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What IS a Jordan-like Algebra?

Jordan Algebras,Generalizations, andAlgebraic Geometry

Exceptional Jordan and LieStructures

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, andCl(8)

Jordan Algebras and SeveriVarieties | UniqueJordan Structures

Here is some early history of myD4-D5-E6-E7-E8 VoDou Physics Model.


To construct a Jordan Algebra: Start with n x n matrices A and B, 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, inhis Week162, has an excellent introduction to Jordan Algebras. John Baezsays:

"... Their classification is nice and succinct. An "ideal" in theJordan algebra A is a subspace B such that if b is in B, a o b liesin B for all a in A. A Jordan algebra A is "simple" if its onlyideals are {0} and A itself. Every formally real Jordan algebra is adirect sum of simple ones. The simple formally real Jordan algebrasconsist of 4 infinite families and one exception:

... think about 2 x 2 hermitian matrices with entries in anyn-dimensional normed division algebra, say K. ... The space h2(K) ofhermitian 2 x 2 matrices with entries in K is a Jordan algebra withthe product x o y = (xy + yx)/2 ... this Jordan algebra is ... a spinfactor There is an isomorphism

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

which sends the hermitian matrix

a+b kk* a-b

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

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

a+b kk* 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 isthe heavenly sphere S(K + R); ... the latter is the projective lineKP1. It follows that these are the same! This shows that:

... it follows ... that:

[ Note that in my signatureconvention SL(2,O) = Spin(1,9) and that theClifford algebra Cl(1,9) = M(32,R) = Cl(2,8). ]

and thus:

 

As John Baez says,"... if the connected component of the group of linearmetric-preserving transformations of R^{p,q} is called SO_0(p,q),then the connected component of the group of conformaltransformations of the conformal compactification of R^{p,q} isSO_0(p+1,q+1) ...[and Spin(p,q) is the double cover ofSO_0(p,q)]...

For example:

Here are three examples that are relevant to theD4-D5-E6-E7-E8 VoDou physics model ( Note that I write V(p,q) fora vector space V with p negative definite dimensions and q positivedefinite dimensions, while some others, including John Baez above,write V(q,p) ) :

 

 

 


 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.If you start with a generalized sphere, 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.If you start with a generalized sphere, the Jordan algebras generate generalized radial expansions and contractions 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.

 JordanAlgebras can be Generalized by processes similar to:

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

 

The FreudenthalAlgebra Fr3(O), the BrownAlgebra Br3(O), and GradedLie Algebras are useful in describing the AlgebraicStructure of the MacroSpace ofMany-Worlds in the D4-D5-E6-E7-E8VoDou Physics model.

 

To see how useful they are, consider theEarly History of the D4-D5-E6-E7-E8 VoDou Physicsmodel.

 


Exceptional Jordan and LieStructures

The following table (table 5.1, of the article Jordan Algebras and their Applicationsby 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 linesin these planes and use the interpretation of these lines inreal 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 ofmotions are the compact groups in the Freudenthal magic squareare locally isomorphic to the groups of motions in thefollowing 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    S15John Baez says"... the complex numbers have a distinct advantage ... Only in this case can we turn any self-adjoint complex matrix into a skew-adjoint one, and vice versa, by multiplying by i.  I.e., only in this casecan 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 independentof other entries with numbers):      Jordan                          Lie    Hermitian                 Skew-Hermitian (Anti-Hermitian)    Self-adjoint                  Skew-adjoint          1    2    2                     1    2    2J3(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)    Self-adjoint                  Skew-adjoint          1    2    2                     1    2    2J3(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)    Self-adjoint                  Skew-adjoint         1    8    8                     7    8    8J3(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  4x8133-dim E7 =    *    -   4x8     + 14 +  3                 *    *   10                14   8x8  8x8248-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 formulaLie 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    8133-dim E7 =    3  +  *    -   24   +   *    -    8   + 14                      *    *    3       *    *    7                       7   56   56       7    8    8248-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 * * 11 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 E7and 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 ofthe 224-dim JE8 plusthe 24-dim Chevalley algebra Chev3(O) of 3x3 Hermitian Octonion matriceswith 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 tothe Vinberg construction of E8 given in Rosenfeld (1997):                8   8x8  8x8248-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                                 24                                 3      ... 3 qubits ...24                                4      ... 4-qubit case ... 128                               5880                               66,816                             760,032                            8589,312                           96,384,384                        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. 
Click here to read about a7-grading structure for E8 noticed by Thomas Larsson on reading apaper by Peter West, which structure has Vector, Lie,Freudenthal (Jordan),and Clifford parts.

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),buthas the structural skeleton of 2 copies of 26-dim traceless J3(O)o.28-dim D4 (plus the finite symmetry group S3) isthe automorphism group of the 24-dim Chevalley algebra Chev3(O):          8          |                                  0    8    8         28           = automorphisms of     *    0    8        /  \                                 *    *    0       8    8If you take the three 8s of Chev3(O) as corresponding tothe 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 alsothe three 8s of Chev3(O) describe D4.For example, you can build the 28-dim adjoint of D4 bytaking the wedge product of two of the 8s, as 8/\8 = 28,which is the same procedure that J. F. Adams usedto construct representations of E8 in his paperThe 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 structuresthat have a lot of common structure.


F4, E6, E7,E8

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-DicksonProcess 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 MinimalRepresentations of E6, E7, and E8 and the Generalized CapelliIdentity (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-DicksonProcess 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 algebraJ4(Q) of 4x4 Hermitian matrices over the Quaternions

p D B AD* q E CB* E* r FA* C* F* t

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

The 4x28 = 112-real dimensionalQuaternification of J4(Q) can be represented as the Symmetric SpaceE8 / E7 x SU(2), with physical interpretation as a theory ofTimelike BraneUniverses.

 

J4(Q) contains the traceless 28-1 = 27-dimensional subalgebraJ4(Q)o that "hasthe unique structure of" the 27-dimensional exceptional Jordanalgebra J3(O) of 3x3 Hermitianmatrices over the Octonions

p B AB* q CA* C* r

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

The 2x27 = 54-realdimensional Complexification of J3(O) = J4(Q)o can be represented asthe Symmetric Space E7 / E6 x U(1), with physical interpretationas a theory of Spacelike BraneUniverses.

 

 

In turn,

J3(O) contains atraceless 27-1 = 26-dimensional subalgebra J3(O)o that can berepresented as the Symmetric Space E6 / F4, with physicalinterpretation as a theory of theBohm Potential of the Many-Worlds Quantum Theory.

 

 

 


 

Note - Considering the Cayley-Dickson construction of Octonions Oas doublings of Quaternions Q, I find it interesting to compare theJordan 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, made by Stephen Adler in his (Oxford 1995) book 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,which has graded structure:1    8   28   56   70   56   28    8    1and 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    8and 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 containingboth E7 Lie and Br3(O) Jordan-like structure,andsince 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 use248-dim E8 = 120-dim bivector adjoint of D8 + 128-dim D8 half-spinorand 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-E8VoDou Physics model:

1971

Physics Today (August 1971, pages 17-18) carries an articledescribing Armand Wyler's calculation ofthe Electromagnetic Fine Structure Constant as 1 / 137.03608... basedon the geometry of Bounded Complex Homogeneous Domains. Although Idid not then, and still do not, understand Wyler's physicsmotivation, ever since I read about Wyler's work I felt that thegeometry of Bounded Complex HomogeneousDomains should prove useful in making a physics model in whichparticle masses and force strengths could be calculated.

 

1980

My first efforts at constructing the model were based on faiththat exceptional structures ought to be useful in physics. The24-cell,

the only regular polytope in any dimension that is both centrallysymmetric and self-dual, impressed me so much that I took it to bethe foundation of the model:

(I have since changed my mind about how the 28 generators ofD4 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 12forming 8 gluons, 3 weak bosons, and 1 U(1) photon.)
 
 1981

Physics Today (May 1981, pages111-112) published a letter dated 26January 1981 from me similar in content to the1980 AMS Notices Query, but adding references to "... the three[quaternionic structures] used by Adler to describe theelectromagnetic, weak, and strong-color fields, plus the one[quaternionic structure] used by Gursey to describe thegravitational 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 Ibanezand Lopez are the first purely theoretical calculations of theT-quark mass to be about 130 GeV:  

 

 1984

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

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

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

 This isa link to how my physics model looked in 1992.

My early physics of the D4-B4-F4 chain of Lie Algebras, with28-dim D4 = Spin(8), 36-dim B4 = Spin(9), and 52-dim F4, used thefibrations

Note that youcould look at F4 / Spin(8) as the flag manifolds of all full flags inOP2.

Since there are no larger rank-4 Lie algebras ( A4 = SU(5) being24-dim and C4 being 36-dim ), the chain stops with F4. The rootvectors of the chain are:

In the D4-D5-E6-E7-E8 VoDou Physicsmodel, whose relationship to the D4-B4-F4 model can be seen byconsidering the symmetric spaceE6/F4,

The Jordan algebra approach gives this Structure:

Automorphism
Groups ---- of -------------------- Jordan-like AlgebrasD4 ---------------- 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 inthe fibrations B4 / D4 for SpaceTime plus Internal Symmetry Space andF4 / B4 for representing Fermion Particles and AntiParticles, if yougo beyond the D4-B4-F4 chain by the fibrationE6 / F4 whose symmetric space E6 / F4 is partof the Shilov boundary of the bounded domain corresponding to E7 /E6xU(1).

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 structurefrom the beginning, giving this morenaturally realistic Structure:

Automorphism
Groups ---- of -------------------- Jordan-like AlgebrasD4 ---------------- 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)P1E6/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 upof 133-dim E7 plus 112-dim Br3(O) plus Quaternionic SU(2).

The lowest dimensional non-trivial representation of E8 is its248-dim Adjoint representation, which can be seen as the sum ofa 224-dimensional tesseract Jordan-likething plus a 24-dimensional thing like Chev3(O).

Therefore

E8 containsboth of the

Geometric andAlgebraicdescriptions of the MacroSpace ofMany-Worlds

Further, the fibrations D5 / D4xU(1) for SpaceTime and InternalSymmetry Space and E6 / D5xU(1) for representing Fermion Particlesand AntiParticles have natural Complex structure for thecalculationally useful geometry of BoundedComplex Homogeneous Domains.

 

These structures are related to GradedLie Algebras.


References:

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

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

The Bookof Involutions, by M.-A. Knus, A.Merkurjev, M.Rost, and J.-P. Tignol (with a preface by J. Tits), AmericanMathematical Society Colloquium Publications, vol. 44 (AmericanMathematical 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 (apseudonym for a group of French mathematicians), Springer, Berlin1978.

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

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

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

On the Role of Division, Jordan and Related Algebras in ParticlePhysics, 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).

J. M. Landsberg has anAlgebraic Geometry point of view of Freudenthal-Titsconstructions.

 

 


  

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