Only the D4 Lie algebra, with Lie group of the type Spin(8), has full TRIALITY, an outer automorphism by which the three 8-dimensional representations of D4 shown on the D4 Coxeter-Dynkin diagram
8 | 28 / \ 8 8
(the vector, the +half-spinor, and the -half-spinor) are isomorphic.
From the point of view of the Cl(8) Clifford algebra with graded structure
whose bivectors represent the 28-dimensional D4 Lie algebra, and whose vectors represent the 8-dimensional D4 vector representation, and whose 8-dimensional + and - half-spinors represent the + and - half-spinor D4 representations, Triality is a symmetry property among components of Cl(8).
J3(O) is useful in the D4-D5-E6-E7-E8 VoDou Physics model and related to the structure of the Freudenthal-Tits Magic Square.
John Baez has a beautiful description of triality at Week 91 in his series This Weeks Finds in Math Physics on the WWW.
The D3 Lie algebra, with Lie group of the type of the Conformal Group Spin(2,4) = SU(2,2), inherits from D4 a non-linear type of triality, described e-mail discussions during August 2001, summarized as follows:
".. John Baez discusses triality ... [saying] ...
"Triality is a cool symmetry of the infinitesimal rotations
in 8-dimensional space.
It was only last night, however, that I figured out what triality
has to do with 3 dimensions. ... Look at the group of all
permutations of {i,j,k}. This is a little 6-element group which
people usually call S_3, the "symmetric group on 3 letters". ..."
... Back to me [John Gonsowski] instead of John Baez:
Is there any way to relate this linear transformation to D3? ...".
The Weyl group of D3 is S3 x (Z2)^2 (of order 6x4 = 24).
The S3 part of the Weyl group of D3 comes from the symmetries
of the 3-Euclidean-dimensional cuboctahedron, which is the
root vector polytope for D3.
The geometric picture that I have in my head is that
the S3 comes from
using reflections through hyperplanes through the origin
to interchange 3 elements of the triangular faces of the cuboctahedron.
Although D3 does not have as obvious a triality of its representations
as does D4 (for which the vector represenation is 8-dimensional,
as are the two half-spinor representations),
here is how I see a similar (but less obvious) version of triality in D3:
D3 has two 4-dimensional half-spinor representations
and
a vector representation that is 6-dimensional.
However,
D3 is the Lie algebra of the conformal group of Minkowski spacetime,
which is 4-dimensional.
Therefore,
although the conventional linear vector representation of D3
is 6-dimensional,
you could (and I do) say that
D3 has a NONlinear 4-dimensional representation due to conformal
transformations,
and
I see a triality among:
two D3 4-dim half-spinor representations
and
one D3 NONlinear conformal (sort of vector-like) 4-dim representation.
(This is related to how I use D3 in my physics model,
after dimensional reduction of spacetime from 8 to 4 dimensions,
but that is a digression that I won't pursue further here.)
For comparison, the D4 Dynkin diagram is,
where v denotes the conventional linear vector representation,
a denotes the adjoint representation,
ad s+ and s- denote the two half-spinor representations:
s+
|
v - a - s-
The dimensionalities of these representations are:
8
|
8 - 28 - 8
The D4 triality is the S3 group of permutations of the outer three
8-dimensional representations of the Dynkin diagram.
-----------------------------------------------
The D3 Dynkin diagram is,
where v denotes the conventional linear vector representation,
a denotes the adjoint representation,
ad s+ and s- denote the two half-spinor representations:
s+
|
v - s-
Note that the adjoint representation "goes away" and is not an
elementary fundamental representation of D3.
The conventional linear dimensionalities of these representations are:
4
|
6 - 4
Note that you can make the 15-dim adjoint representation of D3
by taking the tensor product of 4x4 to make a 16-dim group
that includes the 15-dim adjoint as a subgroup.
By using the conformal non-linear representation c in place of
the conventional linear vector representation,
the D3 diagram becomes
s+
|
c - s-
Using the non-conventional conformal c instead of the vector v,
the dimensionalities of these representations are:
4
|
4 - 4
The D3 triality is the S3 group of permutations of the three
4-dimensional representations of the Dynkin diagram.
Note that the D3 Dynkin diagram can be written equivalently as
4 - 6 - 4 (conventional linear vector)
or
4 - 4 - 4 (conformal)
which shows the isomorphism between D3 and A3.
The A3 viewpoint makes it easier to see that the 4x4 = 16 gives
you 16-dimensional U(4), which is reducible to SU(4)xU(1),
and the 15-dimensional SU(4) is the adjoint of A3 and is
also (by isomorphism) the Spin(6) adjoint of D3.
You could also use signatures such as U(2,2) and Spin(4,2).
The D2 Lie algebra of (1,3) Minkowski physical spacetime and Spin(1,3) = SU(2) x SU(2), inherits from D3 a quaternionic type of triality.
For Cl(1,3) the 2x2 quaternionic matrices have Full Spinors that are 1x2 quaternion column vectors. Each Half-Spinor space is one quaternion variable, which has a 1-2 correspondence with first generation fermions, and also corresponds 1-1 with the (1,3) vector space of physical Minkowski spacetime, resulting in a quaternionic version of triality (diluted by the 1-2 nature of the fermion correspondence) that is related to the reducibility of the D2 Lie algebra Spin(1,3).
hep-th/9306011 takes as fundamental objects Sets, Quivers, and Complex Vector Spaces, and then derives the D4-D5-E6-E7-E8 VoDou Physics model of TRIALITY Spin(0,8) by requiring generalized supersymmetry of the Lie groups of the A-D-E structure of the Quivers. These structures may be related to a real hyperfinite II1 von Neumann algebra factor.
Take as fundamental the Identity finite group of one particle, and use the McKay Correspondence between the A-D-E Lie algebras and finite subgroups of SU(2) = Spin(3) = S3 to get the groups of physics when the finite group is expanded by first allowing the particle to have a (discrete) phase and then by expanding finite groups in a natural way indicated by physical interpretation.
The McKay corresponding Lie group is abelian U(1), with Lie algebra A0.
Expand by Z2, not to get a finite group on 2 elements, but to give the 1 element two phase states +1 and -1.
(The McKay correspondence uses finite subgroups of simply connected SU(2)=Sp(1)=Spin(3)=S3, rather than the SO(3) that is double covered by it.)
The McKay corresponding Lie group is SU(2), with Lie algebra A1.
The McKay corresponding Lie group is SU(4), with Lie algebra A3 = D3.
(The { } notation is due to typographic structure of HTML files.)
{2,2,2} is the group of the 8 quaternions
{+/-1, +/-i, +/-j, +/-k}, which are the 8 vertices of a hyperoctahedron 16-cell.
The McKay corresponding Lie group is Spin(0,8) with Lie algebra D4.
if {G5,G6,G7,G8} are the 4 pseudovectors of that SU(4), then
the 8 Clifford algebra gammas of the Spin(0,8) Clifford algebra are generated by
{(G1+G5), (G2+G6), (G3+G7), (G4+G8), (G1-G5)/i, (G2-G6)/i, (G3-G7)/i, (G4-G8)/i}.
By Spin(0,8) TRIALITY, the construction can be extended from the 8 gammas (which correspond to the vector representation) to either
Denote the two SU(4)'s from the last Z2 expansion by
LgrSU(4) for the Gravity SU(4) and
RsmSU(4) the Standard Model SU(4).
From the left ideal and right ideal half-spinor representations of Spin(0,8), all representations of Spin(0,8) can be constructed by using TRIALITY, the exterior product,and tensor products and sums.
How Does the McKay Correspondence Work? Let n+1 be the dimension of the center of the group algebra of the finite group. There are n conjugacy classes, other than the identity, of the finite group. The McKay correspondence is that their columns in the character table are the eigenvectors of the extended Cartan matrix of the corresponding rank n Lie algebra. The n column eigenvectors define an n-dimensional vector space that is the root vector space of the Lie algebra. In the case of finite group cyclic Z(n+1) - A(n) - SU(n+1) Lie algebra, the center of the group algebra is the entire algebra, and the n vectors, plus the origin, define an n-simplex the symmetries of which form the symmetric group S(n+1) that is the Weyl group of the A(n) Lie Algebra SU((n+1). For the dicyclic groups - D Lie algebras, and the binary tetrahedral, octahedral, and icosahedral groups - E6, E7, and E8 - the nontrivial relations of the finite group algebra define root vector spaces, and therefore Weyl groups, that are more complicated than a simplex, or a symmetric group. In alg-geom/9411010, Ito and Reid extend the McKay correspondence beyond finite subgroups of SU(2) or SL(2,C) to SL(3,C). Their examples 1,2,3 contain the SL(2,C) McKay singularities D4, D5, and E6 corresponding to the D4-D5-E6 physics model.
DIXON considers the division algebras R (Real Numbers), C (Complex Numbers), Q (Quaternions), and O (Octonions) to be fundamental.
HERE is my rough attempt to combine Dixon's approach with Kenichi Horie's Geometric Interpretation of Electromagnetism in a Gravitational Theory with Torsion and Spinorial Matter.
What follows is my understanding of his work. Since my understanding may be incomplete and/or wrong, I encourage interested people to read Dixon's book and papers.
Dixon starts with the 64-dimensional real tensor product T = R x C x Q x O.
He notes that the factor R is redundant for a real tensor product,
and that T = C x Q x O.
Dixon then considers a division algebra to be the spinor space acted upon by the Clifford algebra of adjoint actions of the division algebra on itself.
The spinor space of Cl(0,1) is C.
The algebra of right-adjoint actions of C on C is CR = C = CL.
The combined left-right adjoint actions of C on C is CA = C = CL = CR.
The adjoint actions are not enough to get all R(2) actions on the spinor space C, so:
Add the 4 actions:
Identity(x) = x; Conjugate(x) = x*; i(x) = ix; and i*(x) = ix*.
The last 3 actions are outside all the adjoint structures, and so must be added by forming the tensor product R(2) x C = C(2). Expansion by R(2) takes the real Euclidean 1-dimensional space of Cl(0,1) = C to
the complexification of Cl(1,1) Minkowski 2-dimensional spacetime.
The spinor space of Cl(0,2) is Q.
The algebra of right-adjoint actions of Q on Q is QR = Q, but QR =/= QL.
The combined left-right adjoint algebra QA = R(4) = Cl(3,1).
The action QR must be included to get all R(4) actions on the spinor space Q.
Since QR is inside the adjoint structure, it need not be added in by a tensor product as in the complex case of R(2) x C = C(2).
Since QR = Q is outside Cl(0,2), it can be regared as the SU(2) generated by an outer automorphism symmetry of spinor space between +half-spinor space and -half-spinor space.
The spinor space of Cl(0,6) is O.
Since OL = R(8), no R(2) or outer automorphism symmetry need be added to get R(8) actions on the spinor space O.
The spinor space of PL is P = C x Q = P+ + P-, where P+ and P- are each copies of the Pauli algebra and invariant under PL.
Note that the outer automorphism symmetry of QR acts on {P+,P-} as an SU(2) doublet.
Add in the R(2) factor from the case of the complex division algebra
to form R(2) x PL = R(2) x C(2) = C(4) = C x Cl(3,1) = Dirac algebra.
Expansion by R(2) takes the real Euclidean 3-dimensional space of Cl(3,0) to
the complexification of Cl(3,1) Minkowski 4-dimensional spacetime.
The spinor space of R(2) x PL is P2.
As P decomposes into P+ and P-, so does P2 decompose into P2+ and P2-, each of which is a 4-complex-dimensional Dirac spinor.
Since PL = C(2) and OL = R(8), TL = C(2) x R(8) = C(16).
Since PL = C(2) and OL = R(8), TL = C(2) x R(8) = C(16) = C x R(16) = C x Cl(0,8).
TL = C x Cl(0,8) is complexification of the Clifford algebra Cl(0,8) of the D4 Lie algebra Spin(0,8). Cl(0,8) = Cl(1,7), so the Euclidean and Minkowski 8-dimensional spacetimes are related by Wick rotation.
D4 is Lie algebra of the gauge group Spin(0,8).
Here, Dixon adds in the R(2) factor from the case of the complex division algebra to get:
R(2) x TL = R(2) x C x R(16) = C x R(32) = C x Cl(10,0) = C x Cl(1,9), which is the complexification of the Clifford algebras Cl(0,10) and Cl(1,9) of the D5 Lie algebras Spin(10) (compact) and Spin(1,9) (non-compact).
Dixon then has a 10-dimensional spacetime similar to string theory.
The complexified D4 Clifford algebra C x Cl(0,8) = C x Cl(1,7) already has complexified 8-dimensional spacetime, and C x Cl(0,8) = C x Cl(,7) can be considered to be the expansion by R(2) of the real Euclidean 7-dimensional space of Cl(7,0) = C(8).
The C x R16 decomposes into two 8-complex-dimensional + and - half-spinor spaces of complexified Cl(0,8) = Cl(1,7), which, along with a U(1) related to the complexification, can be added to the D5 to construct E6.
The outer automorphism QR spinor symmetry interchanges the half-spinor spaces. In the special case of D4-D5-E6, it extends by triality to interchange the half-spinor spaces and vector spacetime.
Cl(0,8), including triality and the "opposite algebra" relationship between the +half-spinor fermion particle and -half-spinor fermion antiparticle representations of Cl(0,8), and
Cl(0,6) used in the D4-D5-E6-E7 physics model
have been described by Schray and Manogue. Their algebraic structures are similar to the X-product of Cederwall and Preitschopf and a later paper of Dixon.
The for a given unit (norm = 1) octonion X, the X-product of two octonions A and B is given by (AX)(XtB), where t denotes transpose. The nonassociativity of octonion multiplication means that the X-product is non-trivial. It can be used to define the parallelizing torsion of the 7-sphere, which varies with position on the 7-sphere. It cannot be used to define the structure constants of a 7-sphere Lie algebra product [A,B] because such structure "constants" are not constant, but vary with position on the 7-sphere (unlike the cases of the 1-sphere and the 3-sphere).
IN TERMS OF THE E8 LATTICE DISCRETE VERSION OF THE D4-D5-E6-E7 MODEL:
The 240 elements of the orbit of the permutation group S7 of the 7 imaginaries of the octonion algebra correspond to the discrete octonionic algebra representation of the 240 vertices near the origin of the 8-dimensional E8 spacetime lattice.
The 240 vertices form a 4-complex-dimensional (8-real-dimensional) Witting polytope, with 240 complex 0-cells (vertices), 2160 complex 1-cells, 2160 complex 2-cells, and 240 complex 3-cells (faces of 6 real dimensions).
If w is the cube root of unity in the complex plane, then the 240 vertices are 24 of the form
(X, 0, 0, 0),
(0, X, 0, 0),
(0, 0, X, 0), and
(0, 0, 0, X),
where X = +/- i w^a sqrt(3) and a is in {0,1,2}, and
216 of the form
(0, +/- w^a, -/+ w^b, +/- w^c),
( -/+ w^b, 0, +/- w^a, +/- w^c),
(+/- w^a, -/+ w^b, 0, +/- w^c), and
(-/+ w^a, -/+ w^b, -/+ w^c, 0)
where a,b,c are in {0,1,2}.
In real 8-dimensional coordinates, the 240 vertices can be taken to be
16 of the form
+/- ea , where a is in {0,1,2,3,4,5,6,7}, and
224 of the form
(+/- ea, +/- eb, +/- ec, +/- ed) / 2,
where abcd is one of;
0123 4567 0145 2367 0246 1357 0347
1256 0167 2345 0257 1346 0356 1247
(see Coxeter, Regular Polytopes, Dover 1973 and Coxeter, Regular Complex Polytopes, 2nd ed, Cambridge 1991.)
The 480 elements of the orbit of the group that is the product of the permutation group S7 of the 7 imaginaries of the octonion algebra and the group of reflections of the 7 imaginaries that are consistent with octonionic multiplication correspond to
the discrete octonionic algebra representation of the 240 vertices near the origin of the 8-dimensional fermion particle +half-spinor E8 lattice (The 240 vertices form a 4-complex-dimensional (8-real-dimensional) Witting polytope.) and
the discrete octonionic algebra representation of a second set of 240 vertices near the origin of the 8-dimensional fermion antiparticle -half-spinor E8 lattice (The Witting polytope is self-dual, and the second set of 240 vertices form another 4-complex-dimensional (8-real-dimensional) Witting polytope that is dual to the first Witting polytope.).
In real 8-dimensional coordinates, the 240 dual vertices can be taken to be
112 of the form
(+/- ea +/- eb) , where a,b are in {0,1,2,3,4,5,6,7}, and
128 of the form
(+/- e0 +/- e1 +/- e2 +/- e3 +/- e4 +/- e5 +/- e6 +/- e7) / 2,
where the number of - signs is odd.
Schray and Manogue use the Z2P2 (lines in (Z2)^3 projective space of triples of Z2={0,1}) to define octonion multiplication. If the 7 imaginary octonions are denoted by {e1,...,e7}, 1=(100), 2=(110), 3=(010), 4=(111), 5=(011), 6=(001), and 7=(101) and the real octonion 1=(000) corresponds to the empty set, then Z2P2 can be represented by their figure 1:
giving octonion multiplication by ei ei = -1 and eA eB = eC = -eB eA for ABC collinear in Z2P2, and cyclic identities for ABC collinear in Z2P2.
From their point of view, the algebra and "opposite algebra" describe spinors of opposite chirality, which is consistent with their D4-D5-E6-E7 physics model interpretation as representations of fermion particles and fermion antiparticles.
Only for 5 or fewer control parameters are all catastrophe germs simple (Gilmore, p. 32).
As Gilmore (p. 15) says, "The germ resides between the early [Taylor series] terms which are killed off by the control parameters and the later terms which are killed off by a coordinate transformation."
The number L of variables with vanishing eigenvalue at a non-Morse critical point must be such that L(L+1)/2 is less than K, the number of control parameters, so that L is at most 2 for K at most 5.
For L = 2 and K = 3, the simple non-Morse catastrophe germ is of type D4.
For L = 2 and K = 4, the simple non-Morse catastrophe germ is of type D5.
For L = 2 and K = 5, the simple non-Morse catastrophe germ is of type D6 or E6.
Section 4 of chapter 7 of Gilmore describes the diagrammatic representations of catastrophe germs D4, D5, D6, and E6, showing how they are related to the Coxeter-Dynkin diagrams of the Lie algebras D4, D5, D6, and E6.
Gilmore (pp. 640-641) also analyzes the D4, D5, D6, and E6 catastrophes in terms of the Yang Hui (Pascal) triangle of the terms of Taylor series in 2 variables.
Since the number of variables L = 2, the germs D4, D5, and E6 can be described in terms of surfaces in R^3. In fact, D4, D5, and E6 correspond to umbilics of surfaces in R^3.
Chapter 12 of Porteous, with extensive discussion and nice illustrations, shows the correspondences:
D4 with elliptic (star or monstar) and hyperbolic (star or lemon) umbilics;
D5 with parabolic umbilics; and
E6 with perfect umbilics.
In the D4-D5-E6-E7-E8 VoDou Physics model, E6 (and its substructures D5 and D4) describe quite well Gravity and the Standard Model, including the masses of particles and strengths of forces, so I would describe E6 as modelling the "Material World".
In Quantum Consciouness, the Bohm-Sarfatti Quantum Potential describes the interactions among the possible Worlds of the Many-Worlds, and therefore the evolution of Quantum Consciousness States of the human brain tubulin electrons (which, since they are binary, lend themselves to description by Clifford algebras), and the Quantum Potential (viewed as timelike 1-dim string/membranes in 27-dim MacroSpace of Many-worlds) is mathematically described by the symmetric space E7 / E6xU(1) (which is 27-complex-dimensional).
So, from my view, E7 contains both "Quantum Consciousness" and the "Material World".
As to catastrophe theory, Gilmore, in his book Catastrophe Theory for Scientists and Engineers, Robert Gilmore, Dover 1993 republication of Wiley 1981 edition, says that only for 5 or fewer control parameters are all catastrophe germs simple, and that their simple non-Morse catastrophe germs are of the types D4, D5, D6, and E6, so that from my view
In my view, the Quantum Potential of 1-dim (timelike world-lines) in the Many-Worlds is described by E7 in terms of E7 / E6xU(1) sort of like a 27-complex-dim M-theory over a bosonic 26-dim string theory of string-world-lines in the Many-Worlds and the Quantum World of Space-Like 3-dim brane-worlds is described in the Many-Worlds by E8 in terms of E8 / E7 x SU(2), sort of like a 28-quaternionic-dim F-theory over the bosonic 26-dim string-world-line theory.
The McKay correspondences describe similar structure, if you look at the 24-dim off-diagonal subspace of the 27-dim exceptional Jordan algebra J3(O) of 3x3 octonion matrices (sometimes the 24-dim thing is called the Chevalley algebra). Then:
In other words, I identify E6 and TD with matter and E7/E8 and OD/ID with quantum consciousness, with the Lie group and McKay structures having similar interpretations, seeing the same things from different points of view, and I have E7 (and E8) as monistic structures.
A-D-E Classification not only applies to Catastrophe Singularities, but also to many other phenomena, from crystallography to String Theory.
References:
Catastrophe Theory for Scientists and Engineers, Robert Gilmore, Dover 1993 republication of Wiley 1981 edition;
Geometric Differentiation, Ian Porteous, Cambridge 1994.
The D4-D5-E6-E7 physics model is the natural Feynman Checkerboard theory based on the octonionic 8-real-dimensional E8 lattice with 4-complex-dimensional Witting polytope vertex figure, reduced to a 4-real-dimensional D4 lattice with 24-cell vertex figure.
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