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E8 LieAlgebra from Clifford AlgebraCl(8)

and also D4-D5-E6-E7-E8of my VoDou Physics Model, as well asB4 and F4

The Clifford Algebra Cl(8) has dimension 2^8 = 256.

Since 256 = 16 x 16 = 2^4 x 2^4, the full spinors of Cl(8) are16-dimensional, and the half-spinors of Cl(8) are8-dimensional.

Cl(8) has graded structure

1   8  28  56  70  56  28   8   1

The 28-dimensional grade-2 bivectors of Cl(8) form the LieAlgebra Spin(8).

The E8 Lie Algebra isthe sum (on this page I am using the word "sum" very imprecisely)of

To construct E8 from Cl(8): 

First, construct the 120-dimensional Spin(16) fromCl(8):

Since Cl(16) can be written as the tensor product Cl(16) = Cl(8) xCl(8), the graded structure of Cl(16) can be written in terms of thegraded structure of Cl(8) as follows:

1     8    28    56    70    56    28     8     1      8    64   224   448   560   448   224    64     8           28   224   784  1568  1960  1568   784   224    28                 56   448  1568  3136  3920  3136  1568   448    56                       70   560  1960  3920  4900  3920  1960   560    70                             56   448  1568  3136  3920  3136  1568   448    56                                   28   224   784  1568  1960  1568   784   224    28                                          8    64   224   448   560   448   224    64     8                                                1     8    28    56    70    56    28     8     11    16   120   560  1820  4368  8008 11440 12870 11440  8008  4368  1820   560   120    16     1

Therefore:

Spin(16) = 120 = 28x1 + 8x8 + 1x28 = 28 + 64 + 28, and

Second, construct the 128-dimensional half-spinors ofCl(16) from Cl(8):

The Cl(8) half-spinors are 8-dimensional, so that the Cl(8)full-spinor space is 8e + 8o = 16-dimensional, where 8e is onehalf-spinor 8-dimensional space and 8o is the other mirror imagehalf-spinor 8-dimensional space.

ou can construct the spinor space of Cl(16) as the tensor product( 8e + 8o ) x ( 8e + 8o ) as follows:

8ex8e  +  8ex8o                    8ox8e  +  8ox8o

If ee and oo correspond to e, and if eo and oe correspond to o, wehave:

64e   + 128o  +   64e 

so that the Cl(16) spinors are (64 + 64)e + 128o =256-dimensional, and

Therefore:

248-dimensional E8 is thesum of

two copies of 28-dimSpin(8)

the 64-dim square of the8-dimensional Cl(8) grade-1 vector space

the sum of the 64-dim squares ofthe two 8-dimensional Cl(8) half-spinor spaces.

and

256-dimensional Cl(8) is thesum of

248-dimE8

the 8-dim vector space of Cl(8).

What about D4-D5-E6-E7-E8of my VoDou Physics Model, as well asB4 and F4?

Here is the analogous pattern for 133-dimensional E7:

E7 is the sum of 28-dim Spin(8) plus Quaternionic versions of thetwo 8-dim half-spinor spaces of Cl(8) and the 8-dim vector space ofCl(8), each of which is 4x8 = 32-dimensional.

The central 9 elements correspond to the Quaternionic SU(2)symmetries of each of the two half-spinor spaces and of the vectorspace. Note that distinct Quaternionic SU(2)s are used for each ofthe half-spinor spaces, as is consistent with the fact (noted byGeoffrey Dixon in his book on DivisonAlgebras) that the left-adjoint and right-adjoint actions of theQuaternions are not isomorphic to each other, nor are they 4x4 matrixalgebras over a real 4-dim vector space.

Here is the analogous pattern for 78-dimensional E6:

E6 is the sum of 28-dim Spin(8) plus Complex versions of the two8-dim half-spinor spaces of Cl(8) and the 8-dim vector space ofCl(8), each of which is 2x8 = 16-dimensional.

The central 2 elements correspond to the Complex U(1) symmetriesof the spinor spaces and of the vector space. Note that the sameComplex U(1) is used for both of the half-spinor spaces, as isconsistent with the fact (noted by GeoffreyDixon in his book on Divison Algebras) that the left-adjoint andright-adjoint actions of the Complex numbers are both isomorphic toeach other, but are not 2x2 matrix algebras over a real 2-dim vectorspace.

Here is the analogous pattern for 45-dimensional D5:

D5 is the sum of 28-dim Spin(8) plus a Complex versions of the8-dim vector space of Cl(8), which is 2x8 = 16-dimensional. D5 is theSpin(10) Lie algebra of the Clifford Algebra Cl(10), which factors bytensor product into Cl(10) = Cl(2) x Cl(8). Since the gradedstructure of Cl(2) is 1 + 2 + 1, the graded structure of Cl(10)is

1     8    28    56    70    56    28     8     1      2    16    56   112   140   112    56    16     2            1     8    28    56    70    56    28     8     11    10    45   120   210   252   210   120    45    10     1

The 1 of the 45 corresponds to the grade-2 bivector of Cl(2),acting as U(1) on the 2-dim vector space of Cl(2). It is representedon the pattern by the central element corresponding to the ComplexU(1) symmetry of the Complex version of the Cl(8) vector space.

Here is the analogous pattern for 28-dimensional D4:

D4 is the 28-dim Spin(8) of the Cl(8) Clifford Algebra with gradedstructure

1 8 28 56 70 56 28 8 1

Here is the analogous pattern for 36-dimensional B4:

B4 is the sum of 28-dim Spin(8) plus the 8-dim vector space ofCl(8). B4 is the Spin(9) Lie algebra of the Clifford Algebra Cl(9),which factors by tensor product into Cl(9) = Cl(1) x Cl(8). Since thegraded structure of Cl(1) is 1 + 1, the graded structure of Cl(9)is

1     8    28    56    70    56    28     8     1      1     8    28    56    70    56    28     8     11     9    36    84   126   126    84    36     9     1

Here is the analogous pattern for 52-dimensional F4:

F4 is the sum of 28-dim Spin(8) plus the two 8-dim half-spinorspaces of Cl(8) and the 8-dim vector space of Cl(8).

Note that the 16-dimensional symmetric space F4 / Spin(9) is theOctonion Projective Plane, which isrepresented on the pattern by the two 8-dim half-spinor spaces ofCl(8).

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