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Voudons = 2^8-ions may be a very useful type of 2^N-ions, which include:
For (real Eucliean) Clifford algebras CL(N), nothing really new comes after Cl(8) with 2^8 elements, because of the periodicity theorem
so that the structure of very big Clifford algebras is really just the structure of nested Cl(8) Clifford algebras.
Clifford algebras Cl(N) with 2^N elements have graded structure of the Nth level of the binomial triangle, whose levels 0 through 8 are
1 = 1 2 = 1 1 4 = 1 2 1 8 = 1 3 3 1 16 = 1 4 6 4 1 32 = 1 5 10 10 5 1 64 = 1 6 15 20 15 6 1 128 = 1 7 21 35 35 21 7 1 256 = 1 8 28 56 70 56 28 8 1
and are related to the 2^N-ions with no graded structure by redefining the associative Clifford product which explicitly takes into account the graded structure of the 2^N elements to construct a 2^N-ion product table in which all 2^N elements are treated the same.
Note that an important difference is that for the antisymmetric graded product /\ a/\b = (-1)^pq b/\a where p and q are the grades of a and b, whereas for the antisymmetric ungraded product ab = - ba for all elements.
When you go to octonions, there are a lot of different ways that you can "ungrade" the Clifford algebra, so there are a lot of "different" octonion multiplication tables.
I expect that there exists an 8-fold periodicity theorem for 2^N-ions, differing from 8-fold (real Euclidean) Clifford periodicity only by having a lot of multiplicities for ways to "ungrade" the Clifford algebra, so that
Robert de Marrais in his paper math.RA/0207003 says:
"...Moreno ... determines that the automorphism group of the ZD's of all 2n-ions derived from the Cayley-Dickson process, from the Sedenions on up, obey a simple pattern:
for n > 4, this group has the form G2 x (n-3) x S3
... This says the automorphism group of the Sedenions' ZD's has order 14 x 1 x 6 = 84 ... Let's expand our horizons to the 32-D and 64-D cases and give them names (since they currently don't have any!). Following the convention adopted by Tony Smith, who has called the 2^8-ions Voudons after the 256 deities of the Ifa pantheon of Voodoo or Voudon, we'll call these
- Pathions (for the "32 Paths" of Kabbalah) and
- Chingons (for the 64 Hexagrams of the I Ching or Book of Changes).
- Routons ... the 2^7-ions ... after that legendary source of high-tech innovativeness, Route 128 of the "Massachusetts Miracle" that paralleled Silicon Valley's on the "Left Coast" of this country. ...
- Voudons ... the 2^8-ions Voudons ...
From Reals to Voudons, then, we extend the standard shorthand thus:
R, C, H, O, S, P, X, U, V.
... for 2^n-ions, n>3, the ZD-pairings formula, up to Voudons, gives 6 · ( 2^(n-1) - 4 ) = 24, 72, 168, 360, 744 ...
[ which is a formula that I prefer to write as 6 x ( 2^(n-1) - 4 ) =
The Voudon 2^8-ions have 744 ZD-pairings. As Andrew Ogg wrote in his Bull. AMS 25 (1991) 425-432 review of the book Vertex Operators and the Monster, by Igor Frenkel, James Lepowsky, and Arne Meurman (Academic 1988): "... the elliptic modular invariant j(t) classifies the isomorphism classes of elliptic curves over C, and has the Fourier expansion
where q = e^( 2 pi i t ) and t is in the upper half-plane. The coefficients c(n) are positive integers, growing rapidly, with
... j^(1/3) corresponds to ... a module for the affine Lie algebra ~E8, thereby "explaining" 744 = 3 x dim( E8 ) ... there are twenty-six sporadic groups, the last ... being the Monster M, of order
= 8 x 10^53
... with a nontrivial rational representation of (minimal) degree
... McKay made the striking observation that 196,884 = 196,883 + 1 ... 1 being the degree of the trivial representation ...". Therefore, it seems to me that the 744 ZD pairings of the Voudon 2^8-ions correspond to the constant term of the elliptic modular invariant, and it is interesting that 744 = 3 x 248 for 3 copies of the E8 Lie algebra, and that three copies of E8 lattices make up the 24-dim Leech lattice, and that each vertex of the Leech lattice has 196,560 nearest neighbors, and that the 196,560 Leech lattice units, plus 300 = symmetric part of 24x24, plus 24, are a total of 196,884, which is the sum of the dimensions of the 1-dim trivial representation and the 196,883-dim minimal representation of the Monster.
Robert de Marrais says in his paper math.GM/0011260: "... We know that surprises will keep on coming at least up to the 2^8-ions, due to the 8-cyclical structure of all Clifford Algebras. And we know at least a little about what such surprises will entail: as can already be seen in low-dimensioned Clifford Algebras including non-real units which are square roots of positive one, numbers whose squares and even higher-order powers are 0 will appear. The 8-cycle implies an iterable, hence ever-compoundable pattern, implying in its turn cranking out of numbers which are 2^Nth roots of 0, approaching as a limit-case an analog of the "Argand diagram" whose infinitude of roots form a "loop" of some sort. If we can work with this it all, it could only be by having as backdrop some sort of geometrical environment with an infinite number of symmetries . . . suggesting the "loop" resides on some sort of negative-curvature surface. For the incomparably stable soliton waves which are deployed within such negatively curved arenas also are just about the only concrete wave-forms which meet the "infinite symmetries" (usually interpreted as "infinite number of conservation laws") requirement. ...".
I agree with Robert de Marrais's view of periodicity of the 2^N-ions, and it seems likely to me that the 2^8-ions might be regarded as a basic building block of number theory and group theory, just as I see the 2^8-dim Clifford algebra Cl(8) as a basic building block of physics in the D4-D5-E6-E7-E8 VoDou Physics model.
I conjecture further conjecture that these links might be usable to establish a relationship between the Riemann zeta function and quantum theory.
It is important to note that the ZD-pairings are not the only Zero Divisors, as pointed out explicitly by Robert de Marrais in math.RA/0207003. ]
... If we only consider the first harmonic &endash; all we get with just the Pathions &endash; we have a system of twice the number of ZD-diagonals in a base-line box-kite, all interrelated in the manner just described. This fits quite nicely with Moreno's assessment of the total count of "irreducible" zero-divisors we should have: per his formula, 2^5-ions should have (5-3) x 84 = 168 such entities.
But clearly, the formula breaks down with one more doubling: when the kite-chain midden is extended to 2^6-ions, we get not 3 x 84, but 4 x 84 = 336 ZD-diagonals: the maximal symmetry of Felix Klein's famous figure that Poincaré determined was a tessellation by triangles of the hyperbolic plane. ...
... just as each of the 7 x 3 "8-less" Sedenion triplets of form (O, S, S') produced two Assessor index-pairs &endash; (O, S) and (O, S') &endash; making 42 in all,
so the Pathions have 15 x 7 = 105 "16-less" triplets of form (O or S, P, P'), and which yield 210 "Assessors" organized in 15 ensembles which are the Pathion equivalent of Box-Kites.
In addition, as implied in the above discussion of kite-chain middens, 42 of the (P, P') pairings are ZD's as well, which makes 252 x 2 diagonals in each = 504 "irreducible" ZD's specific to the Pathions. Add in the 84 first showing in the Sedenions, and we have 588 in all: 31/2 times the count Moreno's formula would allow ...".
| CxS = 32 | QxS = 64 | ImQxO = 24 | OxS = 128 | SxS = 256 | del Pezzo Surfaces |
| Spinors | Zero Divisors | Split and Complexified Algebras | Matrices-Bimatrices |
| Complexified Octonions | Sedenions | dim = 24 | dim = 32 | dims 64, 128, 256 | dim = -1 |
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