   |    /
|   / /                                               \  |   /
|  / /                                                 \ |  /
| / /                                                   \| /
1 8 8                                                     |/
* 1 8 --------------------------------------------------- 1
* * 1
```
```It has 8x1 + 8x(1+1+1+8+8+8) = 224 dimensions,
which is twice the 112 dimensions of the Br3(O) structure of E7
and is 4 times the 56 dimensions of Fr3(O) of E6.

Therefore,
the 248-dim Lie algebra E8 might be seen to be made up of
the 224-dim JE8 plus
the 24-dim Chevalley algebra Chev3(O) of 3x3 Hermitian Octonion matrices
with zero diagonal.

That is,

248-dim E8 =    224-dim JE8 + 24-dim Chev3(O)    =

8   8x8  8x8                    *    8    8
=    *    8   8x8     +  8     +     *    *    8
*    *    8                     *    *    *

This can be rearranged to see the relation to
the Vinberg construction of E8 given in Rosenfeld (1997):

8   8x8  8x8
248-dim E8 =    *    8   8x8     +  8     +   24
*    *    8

14   8x8  8x8
=    *    -   8x8     + 14     +   14
*    *   14

----------------------------------------------------------------

In other words,
248-dim E8 does not have the same structural skeleton as 224-dim JE8,
but must have an additional 24-dim Chev3(O).

To see the full structure of 248-dim E8 as a Tesseract Creature:
use 8 copies of 28-dim J(4,Q) instead of 8 copies of 27-dim J(3,O)
and
use 8 copies of a 3-dim thing instead of 8 copies of 1 so
that you get 24-dim Chev(3,O) from

* 1 1     * 8 8
8 x  * * 1  =  * * 8
* * *     * * *
```
` `
```
gold      blue      red      green
28 + 28 + (28+28) + (28+28) + (28+28)
8    +    8    +    8

( D4 + D4 +   64 )  + ( 64    +   64 )

120         +        128

NOTE - NONE of the 8x3 things carry the rank of E8,
ALL of the rank 8 of E8 comes from D4 + D4,
so the 8x3 does NOT represent 8 copies of SU(2)
but rather the 8x3 represents 24-dim Chev(3,O)
and

we have a Tesseract Creature structure for

248-dim E8 = 120-dim Adjoint(D8) + 128-dim Half-Spinor(D8)

E8 is the ultimate Exceptional Lie Group,
and therefore can be seen as a Unique Parallelizable Structure
that can be used to describe realistic physics of Entanglement.
Joy Christian in arXiv 0904.4259
"Disproofs of Bell, GHZ, and Hardy Type Theorems
and the Illusion of Entanglement" says:
"… a [geometrically] correct local-realistic framework … provides exact,
deterministic, and local underpinnings for at least the Bell, GHZ-3, GHZ-4,
and Hardy states. … The alleged non-localities of these states … result
from misidentified [geometries] of the EPR elements of reality. …
The correlations are … the classical correlations among the points of
a 3 or 7-sphere … S3 and S7 … are … parallelizable …
The correlations … can be seen most transparently in
the elegant language of Clifford algebra …".

Entanglement is related to HyperDeterminants, as described
in arxiv quant-ph/0206111 by Akimasa Miyake:
"... multidimensional determinants "hyperdeterminants" ... are
... related to entanglement measures ...
construct DetA_n of format 2^n (n qubits) ...
degree ... of homogeneity ... for n ...
2                                 2
4                                 3      ... 3 qubits ...
24                                4      ... 4-qubit case ...
128                               5
880                               6
6,816                             7
60,032                            8
589,312                           9
6,384,384                        10 ...".

The n = 8 case corresponds to 2^8 = 256-dim Cl(8) Clifford algebra
that can be used to build the Lie group E8 from Cl(8)xCl(8) = Cl(16),
with E8 being the Tesseract Creature of 8x28 + 8x3 = 248 dimensions.
```

In hep-th/0008063, Murat Gunaydin describes

"... a quasiconformal nonlinear realization of E8 on a space of 57 dimensions. This space may be viewed as the quotient of E8 by its maximal parabolic subgroup; there is no Jordan algebra directly associated with it, but it can be related to a certain Freudenthal triple system which itself is associated with the "split" exceptional Jordan algebra J3(OS) where OS denote the split real form of the octonions O .It furthermore admits an E7 invariant norm form N4 , which gets multiplied by a (coordinate dependent) factor under the nonlinearly realized "special conformal" transformations. Therefore the light cone, defined by the condition N4 = 0, is actually invariant under the full E8, which thus plays the role of a generalized conformal group. ... results are based on the following five graded decomposition of E8 with respect to its E7 x D subgroup ... with the one-dimensional group D consisting of dilatations ...
```g(-2)    g(-1)    g(0)    g(1)    g(2)
1        56     133+1     56      1  ```
... D itself is part of an SL(2; R ) group, and the above decomposition thus corresponds to the decomposition ... of E8 under its subgroup E7 x SL(2;R) ...".
```
For other exceptional cases:

133-dim E7, the automorphism group of a 112-dim thing Br3(O),
does not have the same structural skeleton as Br3(O),
but needs to have an additional 21-dim structure.
```
In hep-th/0008063, Murat Gunaydin describes "... a completely explicit conformal Mobius-like nonlinear realization of E7 on the 27-dimensional space associated with the exceptional Jordan algebra J3(OS) ... where OS denote the split real form of the octonions O ... with linearly realized subgroups F4 (the "rotation group") and E6 (the "Lorentz group"). ... [results are] based on a three graded decomposition ... of E7
`133 = 27 + (78+1) + 27`
under its E6 x D subgroup ... with the one-dimensional group D consisting of dilatations ...".
```78-dim E6, the automorphism group of the 56-dim Fr3(O),
does not have the same structural skeleton as Fr3(O),
but needs to have an additional 22-dim structure.

52-dim F4, the automorphism group of 27-dim J3(O),
does not have the same structural skeleton as J3(O),
but
has the structural skeleton of 2 copies of 26-dim traceless J3(O)o.

28-dim D4 (plus the finite symmetry group S3)
is
the automorphism group of the 24-dim Chevalley algebra Chev3(O):

8
|                                  0    8    8
28           = automorphisms of     *    0    8
/  \                                 *    *    0
8    8

If you take the three 8s of Chev3(O) as corresponding to
the three 8-dim fundamental representations of D4,
that is the vector and two half-spinor representations,
you see that,
not only is D4 (up to finite S3 outer automorphism)
the automorphism group of Chev3(O),
but also
the three 8s of Chev3(O) describe D4.

For example, you can build the 28-dim adjoint of D4 by
taking the wedge product of two of the 8s, as 8/\8 = 28,
which is the same procedure that J. F. Adams used
to construct representations of E8 in his paper
The Fundamental Representations of E8,
Contemporary Mathematics 37 (1985) 1-10,
reprinted in vol. 2 of The Selected Works of J. Frank Adams.

Therefore:
```
```D4 and Chev3(O) are Octonionic Lie and Jordan-like structures
that have a lot of common structure.
```

### J3(O)o, J3(O)=J4(Q)o, J4(Q)

```F4 + J3(O)o -> E6 + Cx( J3(O)=J4(Q)o ) + U(1) -> E7 + Qx( J4(Q) ) + SU(2) -> E8
|                                                         |
F4         <-          OP2 + B4          <-         OP1 + D4                   ```

B. N. Allison and J. R. Faulkner, in their paper A Cayley-Dickson Process for a Class of Structurable Algebras (Trans. AMS 283 (1984) 185-210), say:

"... we obtain a procedure for giving the space Bo of trace zero elements of any ... 28-dimensional degree 4 central simple Jordan algebra B ... the structure of a 27-dimensional exceptional Jordan algebra. ... ".

Ranee Brylinski and Bertram Kostant, in their paper Minimal Representations of E6, E7, and E8 and the Generalized Capelli Identity (Proc. Nat. Acad. Sci. 91 (1994) 2469-2472), say:

"... there are exactly three simple Jordan algebras J' of degree 4. All three are classical. They are given as J' = Herm(4,F)c where now F = R, C, or H ... For the three cases we have ...[(using my notation)
```F                          dim( J4(F) )

R (Real J4(R) )                  10
C ( Complex J4(C) )              16
H ( Quaternion J4(Q) )           28    ```

]...".

B. N. Allison and J. R. Faulkner, in their paper A Cayley-Dickson Process for a Class of Structurable Algebras (Trans. AMS 283 (1984) 185-210), say:

"... Suppose B is a 28-dimensional central simple Jordan algebra of degree 4 with generic trace t. Let Bo = { b in B | t(b) = 0 } and choose e in Bo such that t(e^3) =/= 0. Then, Bo has the unique structure of a 27-dimensional exceptional central simple Jordan algebra with identity e ...

... [There] are linear bijections of ... a central simple Jordan algebra of degree 4 ... B ... onto the vector space of all skew-symmetric 8x8 matrices ... ". [ In the case of 28-dimensional B = J4(Q), the corresponding vector space would be the 28-dimensional vector space of real skew-symmetric 8x8 matrices, which can be represented as the 28-dimensional D4 Lie algebra Spin(8). ]

Therefore,

The 28-real-dimensional degree-4 quaternionic Jordan algebra J4(Q) of 4x4 Hermitian matrices over the Quaternions

```p   D   B   A

D*  q   E   C

B*  E*  r   F

A*  C*  F*  t```

where * denotes conjugate and p,q,r,t are in the reals R and A,B,C,D,E,F are in the quaternions Q, and there is a correspondence between the Jordan algebra J4(Q) and the D4 Lie algebra Spin(8).

The 4x28 = 112-real dimensional Quaternification of J4(Q) can be represented as the Symmetric Space E8 / E7 x SU(2), with physical interpretation as a theory of Timelike Brane Universes.

J4(Q) contains the traceless 28-1 = 27-dimensional subalgebra J4(Q)o that "has the unique structure of" the 27-dimensional exceptional Jordan algebra J3(O) of 3x3 Hermitian matrices over the Octonions

```p   B   A

B*  q   C

A*  C*  r```

where * denotes conjugage and p,q,r are in the reals R and A,B,C are in the Octonions O.

The 2x27 = 54-real dimensional Complexification of J3(O) = J4(Q)o can be represented as the Symmetric Space E7 / E6 x U(1), with physical interpretation as a theory of Spacelike Brane Universes.

In turn,

J3(O) contains a traceless 27-1 = 26-dimensional subalgebra J3(O)o that can be represented as the Symmetric Space E6 / F4, with physical interpretation as a theory of the Bohm Potential of the Many-Worlds Quantum Theory.

Note - Considering the Cayley-Dickson construction of Octonions O as doublings of Quaternions Q, I find it interesting to compare the Jordan algebra correspondence

J4(Q)o = J3(O)

with the Lie algebra correspondence

Spin(4) = Spin(3)xSpin(3).

### Jordan, Hilbert, D4, and Cl(8)

``` As John Baez says:
"... the complex numbers have a distinct advantage ..."
and here I will take that remark out of context to mention a point,
Quaternionic Quantum Mechanics and Quantum Fields (pp.10-11):
"...
standard quantum mechanics [is formulated] in a complex Hilbert space
... The special Jordan algebras are equivalent ... to the
Dirac formulation in ... real, complex, or quaternionic Hilbert space
...
the ... exceptional Jordan algebra ... of the 27-dimensional
non-associative algebra of 3x3 octonionic Hermitian matrices
... corresponding to a quantum mechanical system over
a two- (and no higher) dimensional projective geometry that
cannot be given a Hilbert Space formulation ...
...
Zel'manov (1983) ... proved that in the infinite-dimensional case
one finds no new simple exceptional Jordan algebras ...".

Very roughly and non-rigorously,
Adler is stating the conventional wisdom that:

You can't do serious physics with Octonions,
because
Octonion non-associativity prevents you from building
a nice big Hilbert space, with high-order tensor products.

However, if you "think like a Vegan",
you will see that,
although my model has a lot of octonionic structure,
it does not fail due to those conventional objections.

Here is a rough outline (ignoring things such as signature)
of how my model gets high-order tensor products:

My model is based on the D4 Lie algebra,
which is the bivector Lie algebra of the Cl(8) Clifford algebra,

1    8   28   56   70   56   28    8    1

and total dimension 2^2 = 256 = 16x16 = (8+8)(8+8)
with 8-dimensional +half-spinors, 8-dimensional -half-spinors,
8-dimensional vectors, and 28-dim bivector adjoint representation.

Therefore Cl(8) contains all 4 fundamental representations
of the D4 Lie algebra, with Dynkin diagram

8
|
28
/  \
8    8

and I can (and do) embed all my D4 structures into Cl(8),
which is nice and real and associative.
Even further, for any value of N, no matter how large,
Clifford periodicity
lets me decompose Cl(8N) as the tensor product of N copies of Cl(8):

Cl(8N ) = Cl(8) x ...(N times tensor product)... x Cl(8)

Therefore, with my model,
I can build nice big spaces for real physics using Cl(8N).

Since I see the Hyperstructure of E8 as containing
both E7 Lie and Br3(O) Jordan-like structure,
and
since I use Clifford algebras like Cl(8), which contains D4,
to get large tensor products,
I should say how to fit E8 into Clifford structure.

The most straighforward way is to use

248-dim E8 = 120-dim bivector adjoint of D8 + 128-dim D8 half-spinor

and so embed E8 in the Clifford algebra Cl(16), with graded structure
```
`1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1`
```and total dimension 2^16 = 65,536 = (128+128)(128+128)

Since Cl(16) = Cl(2x8) = Cl(8)xCl(8)
( For example, 120 = 1x28 + 8x8 + 28x1 and 128 = 8x8 + 8x8. )

E8 can be represented in a tensor product of Cl(8) algebras,
which is consistent with the structures of my physics model.

```

### Early History of the D4-D5-E6-E7-E8 VoDou Physics model:

1971

Physics Today (August 1971, pages 17-18) carries an article describing Armand Wyler's calculation of the Electromagnetic Fine Structure Constant as 1 / 137.03608... based on the geometry of Bounded Complex Homogeneous Domains. Although I did not then, and still do not, understand Wyler's physics motivation, ever since I read about Wyler's work I felt that the geometry of Bounded Complex Homogeneous Domains should prove useful in making a physics model in which particle masses and force strengths could be calculated.

1980

My first efforts at constructing the model were based on faith that exceptional structures ought to be useful in physics. The 24-cell, the only regular polytope in any dimension that is both centrally symmetric and self-dual, impressed me so much that I took it to be the foundation of the model:

• Since the 24 vertices of the 24-cell describe the vertex figure of the Quaternion Integral D4 lattice, I tried to develop a model with a Lattice SpaceTime having 4-dimensional Quaternionic structure that carried over to its continuum limit as a smooth 4-dim SpaceTime manifold. In January 1980 the Notices of the American Mathematical Society (27 (1980) 67) published a Query (no.208) from me saying: "... J. A. Wolf (J. Math. Mech. 14 (1965) 1033-10470 has shown that there are 4 types of 4-dimensional complete simply connected Riemannian symmetric spaces with quaternionic structure:
• (I) Euclidean 4-space, which has Abelian structure [ the 4-torus T4];
• (II) SU(2) / S(U(1)xU(1)) x SU(2) / S(U(1)xU(1)), ... [ S2 x S2 ] ...;
• (III) SU(3) / S(U(2)xU(1)), ... [ CP2 ] ...; and
• (IV) Sp(2) / Sp(1)xSp(1) ... [ = Spin(5) / Spin(4) = S4 ] ...,

and the noncompact duals of II, III, and IV. ... Can the correspondence between the 4 ty0es of spaces having quaternionic structure and the 4 forces of phyiscs be used to constuct a unified theory ...?".

• Since the 24-cell is the Root Vector Polytope of the 28-dimensional D4 Lie Algebra Spin(8), I tried to develop a model whose Gauge Bosons were the represented by the 28 infinitesimal generators of Spin(8). In my early work,
• 10 formed the Anti-deSitter Lie Algebra B2 = C2 = Spin(2,3) = Sp(2), which could be gauged by the MacDowell-Mansouri mechanism to produce Gravity with a Cosmological Constant and Torsion;
• 8 formed the 8 gluons of Color SU(3)
• 6 formed the 3 weak bosons of the SU(2) Weak force, with
• 3 of the 6 corresponding to 2 of the 4 polarization directions t,x,y,z and
• the other 3 corresponding to the other 2 polarizaitons; and
• 4 formed the U(1) photon of ElectroMagnetism,
• one for each of the 4 covariant polarizations t,x,y,z.
(I have since changed my mind about how the 28 generators of D4 work, in that now I use 16 of them to form U(2,2) = U(1)xSU(2,2) = U(1)xSpin(2,4) and gauge it to get gravity, with the remaining 12 forming 8 gluons, 3 weak bosons, and 1 U(1) photon.)

1981

Physics Today (May 1981, pages111-112) published a letter dated 26 January 1981 from me similar in content to the 1980 AMS Notices Query, but adding references to "... the three [quaternionic structures] used by Adler to describe the electromagnetic, weak, and strong-color fields, plus the one [quaternionic structure] used by Gursey to describe the gravitational field. ... ".

1982-1983

As far as I know, the following works of Inoue, Kakuto, Komatsu, and Takeshita; of Alvarez-Gaume, Polchinski, and Wise; and of Ibanez and Lopez are the first purely theoretical calculations of the T-quark mass to be about 130 GeV:

• Inoue, Kakuto, Komatsu, and Takeshita write Aspects of Grand Unified Models with Softly Broken Suypersymmetry (Prog. Theor. Phys. 68 (1982) 927) (received 10 May 82). They relate supersymmetry to electro-weak symmetry breaking by radiative corrections and renormalization group equations, and find that the renormalization group equations have a fixed point.
• The fixed point is related to a T-quark mass of about 125 GeV, as was explicitly discussed in 1983 by Alvarez-Gaume, Polchinski, and Wise, who wrote Minimal Low-Energy Supergravity (Nuc. Phys. B221 (1983) 495-523) (received 8 Feb 83). Their calculations show that, for electro-weak symmetry breaking to occur, the T-quark mass must be from 100 GeV to 195 GeV.Moreover, they also note (p. 511) that the renormalization group equation "... tends to attract the top quark mass towards a fixed point of about 125 GeV."
• Work similar to that of Alvarez-Gaume, Polchinski, and Wise was done by Ibanez and Lopez in N=1 Supergravity, the Weak Scale and the Low-Energy Spectrum (Nuc. Phys. B233 (1984) 511-544) (received 8 Aug 83).

1984

On 27 February 1984 my paper "Particle Masses, Force Constants, and Spin(8)" was received by the International Journal of Theoretical Physics (Int. J. Theor. Phys. 24 (1985) 155-174). It contained the first publication of my calculation of the truth quark mass as 129.5 GeV. [ More history of Truth Quark mass calculations and experimental analysis can be found here and here. ]

On 16 October 1984 my paper "Spin(8) Gauge Field Theory" was received by the International Journal of Theoretical Physics (Int. J. Theor. Phys. 25 (1986) 355-403). It contained some ideas that have remained useful, such as

• my calculations of force strengths, particle masses, and Kobayashi-Maskawa constants, and
• my initial ideas (learned from David Finkelstein and his student Ernesto Rodriguez at Georgia Tech) about Clifford Algebra Pregeometry,

as well as some ideas about which I have since changed my mind, such as

• details of the Higgs mechanism and weak bosons (for example, I no longer have 3 generations of weak bosons), and
• my thought that the physically important sequence of Lie algebras was D4-B4-F4.
• (Other wrong ideas that I had around that time, such as some about chaos and quantum theory, were mercifully omitted from that paper.)
This is a link to how my physics model looked in 1992.

My early physics of the D4-B4-F4 chain of Lie Algebras, with 28-dim D4 = Spin(8), 36-dim B4 = Spin(9), and 52-dim F4, used the fibrations

• B4 / D4 = Spin(9) / Spin(8) = S8 corresponding to the vector representation of D4
• F4 / B4 = F4 / Spin(9) = OP2 corresponding to the two half-spinor representations of D4
Note that you could look at F4 / Spin(8) as the flag manifolds of all full flags in OP2.

Since there are no larger rank-4 Lie algebras ( A4 = SU(5) being 24-dim and C4 being 36-dim ), the chain stops with F4. The root vectors of the chain are:

• 28-dim D4 has 24 root vectors forming a 24-cell.
```28-dim D4, and the finite triality symmetry group S3,
is
the automorphism group of the 24-dim Chevalley algebra Chev3(O):

8
|                                  0    8    8
28           = automorphisms of     *    0    8
/  \                                 *    *    0
8    8

If you take the three 8s of Chev3(O) as corresponding to
the three 8-dim fundamental representations of D4,
that is the vector and two half-spinor representations,
you see that,
not only is D4 (up to finite S3 outer automorphism)
the automorphism group of Chev3(O),
but also
the three 8s of Chev3(O) describe D4.
For example, you can build the 28-dim adjoint of D4 by
taking the wedge product of two of the 8s, as 8/\8 = 28,
which is the same procedure that J. F. Adams used
to construct representations of E8 in his paper
The Fundamental Representations of E8,
Contemporary Mathematics 37 (1985) 1-10,
reprinted in vol. 2 of The Selected Works of J. Frank Adams.
Therefore, in my opinion:
D4 and Chev3(O) are Octonionic Lie and Jordan-like structures
that are substantially equivalent to each other. ```
• 36-dim B4 has 32 root vectors forming a 24-cell plus 8 vertices.
• 52-dim F4 has 48 root vectors forming one 24-cell and its dual 24-cell.
```52-dim F4, the automorphism group of 27-dim J3(O),
does not have the same structural skeleton as J3(O),
but
has the structural skeleton of
2 copies of 26-dim traceless J3(O)o.
The F4 doubling of J3(O)o may be related to
the fact that the 48-vertex F4 root vector diagram
is made up of two copies of the 4-dim 24-cell,
which are dual to each other. ```

In the D4-D5-E6-E7-E8 VoDou Physics model, whose relationship to the D4-B4-F4 model can be seen by considering the symmetric space E6/F4,

• the 24 vertices of a 24-cell can represent 24 of the 28 gauge boson generators of Spin(8), and
• the 24 vertices of its dual 24-cell can represent
The Jordan algebra approach gives this Structure:
```
Automorphism Groups ---- of -------------------- Jordan-like Algebras

D4 ---------------- Gauge Bosons --------- Chevalley Algebra Chev3(O)
B4 ------- SpaceTime + Internal Symmetry ------- Jordan Algebra J2(O)
F4 ----------------- Fermions ------------------ Jordan Algebra J3(O)```

Although there is no natural Complex Structure in the fibrations B4 / D4 for SpaceTime plus Internal Symmetry Space and F4 / B4 for representing Fermion Particles and AntiParticles, if you go beyond the D4-B4-F4 chain by the fibration E6 / F4 whose symmetric space E6 / F4 is part of the Shilov boundary of the bounded domain corresponding to E7 / E6xU(1).

• E6 / F4 = ( set of OP2s in (CxO)P2 ) with 26-dim structure of J3(O)o

you get a complex structure that can be added to the spaces B4 / D4 and F4 / B4.

`E6 --------------- Complex Structure ----- Freudenthal Algebra Fr3(O)`

If you use the chain D4-D5-E6, you get complex structure from the beginning, giving this more naturally realistic Structure:
```
Automorphism Groups ---- of -------------------- Jordan-like Algebras

D4 ---------------- Gauge Bosons --------- Chevalley Algebra Chev3(O)
D5 ------ SpaceTime + Internal Symmetry -- Freudenthal Algebra Fr2(O)
E6 ---------------- Fermions ------------- Freudenthal Algebra Fr3(O)
E7 --------------- Many-Worlds ----------------- Brown Algebra Br3(O)
----------------------- E8 ------------------------------------------```

```D5/D4xU(1) = (CxO)P1

E6/D5xU(1) = (CxO)P2

E7/E6xU(1) = 54-real-dim = 27-complex-dim set of (CxO)P2s in (QxO)P2 =
= Hermitian symmetric space corresponding to
the bounded symmetric domain of type EVII,
which is "... represented ...
by the 3x3 Hermitian matrices over the Cayley numbers ..."
according to Helgason (1978).

E8/E7xSU(2) = 112-dim) = set of (QxO)P2s in (OxO)P2```
```E7 contains the Superstructure, describing the overall structure
of the Worlds of the ManyWorlds of quantum theory,
and
E8 contains, for both chains, the Hyperstructure. ```
```E8/E7xSU(2) is 112-dim with the structure of Br3(O),
the Jordan-like algebra of which E7 is the automorphism group,
so that:
248-dim E8 contains 112-dim Br3(O)

Since E7 is a local symmetry group of E8/E7xSU(2),
it is also true that
248-dim E8 contains 133-dim E7.

Therefore,
E8 Hyperstructure should be seen as
an amalgam of both the Lie and Jordan-like algebras E7 and Br3(O)
also containing as glue the quaternionic SU(2): ```
248-dim E8 is made up of 133-dim E7 plus 112-dim Br3(O) plus Quaternionic SU(2).

The lowest dimensional non-trivial representation of E8 is its 248-dim Adjoint representation, which can be seen as the sum of a 224-dimensional tesseract Jordan-like thing plus a 24-dimensional thing like Chev3(O).

Therefore

E8 contains both of the

Geometric and Algebraic descriptions of the MacroSpace of Many-Worlds

Further, the fibrations D5 / D4xU(1) for SpaceTime and Internal Symmetry Space and E6 / D5xU(1) for representing Fermion Particles and AntiParticles have natural Complex structure for the calculationally useful geometry of Bounded Complex Homogeneous Domains.

These structures are related to Graded Lie Algebras.

References:

The web site of R. Skip Garibaldi and his papers Structurable Algebras and Groups of Types E6 and E7, math.RA/9811035 and Groups of Type E7 Over Arbitrary Fields, math.AG/9811056.

The Jordan Book by Kevin McCrimmon, apparently no longer (as of 24 January 2003) a preprint on the web, but, according to Amazon.com, to be published in January 2003 under the title "A Taste of Jordan Algebras".

The Book of Involutions, by M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol (with a preface by J. Tits), American Mathematical Society Colloquium Publications, vol. 44 (American Mathematical Society,Providence, RI, 1998); also see http://www.math.ohio-state.edu/~rost/BoI.html.

Manifolds All of Whose Geodesics are Closed, by Arthur L. Besse (a pseudonym for a group of French mathematicians), Springer, Berlin 1978.

Einstein Manifolds, by Arthur L. Besse (a pseudonym for a group of French mathematicians) (Springer-Verlag 1987).

Differential Geometry, Lie Groups, and Symmetric Spaces, by Sigurdur Helgason (Academic 1978).

Geometry of Lie Groups, by Boris Rosenfeld (Kluwer 1997).

On the Role of Division, Jordan and Related Algebras in Particle Physics, by Feza Gursey and Chia-Hsiung Tze (World 1996).

The Selected Works of J. Frank Adams, ed. by. J. P. May and C. B. Thomas (Cambridge 1992).

Jordan Algebras and their Applications, by Kevin McCrimmon (Bull. A.M.S. 84 (1978) 612.

Exceptional Lie Algebras, by N. Jacobson, Dekker, New York, 1971).

```
```