OCTONIONS, other Division Algebras, and ZeroDivisor Algebras,

can be derived from CLIFFORD ALGEBRAS

and are related to Hopf Algebras and Quantum Groups.

- Octonion Products and Lattices
- Split Octonions
- Octonion Mirrorhouse
- Octonion Fractals
- Octonion x-product and xy-product
- X, XY, and WXY Octonion Cross-Products

and is related to **Conformal Group
Structures**.

John Baez has a very nice paper, math.RA/0105155, about Octonions (also mentioned in his week 168), where he, describing algebras over the real number field, says:

"... There are exactly four normed division algebras ....

- The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on.
- The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete.
- The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings.
- But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. ...
... While somewhat neglected ... octonions ... stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry. ...:".

The John Baez paper includes some interesting history.
He also has easily
updatable and expandable html, ps, and pdf versions on his personal
web site. , and he also writes an interesting series of web page
called This Week's
Finds in Mathematical Physics. In his week
190, he says (among other things): "... Lawvere guessed there was
indeed a nice isomorphism T(1)^7 = T(1) In other words: one can ...
construct **a one-to-one correspondence between trees and 7-tuples
of trees!** For a good treatment see ... Andreas Blass, Seven trees
in one, Jour. Pure Appl. Alg. 103 (1995), 1-21. Also available at
http://www.math.lsa.umich.edu/~ablass/cat.html
..". On that web page, about that paper, Blass says: "... Following a
remark of Lawvere, we explicitly exhibit a particularly elementary
bijection between the set T of finite binary trees and the set T^7 of
seven-tuples of such trees. "Particularly elementary" means that
**the application of the bijection to a seven-tuple of trees
involves case distinctions only down to a fixed depth (namely four)
in the given seven-tuple**. We clarify how this and similar
bijections are related to the free commutative semiring on one
generator X subject to **X=1+X^2**. Finally, our main theorem is
that the existence of particularly elementary bijections can be
deduced from the provable existence, in intuitionistic type theory,
of any bijections at all. ...". Compare:

- the 4-depth of the trees with the 4-depth = number of basis elements in the 224 terms like (±i ±j ±ie ±je)/2 of the E8 lattice and
- the X=1+X^2 of the set of all trees with X = -1 + X^2 of the Golden Mean.

There are 480 different ways to write an octonion multiplication table. Here is a geometric representation of the way preferred by Geoffrey Dixon:

In the heptagon of imaginary octonions {e1,e2,e3,e4,e5,e6,e7}, there are 7 triangles (6 colors and 1 black). The product of any two imaginary octonions is the third imaginary octonion in their triangle, with + sign if the product is a clockwise rotation and - sign if counterclockwise. The algebraic rule for this product is determined by e(a)e(a+1) = e(a+5). If (a+5) is greater than 7, use (a-7).

O1 O4 O5 O7 O2 O6 O8 O9 O3 In some rough sense, they correspond to the vertices of the E6 root vector polytope.

Re(O1) O4 O5 O4* Re(O2) O6 O5* O6* Re(O3) They form the exceptional Jordan algebra J3(O), which represents a Nearest Neighbor approximation to 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.

Im(O1) O4 O5 -O4* Im(O2) O6 -O5* -O6* Im(O3) To make a Lie algebra out of them, you must restrict to traceless matrices (45-7 = 38-dim) and add the 14-dimensional octonion derivation algebra G2: Im(O1) O4 O5 -O4* Im(O2) O6 -O5* -O6* - x G2

Im(O1) O4 O5 -O4* Im(O2) O6 -O5* -O6* - x G2 x Re(O1) O4 O5 O4* Re(O2) O6 O5* O6* -

My talk at Corvallis 97 gives some further details.

Here is a rough outline of the structure (ignoring some matters of signature) of the D4-D5-E6-E7-E8 VoDou physics model:

- The complex symmetric space E6
/ Spin(10)xU(1) corresponds to a complex domain whose
Shilov boundary is an 8+8 =
16-dim representation space for
**8 (first-generation) fermion particles and their 8 antiparticles**. Since 78-dim E6 can be seen as being made up of (ignoring signature) 45-dim adjoint representation of Spin(10) plus 32-dim full spinor representation of Spin(10) plus 1-dim U(1) for complex structure, it is clear that the fermion representation space E6 / Spin(10)xU(1) corresponds to the 32-dim (16-complex-dim) spinors of Spin(10), and it is natural that fermions act like spinors. After dimensional reduction, there are 3 generations of fermions, including the Truth Quark. - as to the Spin(10) inside E6, up to signature it is
**Spin(1,9) = SL(2,O)**. It contains a complex version of 8-dim spacetime = symmetric space Spin(10) / Spin(8)xU(1) with the corresponding Shilov boundary = S1xS7 acting as**8-dim physical spacetime**;. After dimensional reduction, there is a 4-dimensional physical spacetime and 4-dimensional internal symmetry space. - as to the Spin(8) inside the Spin(10) ( or Spin(1,7) inside Spin(1,9) ) its 28 generators act as 8-dim gauge bosons. After dimensional reduction, 16 generators form U(2,2) = UY(1)xSpin(2,4) for Conformal Gravity and the other 12 form the U(1)xSU(2)xSU(3) of the Standard Model.

In terms of 3x3 Octonion matrices,

The fact that SL(3,O) = E6 is mentioned in the paper Division algebras, (pseudo)orthogonal groups and spinors by A. Sudbery, J. Phys. A: Math. Gen. 17 (1984) 939-955 at page 950 and is used in describing interesting math structures in such papers as The Chow Ring of the Cayley Plane, by A. Iliev and L. Manivel, math.AG/0306329 where they say (on page 2)

"... **the subgroup SL3(O) of GL(J3(O)) consisting in
automorphisms preserving the determinant is the adjoint group of type
E6. The Jordan algebra J3(O) and its dual are the minimal
representations of this group**. ...".

There is also a paper by C.H. Barton and A. Sudbery at math.RA/0001083 in which they say:

"... Tits ... showed ... the so-called magic square of Lie algebras of 3 x 3 matrices whose complexifications are

R C H O R A1 A2 C3 F4 C A2 A2xA2 A5 E6 H C3 A5 B6 E7 O F4 E6 E7 E8

... the [first three] rows can be interpreted as analogues of the matrix Lie algebras su(3), sl(3) and sp(6) defined for each division algebra. ...

... most exceptional Lie algebras are related to the exceptional Jordan algebra of 3 x 3 hermitian matrices with entries from the octonions, O. ... this relation yields descriptions of certain real forms of the complex Lie algebras

- F4 ... which can be interpreted as octonionic versions of ... the Lie algebra of antihermitian 3 x 3 matrices ...
- E6 ... which can be interpreted as octonionic versions of ... the Lie algebra of .... special linear 3 x 3 matrices and ...
- E7 which can be interpreted as octonionic versions of ... the Lie algebra of ... symplectic 6 x 6 matrices. ...".

A revised version of their paper is at math.RA/0203010 With respect to E6 and E7, A. Sudbery says on page 950 of his J. Phys. A: Math. Gen. artice cited above:

"... sl(3,K) ... when K = O ... is a non-compact form of the exceptional algebra E6, the maximal compact subalgebra being F4.

... sp(6,K) ... when K = O ... is a non-compact form of E7, the maximal compact subalgebra being E6 (+) SO(2). ...".

My physical interpretation of the symmetric spaces based on E6 and E7 and the above maximal compact subalgebras is summarized in my paper at physics/0207095 (the last paper I was able to put on arXiv before I was blacklisted by Cornell) with more relevant details given here.

- I interpret the 78 - 52 = 26-dimensional symmetric space E6 / F4 as the 26-dimensional space of bosonic string theory, with strings being interpreted as world-lines of Everett Many-Worlds and
- the 133 - 78 - 1 = 54-dimensional symmetric space E7 / E6 (+) SO(2) is seen as a 27-complex-dimensional space of a related M-theory.

The fermion creation and annihilation operators in the 3x3 octonion nilpotent matrices are related to Spin Networks.

Octonion x-product, xy-product, and

Start with the the division algebras C (complex), Q (quaternion), and O (octonion). Consider the iterated map from z to (z X) (X^(-1) z) - 1 For associative C and Q, since X X^(-1) = 1, the iterated map becomes z to z z - 1 This is the iterated map that produces a conventional z-space Julia set in C and in Q . Since, for the map z to zz - 1 , the multiplicative factor is fixed at 1 and the additive factor is also fixed at 1 , we do not at this stage have a variable Mandelbrot parameter, either multiplicative or additive, and we have only one Julia set. Now, let z and X be in O. Since octonions are non-associative, the product (z X) (X^(-1) z) is not z z , but is the nontrivial octonion X-product of Martin Cederwall. The iterated octonion map from z to (z X) (X^(-1) z) - 1 produces octonion Mandelbrot sets and Julia sets that have a non-trivial multiplicative Mandelbrot parameter, the space of the octonion variable X. Next, consider the additive factor 1 from the point of view of the non-associative octonions. Due to their non-anssociativity, the octonions have 480 different rules of multiplication. Unlike the associative algebras C and Q, in which the multiplicative identity 1 is always the same, the octonion multiplicative identity 1 can be shifted. If we can shift 1 then we can change the additive factor 1 and get a non-trivial additive Mandelbrot parameter, which should be the space of another octonion variable Y. Explicitly, we can use the XY-product of Geoffrey Dixon to do this for octonion X and Y. The XY-product shifts the multiplicative identity from 1 to Y X^(-1) and we get for our iterated map: z to (z X) (Y^(-1) z) - Y X^(-1) Now the (X,Y)-space of 2 octonion variables is the Mandelbrot parameter space for both a non-trivial multiplicative factor and a non-trivial additive factor. THE MULTIPLICATIVE AND ADDITIVE MANDELBROT SETS ARISE NATURALLY FROM NON-ASSOCIATIVE OCTONION X-PRODUCT AND XY-PRODUCT, respectively. The WXY-product might be used to get the map z to W (( z X ) ( Y^(-1) z )) - W ( Y X^(-1) ) Onar Aam uses these properties of the non-associative octonions to produce octonion fractal images. Since the X-product and XY-product were constructed to study octonions of unit norm, that is, the unit 7-spheres in octonionic X-space and in octonionic Y-space, it is natural that some of the most interesting fractal images occur for values of X and Y on or near the two unit 7-spheres.

Girish Joshi and his co-workers C. J. Griffin and Andrew Kricker at the University of Melbourne School of Physics have studied octonionic Julia sets in the articles: Octonionic Julia sets - - Chaos, Solitions & Fractals 2 (1992) 11-24 Transition Points in Octonionic Julia Sets - - Chaos, Solitions & Fractals 3 (1993) 67-88 Associators in Generalized Octonionic Maps - Bifurcation Phenomena of the Non-associative Octonionic Quadratic - - Chaos, Solitions & Fractals 5 (1995) 761-782 They find, with respect to Julia sets, that: "In summary, non-associativity has been shown to directly dictate the gross structure of the Julia sets of a modified quadratic map z->z^2+c+c(Za)-(cZ)a, demonstrating the tendency to squash the set onto the quaternionic subspace framed by the imaginary parts of the parameters a,c and ac, and the real line. [in disconnected Julia sets] the transition from one region to another is sudden and visually resembles a condensation process. The critical value about which the transition occurs is dependent on c, but is otherwise uniform over the whole set." They find, with respect to attractors in phase space of Mandelbrot maps, that the attractors consist of KNOTS, TORI and LOOPS, and that with the non-associative factor one can forge loop doublings, knot triplings,

and complexifications of the attractors, and drive the attractor into the chaotic regime. The resulting attractor is hyper-chaotic,

in that it displays local chaos while the overall structure is ordered, and the structure looks like A MOBIUS STRIP with a chaotic surface. They say: "We found that this system is wealthy with nonlinear phenomena. All classic nonlinear mechanics are represented and more besides. Yet the importance of this system is that the mechanics appear recursively, and build constructively. ..." HERE IS WHAT I THINK ABOUT HOW IT WORKS: The phase space is the boundary of the Julia set, or a lower-dimensional transformation of it. The boundary Julia set is a stable attractor. For example, in the complex case it is a deformation of the 1-sphere unit circle S1. For quaternions, the boundary Julia set should be a deformation of the 3-sphere S3. For octonions, as they used, the boundary Julia set should be a deformation of the 7-sphere S7. The 7-sphere S7 fibres into an S4 and an S3. Look at the S3. As a 3-dim space, it naturally contains S1 knots. That is where their knots come from, I think - dynamical flows on S3. Relationships between S3 flows and knots are discussed in the AMS ERA paper 1995-01-02, Flows on S3 supporting all links as orbits, by Robert W. Ghrist. Ghrist shows examples of flows on S3 containing closed orbits of all knot and link types simultaneously. Particularly, the set of closed orbits of any flow transverse to a fibration of the complement of the figure-eight knot in S3 over S1 contains representatives of every (tame) knot and link isotopy class.

The abstract of math.QA/9802116,

by Helena Albuquerque and Shahn Majid, states:

"We show that the octonions are a twisting of the group algebra of Z2xZ2xZ2 in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle. We consider general quasi-associative algebras of this type and some general constructions for them, including quasi-linear algebra and representation theory, and an automorphism quasi-Hopf algebra. Other examples include the higher 2^n -onion Cayley algebras and examples associated to Hadamard matrices."

According to their paper, their general construction of the octonions mirrors, for discrete groups, Drinfeld's construction of the quantum groups Uq(g).

Their construction of Cayley-Dickson algebras begins with the group algebra kG of a group.

This has coproduct etc. forming a Hopf algebra.

They define k_F G as kG with a modified product x *_F y = xy F(x;,y) for all x,y in G and

show that k_F G is a coboundary G-graded quasialgebra, where the degree of x in G is x, and F is any 2-cochain on G.

They show that the `complex number' algebra, the quaternion algebra, the octonion algebra and the higher Cayley algebras are all G-graded quasialgebras the form k_F G for suitable G and F, which they construct.

To describe their cochains for the `complex number' algebra, the quaternion algebra, the octonion algebra and the higher Cayley algebras, they consider the special case where

G = (Z2)^n and F is of the form F(x,y) = ( -1)^f(x,y) for some Z2-valued function f on GxG.

In all these cases (and for the who 2^n -onion family generated in this way) f has a bilinear part defined by the bilinear form

1 1 . . . 1 0 1 . . . 1 . . . . . . 0 . . . 1 1 0 . . . 0 1

They show that for the complex number and quaternion algebras this is the only part. The f for the octonions has this bilinear part, which does not change associativity, plus a cubic term. The 16-onion has additional cubic and quartic terms.

When I asked Shahn Majid by e-mail whether a similar construction (restricting their function f to the bilinear form and omitting cubic, quartic and higher parts) would give you the Clifford algebras Cl(n), he replied:

"... yes, sure, this is clear since such $F(x,x)=-1$ and $F(x,y)=-F(y,x) for $x\ne y$ so

x * y + y * x = xy (F(x,y)+F(y,x))=-2 when x=y and 0 otherwise.

Here * is the product in k_FG and G=(Z_2)^n, so that xy (product in G) means the addition law of `Z_2 vectors', which means in particular that xx=e (the identity of G) which is also the 1 of k_FG

We were interested in nonassociativity but sure you can construct associative clifford algebras this way, or at least realisations of them. In the present case we have a (Z_2)^n -dimensional realisation of the clifford algebra in n dimensions and the Euclidean metric. You can make a similar construction in general starting from a symmetric bilinear form. ..."

(Based on Reese Harvey's book, Spinors and Calibrations, Academic Press, 1990, chapter 6)

1 - For unit length octonion X, X^(-1) = X* (* = octonion conjugate) 2 - For orthogonal octonions X and Y, X Y* = - Y X* 3 - For octonions X and Y, their cross-product is X x Y = (1/2) ( Y* X - X* Y ) = Im( Y* X ) 4 - For octonions X and Y, Re( X x Y ) = 0 5 - For unit orthogonal octonions X and Y, X x Y = Y^(-1) X = - X Y^(-1) WHICH GIVES (the negative of) the XY-PRODUCT a b = (a X) (Y^(-1) b) (the X-product is the XY-product for X = Y) 6 - For imaginary octonions X and Y, X x Y = X Y + { Y , X } (where { , } is the inner product) 7 - For octonions W, X, and Y, their triple-cross-product is W x X x Y = (1/2) ( W ( X* Y ) - Y ( X* W ) ) 8 - For unit orthogonal octonions W, X, and Y, W x X x Y = W ( X^(-1) Y ) = - W ( Y X^(-1) ) WHICH GIVES (the negative of) the WXY-PRODUCT a b = W ((a X) (Y^(-1) b)) 9 - For imaginary octonions W, X, and Y, Re( W x X x Y ) = PHI( W /\ X /\ Y ) (where PHI is the associative 3-form for the octonions) and Im( W x X x Y ) = (1/2) [ W, X, Y ] (where [ W, X, Y ] = (W X) Y - W (X Y) is the associator) 10 - G2 = Aut(O) preserves PHI and the coassociative 4-form PSI. G2 also preserves the cross-product on Im(O). Therefore: for the E8 lattice, use the X-product a b = (a X) ( X^(-1) b) for the /\16 lattice, use the XY-product a b = (a X) ( Y^(-1) b) for the /\24 lattice, use the WXY-product a b = W ((a X) ( Y^(-1) b))

**Conformal Groups are related to ****Moebius
Transformations****. **

**The ****D4-D5-E6-E7-E8 VoDou
Physics model**** coset spaces ****E7****
/ (****E6**** x U(1)) and
****E6**** / (****D5****
x U(1)) and ****D5**** /
(****D4**** x U(1)) are
Conformal Spaces. ** You can continue the chain to D4
/ (D3 x U(1)) where D3 is the
15-dimensional Conformal Group whose compact version is Spin(6), and
to D3 / (D2
x U(1)) where D2 is the 6-dimensional Lorentz Group whose compact
version is Spin(4). Electromagnetism, Gravity,
and the ZPF all have in common
the symmetry of the 15-dimensional D3 Conformal Group whose compact
version is Spin(6), as can be seen by the following structures with
D3 Conformal Group symmetry:

- Maxwell's equations of Electromagnetism
- the canonical Dirac Lagrangian for massive fermions
- Gravity derived from the Conformal Group using the MacDowell-Mansouri mechanism
- the Quantum Theoretical Hydrogen atom
- the Lie Sphere geometry of SpaceTime Correlations in the Many-Worlds picture

Further, the 12-dimensional Standard Model Lie Algebra U(1)xSU(2)xSU(3) may be related to the D3 Conformal Group Lie Algebra in the same way that the 12-dimensional Schrodinger Lie Algebra is related to the D3 Conformal Group Lie Algebra.

The physical 4-dimensional SpaceTime of the D4-D5-E6-E7-E8 VoDou Physics model is a 4-dimensional HyperDiamond lattice SpaceTime that is continuously approximated globally by RP1 x S3 and locally by Minkowski SpaceTime, with Gravity coming from the 15-dimensional Conformal Group Spin(2,4) by the MacDowell-Mansouri mechanism. The curved SpaceTime of General Relativity is not considered fundamental, but is produced by by starting with a linear spin-2 field theory in flat spacetime, and then adding higher-order terms to get Einstein-Hilbert gravity. The observed curved SpaceTime is therefore based on an unobservable flat Minkowski SpaceTime. (See Feynman, Lectures on Gravitation, Caltech 1971 and Addison-Wesley 1995, and see Deser, Gen. Rel. Grav. 1 (1970) 9-18 as described in Misner, Thorne, and Wheeler, Gravitation, Freeman 1973, pp. 424-425.)

If you were to start, not with locally Minkowski SpaceTime, but with the curved SpaceTime of General Relativity, then you would see that the Conformal transformations of Minkowski SpaceTime by the 15-dimensional Conformal Group Spin(2,4) corresponds to the Conformal transfomations of the curved SpaceTime by the infinite-dimensional Conformal subgroup of the group Diff(M4) of General Relativistic coordinate transformations of the 4-dimensional SpaceTime M4 of General Relativity, which Conformal subgroup is defined as those General Relativistic coordinate transformations that preserve conformal structure and which infinite-dimensional Conformal subgroup can be called the Weyl Conformal Group. (See Ward and Wells, Twistor Geometry and Field Theory, Cambridge 1991, p. 261.)

Robert Neil Boyd has told me about structures that Alexander Shpilman's calls Overtime denoted by a space with signature ((3,1),1). Such structures may be related to the two timelike dimensions of the Conformal Group Spin(4,2).

D2=Spin(4)=Spin(3)xSpin(3)=SU(2)xSU(2)=Sp(1)xSp(1)=S3xS3 is compact version of Lorentz rotations and boosts. D3=Spin(2,4)=SU(2,2) is conformal group over D2 D3/(D2xU(1)) is 4 Translations and 4 Conformal Transformations. D4=Spin(2,6) is conformal group over D3 D4/(D3xU(1)) is 12 gauge bosons of SU(3)xSU(2)xU(1). D3=Spin(2,4) contains Spin(2,3) anti-de Sitter group, which produces gravity by MacDowell-Mansouri mechanism, and contains 4 Conformal Transformations and 1 Scale transformation for Higgs symmetry breaking and mass generation. U(1) is complex phase of propagators. D5=Spin(1,9) = SL(2,O) is conformal group over D4 D5/(D4xU(1)) is complex 8-dim spacetime. E6 is conformal group over D5 E6/(D5xU(1)) is complex 16-dim 1st generation fermions. There is only 1 copy of the traceless Jordan algebra J3(O)o in E6. The 26-dim J3(O)o in E6 corresponds to the single octonion that represents 1st-generation fermions. E7 is conformal group over E6 E7/(E6xU(1)) represents the MacroSpace of ManyWorlds. It can be seen as the complexification of 27-dim J3(O), or as 2 copies of J3(O)o plus SU(2)/U(1). There are 3 copies of J3(O)o in E7. The two algebraically independent copies of J3(O)o in E7 correspond to the pairs of octonions that represent 2nd-generation fermions. E8 is not a traditional conformal group over E7, but E8/(E7xSU(2)) is 4 copies of J3(O)o plus SU(3)=G2/S6. There are 7 copies of J3(O)o in E8. The three algebraically independent copies of J3(O)o in E8 correspond to the triples of octonions that represent 3rd-generation fermions.

At each level of Conformal Structure, Physical Wavelets provide a connection between the World of Physics and the World of Information.

The Geometry of those connections is that of Bounded Complex Domains. A good introductory paper is Conformal Theories, Curved Phase Spaces Relativistic Wavelets and the Geometry of Complex Domains, by R. Coquereaux and A. Jadczyk, Reviews in Mathematical Physics, Volume 2, No 1 (1990) 1-44, which can be downloaded from the web as a 1.98 MB pdf file.

D2=Spin(4)=Spin(3)xSpin(3)=SU(2)xSU(2)=Sp(1)xSp(1)=S3xS3 is compact version of Lorentz rotations and boosts. Noncompact version Spin(2,2) of D2 is conformal over D1=Spin(2,0)=U(1)=S1. D2/(D1xU(1)) is 2 copies of S3/S1 = S2, or S2xS2. D1=Spin(2,0)=U(1)=S1 is conformal over D0=Spin(0,0)=IDENTITY. D1/(D0xU(1)) = IDENTITY = D0. D0=Spin(0,0)=IDENTITY is the BEGINNING of the chain. Between D0=IDENTITY and D1=S1=CIRCLE is B0=Spin(1,0)=Z2={+1,-1}=YIN-YANG:

The flag of South Korea has red-blue yin and yang in a circle, along with 4 of the 8 trigrams, on a white background. White Background - void. One I Ching bar - yin or yang - C. Two I Ching bars - 4 forces - Q - 4-dim physical spacetime. Three I Ching bars - 8 trigrams - O - - each of the three 8-dim reps of D4. 1x2x3=6 I Ching bars - 64-dim Clifford algebra of the Conformal Group, or each one of the two irreducible parts of the D4 Clifford even subalgebra.

The E series of Lie algebras ends with E8, so extensions would not have Lie algebra structure, and it would probably be hard to build an action. If the chain is modified to the infinite chain D0-D1-...-Dn-... , then the adjoint representation does not contain any fermion half-spinor representations. You only have more vector spacetime type representations, so you would have physics without fermions.

Ye-Lin Ou and John C. Wood have written a series of papers dg-ga/9511001 dg-ga/9511002 dg-ga/9511003 dealing with harmonic morphisms of Euclidean n-dim space that preserve the structure of harmonic functions and Brownian paths. That is, how you can map n-dim space into itself so that harmonic functions get pulled back into harmonic functions and Brownian paths get mapped into Brownian paths. Sigmundur Gudmundsson and Stefano Montaldo have a WWW site, THE ATLAS OF HARMONIC MORPHISMS, and Sigmundur Gudmundsson is also editor of THE HARMONIC MORPHISMS BIBLIOGRAPHY. Ou and Wood note that Baird proved in 1983 in his book Harmonic maps with symmetry, harmonic morphisms, and deformation of metrics (Pitman Res. Notes Math. Ser.) that the only possible dimensions for orthogonal multiplication to produce harmonic morphisms are 1, 2, 4, and 8: Real numbers, Complex numbers, Quaternions, and Octonions. Such harmonic morphisms are maps that are both harmonic and horizontally weakly conformal. Since harmonic functions are the key to Greens functions, and Greens functions give particle propagators, these results show how division algebras are important in building particle physics models. Since harmonic functions are related to bounded complex homogeneous domains used in the D4-D5-E6 model, and also are related to Brownian paths used in the physics models of Michael Gibbs, the work of Ou and Wood may be useful in showing that those physics models are indeed equivalent.

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