| Complexified Octonions | dim = 16 | dim = 24 | dim = 32 | dims 64, 128, 256 | dim = -1 |
Charles Muses has described some algebras,
containing Zero Divisors,
which he considers to be a more fundamental concept
than Division Algebras.
The Alternative Division Algebras over the Real Numbers are:
(here a is nonzero)
R - dimension 2^0 = 1 - real numbers, with a^2 = 1;
C - dimension 2^1 = 2 - complex numbers, with a^2 = -1;
Q - dimension 2^2 = 4 - quaternions;
O - dimension 2^3 = 8 - octonions.
The above Division Algebras have no Zero Divisors.
Similar Algebras in higher dimensions have Zero Divisors:
(here a and b are nonzero and noninfinite)
S - dimension 2^4 = 16 - sedenions S with ab = 0;
SC - dimension 2^5 = 32 - complexified S with a0 = b;
dimension 2^6 = 64 - M(8,R)
dimension 2^7 = 128 - M(8,R)+M(8,R) with a^2 = 0;
dimension 2^8 = 256 - M(16,R) with a^4 = 0
and a^2 =/= 0 and a^3 =/= 0.
For S, normalized zero divisors are homeomorphic to G2 = Aut(O);
M(8,R)+M(8,R) has spinors that are not pure;
M(16,R) has half-spinors that are not pure, and
is the basis for all higher algebras,
because M(16,R) is based on Cl(8) and, by periodicity,
Cl(8N) = Cl(8) x...(N times)...x Cl(8)
and Cl(8N) is N tensor products of Cl(8)
Here is a chart of relationships between these algebras
and Clifford Algebras:
Cl(0) 1 1 R a^2 = 1
Cl(1) 1 1 2 C a^2 = -1
Cl(2) 1 2 1 4 Q
Cl(3) 1 3 3 1 8 O
Cl(4) 1 4 6 4 1 16 S ab=0
Cl(5) 1 5 10 10 5 1 32 SC a0=b
Cl(6) 1 6 15 20 15 6 1 64 M(8,R)
Cl(7) 1 7 21 35 35 21 7 1 128 M(8,R)+M(8,R) a^2=0
Cl(8) 1 8 28 56 70 56 28 8 1 256 M(16,R) a^4=0
By Periodicity, dimension 7 corresponds to dimension -1.
Pure spinors in Cl(N) are those spinors that can be
represented by simple exterior wedge products of vectors.
For odd N, any pure spinor can be represented
by a (1/2)(N-1)-dimensional hyperplane
in the N-dimensional vector space of Cl(N)
and its (1/2)(N+1)-dimensional orthogonal complement hyperplane.
By representing the pure spinors in terms of
the (N-1)-dimensional complex projective space associated
with the vector space of Cl(N), the pure spinors can be seen to be
of dimensionality (1/8)(N^2 - 1) + 1 (see Penrose and Rindler).
Since the dimensionality of spinors is 2^(1/2)(N-1)
and (1/8)(49-1)+1 = 7 and 2^(1/2)(7-1) = 8,
non-pure spinors exist if N = 7 or greater.
Such non-pure spinors seem to me to
be like nonzero a such that a^2 = 0 for Cl(7).
For even N, any pure spinor can be represented
by a (1/2)N-dimensional hyperplane in the N-dimensional
vector space of Cl(N) and its self-dual or anti-self-dual
(1/2)N-dimensional orthogonal complement hyperplane.
By representing the pure spinors in terms of
the (N-1)-dimensional complex projective space associated
with the vector space of Cl(N), the pure spinors can be seen to be
of dimensionality (1/8)(N(N-2)) + 1 (see Penrose and Rindler).
Since the dimensionality of half-spinors is 2^((1/2)N-1)
and (1/8)(8x6)+1 = 7 and 2^((1/2)8-1) = 8,
non-pure half-spinors exist if N = 8 or greater.
Such non-pure half-spinors seem to me to
be like nonzero a such that a^2 = 0 for Cl(8).
Perhaps non-pure full spinors are like nonzero a such that a^4 = 0
but a^3 and a^2 nonzero.
Carlos Castro mentioned pure spinors as described in
a paper by Nathan Berkovits and Sergey A. Cherkis at
http://xxx.lanl.gov/abs/hep-th/0409243
Similar material is discussed by Penrose and Rindler
in vol. 2 of Spinors and spacetime (Cambridge 1986),
the Appendix, particularly pp. 451-454
where they say (I quote mostly about even n because
in my model the even dimensions are more physically relevant):
"... pure spinors ... We use the term 'pure spinor' to
imply reduced [what I call half-spinor], when n is even ...
the spinor ... is necessarily pure if n < 7 ...
any pure spinor can be represented, up to proportionality,
by a (1/2) n plane through the origin in the vector
space Vn (n even) ...
... Geometry of pure spinors ...
... in terms of the (n-1)-dimensional (complex) projective
space PV associated with Va (i.e. the space of one-dimensional
linear subspaces of Va) ... the (non-zero) null vectors
va of Va ... defining the null cone in Va ... provide the
points of a non-singular quadric (n-2) surface Q in PV.
... any pure spinor determines .. a projective (1/2)(n-2)-plane
in Q if n is even. ... Acccording to the theory of
(complex projective) quadric surfaces ... the maximum dimension
of a linear projective space lying on a non-singular (n-1)-quadric
... when n is even ...is ... (1/2)(n-2),
these (1/2)(n-2) planes forming two disjoint (1/8)n(n-2)-dimensional
families. ... These two families, in the even case, correspond
to the unprimed and primed pure spinors, respectively, and
a (1/2)(n-2)-plane of the first family is frequently called
an alpha-pane and one of the second family a beta-plane. ...
... the dimension ... of the space of pure spinors
is (1/8)n(n-2) + 1 in the even case ...
... n ... pure spinors .... ... reduced [or half] spinors
not necesssarily... [pure]...
2 1 1
4 2 2
6 4 4
8 7 8
10 11 16
12 16 32
14 22 64
16 29 128
... whereas purity represents just a single condition
where this first appears at n = 7, 8, the number of
conditions increases rapidly with n thereafter.
... It is clear from all this that for large n,
the structure of the various spin-spaces can get
very complicated. ...".
The paper of Berkovits and Cherkis, where they say
"... ... In any even (Euclidean) dimension d = 2n,
projective pure spinors parameterize the coset space SO(2n)/U(n) ...",
seems to be looking at the total dimension of the spaces of
the two families of planes (alpha and beta), which
total dimension according to Penrose and Rindler is
2(1/8)(2n)(2n-2) =(1/4)(2n)(2n-2) = n(n-2) = n^2 - n.
Consider that the dimension of SO(2n) is n(2n-1) = 2n^2 - n
and the dimension of U(n) is n^2,
so that the dimension of SO(2n)/U(n) = 2n^2 - n - n^2 = n^2 - n
(as is consistent with the Penrose-Rindler formula)
and construct a table similar to the above Penrose-Rindler table,
but with full spinors instead of half-spinors:
... 2n ... dim SO(2n)/U(n) .... ... [full spinors]...
2 2-1-1 = 0 2
4 8-2-4 = 2 4
6 18-3-9 = 6 8
8 32-4-16 = 12 16
10 50-5-25 = 20 32
12 72-6-36 = 30 64
14 98-7-49 = 42 128
16 128-8-64 = 56 256
It seems to me that Berkovits and Cherkis are emphasizing the
study of what I might call full pure spinors (both alpha and
beta planes) instead of what I might call pure half-spinors,
or pure reduced spinors, on which Penrose and Rindler are concentrating.
I guess that Berkovits and Cherkis are doing that in order
to focus on the coset space structure SO(2n)/U(n),
which they see as being
"... the ... space of ... all complex structures on R(2n) ...".
According to the book Einstein Manifolds (Besse, Springer 1987),
SO(2n)/U(n) can be characterized as the
"... Set of complex structures of R(2n) compatible with its
Euclidean structures or the set of the metric-compatible
fibrations S1 --> RP(2n-1) --> CP(n-1) ..."
[where I presume S1 is shorthand for RP1 = S1/S0].
It seems to me that Penrose and Rindler are looking more
closely at what I might call pure half-spinors,
which they say are half of the dimension of SO(2n)/U(n), plus 1.
From the Penrose and Rindler point of view,
you can see that it is at dim = 8 that the dimensionality
of all half-spinors begins to exceed that of pure half-spinors,
and
that if you use Clifford 8-periodicity to factor high-dim
Clifford algebras into tensor products of 8-dim Clifford algebras,
you might be able to break down the "very complicated" spin
structure of high-dim Clifford algebras into something
that may be manageable (tensor product of Cl(8) spinor
structure, where "purity represents just a single condition").
That is something that I think is important with respect
to my physics model based on 8-periodicity of real Clifford algebras.
From the Berkovits and Cherkis point of view,
the dimensionality of SO(2n)/U(n) is always less than
the dimensionality of the full spinors = 2 x dim of half-spinors,
so the relation to 8-periodicity is not as clear to me,
but
it is interesting to me that at dimension 8 you have
... 2n ... dim SO(2n)/U(n) .... ... [full spinors]...
8 28-16 = 12 16
and
16 is the dimension of both U(4) and the full spinors of Cl(8)
where in my model U(4) describes conformal gravity etc
and
12 is the dimension of SO(8)/U(4) and also
of the Standard Model SU(3)xXU(2)xU(1).
In other words,
the Berkovits and Cherkis point of view seems to be related
to the dimensional reduction of my SO(8) 8-dim bivector gauge
boson Lie algebra,
which dimensional reduction is related to selecting a quaternionic
substructure of the octonionic 8-dim vector spacetime.
The quaternionic connection is made clearer when you consider
the book Einstein Manifolds (Besse, Springer 1987),
which describes a hyperbolic symmetric space related to the
Euclidean symmetric space used by Berkovits and Cherkis as
SO*(2n)/U(n) = SO(n,H)/U(n) being
the set of quaternionic quadratic forms on R(2n),
or set of the CP(n-1)hyp's in RP(2n-1)hyp
(where H denotes quaternions).
In my opinion,
the complex projective space construction of twistors
is not as physically important as is the
(mathematically equivalent in 4-dim spacetime)
quaternionic construction based on conformal quadric structures,
which to me is more clearly connected with the Klein correspondence
and with Lie sphere geometry of conformal structures,
so
I prefer to see the material in the Berkovits and Cherkis paper
more from a quaternionic point of view,
which I think would indicate that,
instead of just going up to more and more complicated stuff
in higher dimensions,
you would break the higher dimension structures down
into Cl(8) factor using real 8-periodicity,
and
then use quaternionic structure to understand dimensional reduction
to 4-dim physical spacetime plus 4-dim internal symmetry space.
Robert de Marrais has constructed detailed models
for Zero Divisors of algebras of dimension 2^N,
with explicit results through N = 8.
Guillermo Moreno has shown that
the zero divisors of the sedenions S whose basis entries are norm one
are homeomorphic to the 14-dimensional Lie group G2 = Aut(O).
Charles Muses pictures the nonzero elements a,b
such that ab = 0 as mutually annihilating,
or as creating zero,
or as interference of two waves.
Perhaps another viewpoint
might be to see them as soliton breathers.
Guillermo Moreno, in later papers,
shows for Cayley-Dickson algebra A(n) that:
1 - the automorphism group is G2 x (n-3)S3,
where S3 is the symmetric group on 3 elements, of order 3x2x1 = 6
2 - the zero divisors in A(n + 1) are related to
the real Stiefel manifold V(2^n - 1, 2)
3 - 2x2 matrices of O(2) are related to the Cayley-Dickson construction
of A(n+1) from An, and to 0An x oAn (where oAn is pure imaginary)
4 - all the elements of An are on the same footing,
with no obvious graded structure (although you can introduce
a grading by looking at which elements come from which
level of the Cayley-Dickson process)
Corresponding things for Clifford Algebras Cl(n) might be:
1 - the automorphism group is related to O(n), which
is related to the Clifford Group CG(n)
2 - the spinors of Cl(n), of dimension 2^(n/2)
3 - 2x2 matrices of SL(2,R) generalized to
have Clifford group CG(n) entries
are SL(2,CG(n-2)) = Spino(n,1)
and Spino(n,1) is a 2-fold covering of
the Conformal Group of R(n-1) plus the point at infinity
4 - the elements of Cl(n) have an obvious graded structure
Here are some comments:
1 - G2 x (n-3)S3 just basically stays as lots of copies of G2
while
O(n) gets bigger and bigger as n increases, BUT there is
a sense in which O(n) does sort of stay the same
and that is due to Clifford periodicity,
by which Cl(8N) = Cl(8) x ...(tensor product N times) ... x Cl(8),
so that
In some sense G2 is a fundamental group of the An
and Spin(8) is a fundamental group of Cl(n),
which makes some sense because G2 is part of Spin(8) by
the fibrations S7 = Spin(7) / G2 and S7 = Spin(8) / Spin(7).
2 - The zero divisors grow very fast as n gets large,
beginning after A3 = octonions.
and
spinors grow very fast as n gets large,
becomeing much bigger than the the vector space after Cl(8),
which is an algebra with octonionic vector space
Split Clifford Algebras, Cl(p,p), are Clifford algebras whose vector space is of signature (p,p) so that there are an equal number p of dimensions of signature -1 and of +1. Algebras of signature (p,1) can be said to be of Lorentz type, with p dimensions of signature -1 and one real axis of signature +1. Conventional number-algebras such as complex numbers, quaternions, octonions, and sedenions are of Lorentz type. Split number-algebras, such as split complex numbers, split quaternions, Split Octonions, and split sedenions, are number-algebras of signature (p-1, p+1) with p dimensions of signature -1 plus a real 1 and p dimensions, other than the real axis, of signature +1. Charles Muses and Kevin Carmody use the terminologies counterimaginary and countercomplex to describe the dimensions of signature +1, other than the real 1, of a basis of a Complexified Algebra, which is a tensor product CxALG of the complex numbers C and the algebra ALG. If ALG is N-dimensional, the Complexified Algebra CxALG is the 2N-real-dimensional algebra formed by REAL linear combinations of the original N-dimensional ALG with signature (N-1,1) and an imaginary N-dimensional algebra with signature (1,N-1). As these REAL linear combinations are equivalent to COMPLEX linear combinations of the original ALG basis, the signature of the Complexified Algebra CxALG is indeterminate, and not well defined. For example, note that both a conventional algebra and the corresponding split algebra are subalgebras of the Complexified Algebra.
Charles Muses represents algebras in terms of matrices and bimatrices.
They represent the multiplication rules of the unit basis elements
of the algebra.
To take the simplest example, consider the 1x1 matrix 1.
The algebra R of the real numbers consists of REAL linear
combinations of that 1 basis element.
The 2 elements +/-1
form a discrete subalgebra of the real numbers R.
TO EXTEND BEYOND the real numbers R,
consider real 2x2 matrices, a basis for which is:
1 0 0 1 1 0 0 1
0 1 1 0 0 -1 -1 0
1 0 0 1
The 2 matrices 0 1 = 1 1 0 = e
are
a basis for the split complex numbers,
where 1^2 = + 1 and e^2 = +1.
1 0 0 1
The 2 matrices 0 1 = 1 -1 0 = i
are
a basis for the complex numbers C,
where 1^2 = +1 and i^2 = -1.
(The algebra C of complex numbers consists of REAL liner
combinations of those 2 basis elements,
and are the Complexified Real numbers, the tensor product CxR.
The 4 elements +/-i, +/-i
form a discrete subalgebra of the complex numbers C.
Also, the 3 elements +/-1, (1/2)(-1 +/- sqrt(3)i)
form another discrete subalgebra of the complex numbers C.)
1 0 0 1 1 0 0 1
All 4 matrices 0 1 = 1 1 0 = e 0 -1 = f -1 0 = i
are
a basis for the split quaternions,
where 1^2 =+1 and i^2 = -1 and e^2 = f^2 = +1.
TO EXTEND BEYOND the split quaternions use 2x2 matrices,
ALL of whose nonzero entries are EITHER
1 0 0 1 1 0 0 1
0 1 = 1 OR 1 0 = e OR 0 -1 = f OR -1 0 = i
At first, use as nonzero entries only either all 1 or all i,
called pure 1 matrices or pure i matrices,
so that you have a basis of 2x4 = 8 matrices.
Notice that any matrix with mixed 1 and i entries
can be formed as linear combinations of these 8 matrices,
so that the 8 pure 1 and pure i matrices
effectively span all 2^4 = 16 types
of 2x2 matrices with entries either 1 or i.
Using pure 1 and pure i matrices gives you
Complexified Algebras.
Such a Complexified Algebra is represented by Charles Muses
as REAL linear combinations of
a pure 1 real subspace and a pure i imaginary subspace.
Such a Complexified Algebra contains as subalgebras
both the underlying real algebra
and its corresponding split algebra.
The 8 pure 1 and pure i matrices are enough
for Charles Muses to construct:
Quaternions from
1 0
0 1 = e0
0 -1
i1 = 1 0
i 0
i2 = 0 -i
0 i
i3 = i 0
(Here, the matrices e0, i1, i2, i3
represent the unit quaternion basis elements.
The entire quaternion algebra consists of REAL linear
combinations of those 4 basis elements.
The 8 elements +/-i0, +/-i1, +/-i2, +/-i3
form a discrete subalgebra of the quaternions.
The 24 elements +/-i0, +/-i1, +/-i2, +/-i3,
(1/2)(+/-i0 +/- i1 +/- i2 +/- i3)
also form a discrete subalgebra of the quaternions,
and represent the vertices of a 24-cell in quaternion 4-space,
which are the 24 nearest neighbors of the origin in a D4 lattice.)
and
Complexified Quaternions, the tensor product CxQ, from
1 0 i 0
0 1 = e0 i0 = 0 i
0 -1 0 i
i1 = 1 0 -i 0 = e1
i 0 1 0
i2 = 0 -i 0 -1 = e2
0 i 0 1
i3 = i 0 1 0 = e3
Note that the Complexified Quaternions are 8-real-dimensional.
Note that e0 = 1, i1 = i, e2 = f, and e3 = e
form the Split Quaternions of signature (1,3).
The 2x4 = 2^3 = 8-dimensional Complexified Quaternions CxQ
are as far as we can get with
2x2 matrices with 2 types of entries (1 and i).
TO EXTEND BEYOND complexified quaternions CxQ, use 2x2 matrices,
ALL of whose nonzero entries are EITHER
1 0 0 1 1 0 0 1
0 1 =1=e0 OR 1 0 =e=e3 OR 0 -1 =f=e2 OR -1 0 =i=i1
using as entries not only pure 1=e0 or pure i=i1,
but also pure e=e3 or pure f=e2.
While it was OK to use only 1 and i as entries,
because 1 and i close algebraically to form the complex numbers C,
it is NOT OK to use only e or f in addition to 1 and i,
because ie = f and fi = e,
so algebraically you must extend from 1 and i
to 1,i,e,f (thus forming the split quaternions).
That gives you a basis of 4x4 = 16 matrices.
Notice that any matrix with mixed 1, i, e, and f entries
can be formed as linear combinations of these 16 matrices,
so that the 16 pure 1, pure i, pure e, and pure f matrices
effectively span all 4^4 = 256 types of
2x2 matrices with entries either 1, i, e, or f.
To maintain algebraic consistency
with the split-quaternionic structure of 1, i, e, and f,
pure 1 and pure i matrices should be in the same subspace
and
pure e and pure f matrices should be in the same subspace.
The bimatrices of Charles Muses, as pairs A|B of 2x2 matrices,
maintain that algebraic consistency
by having the left-hand matrix A of each bimatrix made up of
the 8 matrices of the subspace spanned by 1 and i
and
the right-hand matrix B of each bimatrix made up of
the 8 matrices of the subspace spanned by e and f,
and
also
by requiring that the left-hand matrix of a bimatrix
be of quaternionic form, either 1=e0, i1, i2, or i3.
If the distinction between left-hand and right-hand matrices
of a bimatrix is consistently maintained,
then it is OK to denote the subspace spanned by e and f
as
the right-hand matrix, with entries either pure 1 or pure i,
where right-hand 1 corresponds to e
and right-hand i corresponds to f.
With the left-hand matrices of a bimatrix restricted to
the 4 matrices of quaternionic form,
and with the right-handed matrices of a bimatrix
being regarded as an independent subspace with 8 basis elements,
there are 4x8 = 32 distinct bimatrix elements,
so that Charles Muses can use his bimatrices
to construct algebras up to and including
the 32-dimensional Complexified Sedenion Algebra.
Rules for bimatrix calculations are given by Charles Muses
in his 1960 paper Hypernumbers and Quantum Field Theory.
Some of the rules are, for any X, for N = +/- 1 or +/- i0,
and for A, B, C, D =/= +/- 1 or +/- i0 such
that N commutes with everything
and A,B,C,D anticommute with themselves except
that B and D commute:
N(A|B) = NA|B = AN|B = A|NB = A|BN
|(+/-N) = +/-N
(|X)^2 = |X^2
X = X|1 = 1|X
X(|B} = X|B
(A|B)^2 = -A^2|B^2
(A|A}^2 = -A^2|A^2
(N|B)^2 = N^2|B^2
A(|B) = -(|B)A = A|B
A(C|D) = -AC|D = -(C|D)A = CA|D
(C|D)A = -A(C|D) = +AC|D
A(A|B) = -(A|B)A = -A^2|B
(A|B)(A|D) = -A^2|BD
(A|B)(C|D) = AC|BD
(C|D)(A|B) = CA|DB
(N|B)C = -NC|B = - CN|B
(|B)(|D) = |(BD) = |BD
(A|B)(N|D) = AN|BD
(N|D)(A|B) = -AN|BD
Using bimatrices, Charles Muses constructed:
Octonions from
1 0
0 1 = e0
0 -1
i1 = 1 0
i 0
i2 = 0 -i
0 i
i3 = i 0
1 0 | 0 -1
i4 = 0 1 | 1 0
0 -1 | 0 -1
i5 = 1 0 | 1 0
-i 0 | 0 -1
i6 = 0 i | 1 0
0 i | 0 -1
i7 = i 0 | 1 0
(Here, the matrices e0, i1, i2, i3, i4, i5, i6, i7
represent the unit octonion basis elements.
The entire octonion algebra consists of REAL linear
combinations of those 8 basis elements.
The 240 vertices of a Witting polytope in octonion 8-space,
are the 240 nearest neighbors of the origin in an E8 lattice.)
as well as
Complexified Octonions, Sedenions, the 24-dimensional Algebra,
and the 32-dimensional Complexified Sedenion Algebra.
To go beyond the 32-dimensional Complexified Sedenion Algebra,
you have to give up the quaternionic and complex algebraic rules
used to get 4x8 = 32 distinct bimatrix elements,
and
look at pairs of 2x2 matrices each entry of which
is itself a real 2x2 matrix,
giving a 4x4 x 4x4 = 16x16 = 256-dimensional algebra.
As Charles Muses indicates, his matrices and bimatrices
cannot all be amalgamatively added, so it is important to
consider them as representing ONLY the multiplication rules
of the unit basis elements of the algebra.
Also, it is important to realize that, even though the
imaginary i is used in the matrices and bimatrices,
the algebra is made up of REAL (not complex) linear combinations
of the basis elements.
Charles Muses constructs two distinct 16-real-dimensional Algebras,
so it is important to realize that one of them
(containing a real 1, 7 complex numbers,
an imaginary i, and 7 countercomplex numbers)
is a Complexified Octonion algebra,
while
the other (containing a real axis and 15 complex numbers)
is the conventional Sedenion algebra.
Charles Muses represents his Complexified Octonion Algebra
in terms of matrices and bimatrices. They represent the
multiplication rules of the unit basis elements of the algebra.
1 0 i 0
0 1 = e0 i0 = 0 i
0 -1 0 i
i1 = 1 0 -i 0 = e1
i 0 1 0
i2 = 0 -i 0 -1 = e2
0 i 0 1
i3 = i 0 1 0 = e3
1 0 | 0 -1 1 0 | 0 i
i4 = 0 1 | 1 0 0 1 | -i 0 = e4
0 -1 | 0 -1 0 1 | 0 i
i5 = 1 0 | 1 0 -1 0 | -i 0 = e5
-i 0 | 0 -1 -i 0 | 0 i
i6 = 0 i | 1 0 0 i | -i 0 = e6
0 i | 0 -1 0 i | 0 i
i7 = i 0 | 1 0 i 0 | -i 0 = e7
The red numbers e0 and i1 through i7 are the octonion subalgebra,
to which the green numbers i0 and e1 through e7 are adjoined
to form the Complexified Octonions.
The 8 numbers i0, i1, i2, i3, e0, e1, e2, e3
are the imaginary part of the Complexified Octonions.
The 7 numbers i1 though i7 are all square roots of -1,
and are called i-numbers by Charles Muses.
The 7 numbers e1 through e7 are all square roots of +1,
and are called epsilon-numbers by Charles Muses.
For typographical reasons, I call them e-numbers.
Note that e0, i1, i4, i5, e2, e3, e4, and e5
form the Split Octonions of signature (3,5).
1 0 | 0 -1
Numbers such as 0 1 | 1 0 are represented by bimatrices
rather than matrices because they are nonassociative.
Rules for bimatrix calculations are given by Charles Muses
in his 1960 paper Hypernumbers and Quantum Field Theory.
The 16-dimensional Complexified Octonion algebra
not only represents
a modification of the the Clifford algebra Cl(4),
at which the phenomenon ab = 0 appears,
but also represents
the 8-complex-dimensional Hermitian Symmetric Space
Spin(10) / Spin(8) x U(1)
with 45 - 28 - 1 = 16 real dimensions
whose corresponding bounded complex domain
has an 8-real dimensional Shilov boundary
that represents the vector space of Cl(8) and Spin(8)
and
represents octonion 8-dimensional vector spacetime
in the D4-D5-E6-E7 physics model.
Charles Muses constructs various 16-dimensional Algebras,
so it is important to realize that one of them
(containing a real 1, 7 complex numbers,
an imaginary i, and 8 countercomplex numbers),
is a Complexified Octonion algebra,
while
another (containing a real axis and 15 complex numbers)
is similar to the conventional Sedenion algebra
and
yet another has non-distributivity.
Charles Muses represents his conventional Sedenion Algebra
in terms of matrices and bimatrices. They represent the
multiplication rules of the unit basis elements of the algebra.
1 0
0 1 = e0
0 -1
i1 = 1 0
i 0
i2 = 0 -i
0 i
i3 = i 0
1 0 | 0 -1
i4 = 0 1 | 1 0
0 -1 | 0 -1
i5 = 1 0 | 1 0
-i 0 | 0 -1
i6 = 0 i | 1 0
0 i | 0 -1
i7 = i 0 | 1 0
1 0 | i 0
i8 = 0 1 | 0 -i
0 -1 | i 0
i9 = 1 0 | 0 -i
-i 0 | i 0
i10 = 0 i | 0 -i
0 i | i 0
i11 = i 0 | 0 -i
1 0 | 0 i
i12 = 0 1 | i 0
0 -1 | 0 i
i13 = 1 0 | i 0
-i 0 | 0 i
i14 = 0 i | i 0
0 i | 0 i
i15 = i 0 | i 0
The red numbers e0 and i1 through i7 are the octonion subalgebra,
to which the blue numbers i8 through i15 are adjoined
to form the sedenions.
The 15 numbers i1 through i15 are all square roots of -1,
and are called i-numbers by Charles Muses.
The number on the right-hand side is the real square root of +1,
and is called an epsilon-number by Charles Muses.
For typographical reasons, I call such numbers e-numbers.
1 0 | 0 -1
Numbers such as 0 1 | 1 0 are represented by bimatrices
rather than matrices because they are nonassociative.
Rules for bimatrix calculations are given by Charles Muses
in his 1960 paper Hypernumbers and Quantum Field Theory.
Here is a sedenion multiplication table:
1 i j k E I J K S T U V W X Y Z
1 1 i j k E I J K S T U V W X Y Z
i i -1 k -j I -E -K J T -S -V U -X W Z -Y
j j -k -1 i J K -E -I U V -S -T -Y -Z W X
k k j -i -1 K -J I -E V -U T -S -Z Y -X W
E E -I -J -K -1 i j k W X Y Z -S -T -U -V
I I E -K J -i -1 -k j X -W Z -Y T -S V -U
J J K E -I -j k -1 -i Y -Z -W X U -V -S T
K K -J I E -k -j i -1 Z Y -X -W V U -T -S
S S -T -U -V -W -X -Y -Z -1 i j k E I J K
T T S -V U -X W Z -Y -i -1 -k j -I E K -J
U U V S -T -Y -Z W X -j k -1 -i -J -K E I
V V -U T S -Z Y -X W -k -j i -1 -K J -I E
W W X Y Z S -T -U -V -E I J K -1 -i -j -k
X X -W Z -Y T S V -U -I -E K -J i -1 k -j
Y Y -Z -W X U -V S T -J -K -E I j -k -1 i
Z Z Y -X -W V U -T S -K J -I -E k j -i -1
The 16-dimensional Sedenion algebra
not only represents
a modification of the Clifford algebra Cl(4),
at which the phenomenon ab = 0 appears,
but also represents
the full spinor space of the Clifford algebra Cl(8),
whose Lie algebra is the D4 Lie algebra Spin(8)
and also represents
the 16-real-dimensional Shilov boundary of
the bounded complex domain corresponding to
the 16-complex-dimensional Hermitian Symmetric Space
E6 / Spin(10) x U(1)
with 78 - 45 - 1 = 32 real dimensions
all of which
represents spinor fermions in the D4-D5-E6-E7 physics model.
Sedenions may also be a good way to express
the structure of the last Hopf fibration
S7 --> S15 --> S8.
Since there are no Hopf fibrations other than
S0 --> S1 --> RP1 = Spin(2)/Spin(1)
S1 --> S3 --> S2 = CP1 = SU(2)/U(1) = Sp(1)/U(1) = Spin(3)/Spin(2)
S3 --> S7 --> S4 = HP1 = Sp(2)/(Sp(1)xSp(1)) = Spin(5)/Spin(4)
S7 --> S15 --> S8 = OP1 = Spin(9)/Spin(8)
sedenion structure may show why the sequence terminates.
Also,
it is possible that the sedenions may show the structure
of the 16-dimensional octonionic projective plane OP2 = F4/Spin(9)
and may also show
why there is no higher-dimensional octonionic projective space.
The w hypernumbers of Charles Muses are described in his paper Hypernumbers II, Appl. Math. and Comput. 4:45-66 (1978). In that paper (at pages 61-62), Charles Muses says: "... When we come to w defined by w^2 = - 1 + w and (-w)^2 = - 1 - w, ... we are ... able to distinguish arithmetically between (+x)^2 and (-x)^2. The power orbits of +/- w are ellipses ... ... note that i^(2/3) must not be confused with w, for the power orbit of boht +/- i^(2/3) is the same unit circle that we saw was the power orbit of i. But the power orbit of w and -w is elliptical and moreover a different ellipse for each. ... we further define -w ... to mean (+1)[(-1)w] and not (-1)[(+1)w] ... Hence w( w - w ) = - 1 + w + 1 + w = 2w by distribution. But w - w = 0 and hence, without distribution, w(0) = 0. Thus w-arithmetic is nondistributive ... Thus the product (w - w)w is distributive, but w( w - w ) is not. ... True nondistribution ... did not arise until we found a hypernumber which had a power orbit distinct from its negative ...". In Applied Mathematics and Computation 60:25-36 (1994), Charles Muses describes the hypernumber w and says (at pages 33-34): "... The orbits of +w and -w are two orthogonal, but otherwise, identical ellipses, the former traversed counterclockwise and the latter clockwise with increasing real powers of +w and -w, respectively. ... the concept of the power orbit of a number, as well as the related one of a multiplication trajectory, were introduced by the present author into number theory in 1977 ... This term indicates the locus of u^k where u is an elemental number unit and k is real. ...".
Charles Muses's ZeroDivisor Algebras also include
a 24-dimensional algebra that is a combination
of Conventional Sedenions and Complexified Octonions.
Charles Muses represents his 24-dimensional Algebra
in terms of matrices and bimatrices. They represent the
multiplication rules of the unit basis elements of the algebra.
1 0 i 0
0 1 = e0 i0 = 0 i
0 -1 0 i
i1 = 1 0 -i 0 = e1
i 0 1 0
i2 = 0 -i 0 -1 = e2
0 i 0 1
i3 = i 0 1 0 = e3
1 0 | 0 -1 1 0 | 0 i
i4 = 0 1 | 1 0 0 1 | -i 0 = e4
0 -1 | 0 -1 0 1 | 0 i
i5 = 1 0 | 1 0 -1 0 | -i 0 = e5
-i 0 | 0 -1 -i 0 | 0 i
i6 = 0 i | 1 0 0 i | -i 0 = e6
0 i | 0 -1 0 i | 0 i
i7 = i 0 | 1 0 i 0 | -i 0 = e7
1 0 | i 0
i8 = 0 1 | 0 -i
0 -1 | i 0
i9 = 1 0 | 0 -i
-i 0 | i 0
i10 = 0 i | 0 -i
0 i | i 0
i11 = i 0 | 0 -i
1 0 | 0 i
i12 = 0 1 | i 0
0 -1 | 0 i
i13 = 1 0 | i 0
-i 0 | 0 i
i14 = 0 i | i 0
0 i | 0 i
i15 = i 0 | i 0
The red numbers e0 and i1 through i7 are the octonion subalgebra,
to which the green numbers i0 and e1 through e7 are adjoined
to form the Complexified Octonions,
and to which the blue numbers i8 though i15 are adjoined
to form the sedenions.
The split sedenions and sedenions, when combined,
form the 24-dimensional algebra.
The 15 numbers i1 though i15 are all square roots of -1,
and are called i-numbers by Charles Muses.
The 7 numbers e1 through e7 are all square roots of +1,
and are called epsilon-numbers by Charles Muses.
For typographical reasons, I call them e-numbers.
1 0 | 0 -1
Numbers such as 0 1 | 1 0 are represented by bimatrices
rather than matrices because they are nonassociative.
Rules for bimatrix calculations are given by Charles Muses
in his 1960 paper Hypernumbers and Quantum Field Theory.
Charles Muses corresponded with John Leech
when Leech was working on the 24-dimensional Leech Lattice.
Charles Muses notes that
the density of the 24-dimensional Leech Lattice is unity,
the same as the density of the 0-dimensional point lattice,
so that Leech Lattice structures, such as elliptic modular functions,
should be related to point structures, such as number theory.
The 24-dimensional algebra, being a combination of both
the sedenion algebra and the Complexified Octonion algebra,
represents
the Complexified Octonion 8-complex-dimensional
Hermitian Symmetric Space
Spin(10) / Spin(8) x U(1)
with 45 - 28 - 1 = 16 real dimensions
whose corresponding bounded complex domain
has an 8-real dimensional Shilov boundary
that represents the vector space of Cl(8) and Spin(8)
and
represents octonion 8-dimensional vector spacetime
in the D4-D5-E6-E7 physics model.
and also represents
the sedenion 16-real-dimensional full spinor space
of the Clifford algebra Cl(8),
whose Lie algebra is the D4 Lie algebra Spin(8)
and
the 16-real-dimensional Shilov boundary of
the bounded complex domain corresponding to
the 16-complex-dimensional Hermitian Symmetric Space
E6 / Spin(10) x U(1)
with 78 - 45 - 1 = 32 real dimensions
all of which
represents spinor fermions in the D4-D5-E6-E7 physics model.
Therefore,
the 24-dimensional algebra represents
the three 8-dimensional (vector and two half-spinor)
representations of the Spin(8) Lie algebra,
and so represents triality
and
the three 8-dimensional spaces of the 27-dimensional
exceptional Jordan algebra used
in the octonionic D4-D5-E6-E7 physics model
and
also represents the 24 root vectors
of the 28-dimensional Spin(8) Lie algebra
and
the the 4-dimensional 24-cell
and
the 24+4 = 28-dimensional adjoint representation of Spin(8),
and
the 28 bivector gauge bosons of the D4-D5-E6-E7 physics model.
There is an alternative way to look at
the 24-dimensional ZeroDivisor Algebra.
Charles Muses's ZeroDivisor Algebras also include
a 32-dimensional algebra that is a combination
of Conventional Sedenions, Complexified Octonions, and
an extension of his 24-dimensional algbra by adding 8 e-numbers.
As it has 16 square roots of -1 and 16 square roots of +1,
it is a Complexified Sedenion algebra.
Charles Muses represents his
32-dimensional Complexified Sedenion Algebra
in terms of matrices and bimatrices.
They represent the multiplication rules of
the unit basis elements of the algebra.
1 0 i 0
0 1 = e0 i0 = 0 i
0 -1 0 i
i1 = 1 0 -i 0 = e1
i 0 1 0
i2 = 0 -i 0 -1 = e2
0 i 0 1
i3 = i 0 1 0 = e3
1 0 | 0 -1 1 0 | 0 i
i4 = 0 1 | 1 0 0 1 | -i 0 = e4
0 -1 | 0 -1 0 1 | 0 i
i5 = 1 0 | 1 0 -1 0 | -i 0 = e5
-i 0 | 0 -1 -i 0 | 0 i
i6 = 0 i | 1 0 0 i | -i 0 = e6
0 i | 0 -1 0 i | 0 i
i7 = i 0 | 1 0 i 0 | -i 0 = e7
1 0 | i 0 1 0 | 1 0
i8 = 0 1 | 0 -i 0 1 | 0 -1 = e8
0 -1 | i 0 0 1 | 1 0
i9 = 1 0 | 0 -i -1 0 | 0 -1 = e9
-i 0 | i 0 -i 0 | 1 0
i10 = 0 i | 0 -i 0 i | 0 -1 = e10
0 i | i 0 0 i | 1 0
i11 = i 0 | 0 -i i 0 | 0 -1 = e11
1 0 | 0 i 1 0 | 0 1
i12 = 0 1 | i 0 0 1 | 1 0 = e12
0 -1 | 0 i 0 1 | 0 1
i13 = 1 0 | i 0 -1 0 | 1 0 = e13
-i 0 | 0 i -i 0 | 0 1
i14 = 0 i | i 0 0 i | 1 0 = e14
0 i | 0 i 0 i | 0 1
i15 = i 0 | i 0 i 0 | 1 0 = e15
The red numbers e0 and i1 through i7 are the octonion subalgebra,
to which the green numbers 10 and e1 through e7 are adjoined
to form the Complexified Octonions,
and to which the blue numbers i8 though i15 are adjoined
to form the sedenions.
The complexified octonions and sedenions, when combined,
form the 24-dimensional algebra.
When the yellow numbers e8 through e15 are adjoined,
the 32-dimensional Complexified Sedenion algebra is formed.
The 15 numbers i1 though i15 are all square roots of -1,
and are called i-numbers by Charles Muses.
The 15 numbers e1 through e15 are all square roots of +1,
and are called epsilon-numbers by Charles Muses.
For typographical reasons, I call them e-numbers.
Note that e0, i1, i4, i5, i8, i9, i12, i13,
e2, e3, e4, e5, e8, e9, e12, and e13
form the Split Sedenions of signature (7,9).
1 0 | 0 -1
Numbers such as 0 1 | 1 0 are represented by bimatrices
rather than matrices because they are nonassociative.
Rules for bimatrix calculations are given by Charles Muses
in his 1960 paper Hypernumbers and Quantum Field Theory.
The 32-dimensional Complexified Sedenion algebra
not only represents
a modification of the the Clifford algebra Cl(5),
at which the phenomenon a0 = b appears,
but also represents
the full spinor space of the Clifford algebra Cl(10),
whose Lie algebra is the D5 Lie algebra Spin(10)
and also represents
the 16-complex-dimensional Hermitian Symmetric Space
E6 / Spin(10) x U(1)
with 78 - 45 - 1 = 32 real dimensions
whose corresponding bounded complex domain
has a 16-real-dimensional Shilov boundary
that represents
the Sedenion full spinor space of Cl(8) and Spin(8)
and
Rosenfeld's elliptic projective plane (CxO)P2
and
represents spinor fermions in the D4-D5-E6-E7 physics model.
To go beyond the 32-dimensional Complexified Sedenion Algebra, you have to give up the quaternionic and complex algebraic rules used to get 4x8 = 32 distinct bimatrix elements, and look at pairs of 2x2 matrices each entry of which is itself a REAL 2x2 matrix, giving a 4x4 x 4x4 = 16x16 = 256-dimensional algebra. The 256-dimensional algebra is the Clifford Algebra M(16,R): Cl(0) 1 1 R Real Cl(1) 1 1 2 C Complex Cl(2) 1 2 1 4 Q Quaternion Cl(3) 1 3 3 1 8 O Q+Q Cl(4) 1 4 6 4 1 16 S Quaternion Cl(5) 1 5 10 10 5 1 32 SC Complex Cl(6) 1 6 15 20 15 6 1 64 M(8,R) Real Cl(7) 1 7 21 35 35 21 7 1 128 M(8,R)+M(8,R) R+R Cl(8) 1 8 28 56 70 56 28 8 1 256 M(16,R) Real The underlying Clifford structure of Octonions is Q+Q, or pairs of quaternions, as 2x4 = 8. The underlying Clifford structure of Sedenions is quaternionic, or a quaternionic set of quaternions, as 4x4 = 16. The underlying Clifford structure of Complexified Sedenions is complex. The Clifford algebras Cl(1), Cl(2), Cl(3), Cl(4), and Cl(5) of 2, 4, 8, 16, and 32 dimensions all have complex or quaternionic structure, so that the quaternionic and complex algebraic rules used to get 4x8 = 32 distinct bimatrix elements are needed to construct the algebras C, Q, O, S, and SC. However, when you go beyond 32 dimensions, the Clifford algebras Cl(6), Cl(7), and Cl(8) of 64, 128, and 256 dimensions all have Real structure, so that the quaternionic and complex algebraic rules used to get 4x8 = 32 distinct bimatrix elements no longer are applicable and you have to look at pairs of 2x2 matrices each entry of which is itself a REAL 2x2 matrix, giving a 4x4 x 4x4 = 16x16 = 256-dimensional algebra. By Periodicity, Cl(8N) = Cl(8) x...(N times)...x Cl(8) and Cl(8N) is N tensor products of Cl(8) so that nothing really new happens above Cl(8), and these are all the algebras that are needed to construct the D4-D5-E6-E7 physics model. Also by Periodicity, dimension 8 corresponds to dimension 0, and dimension 7 corresponds to dimension -1. New algebraic phenomena that appear in dimensions 128 and 256, the existence of nonzero a such that a^2 = 0 in 128 dimensions and a^4 = 0 for a^2 =/= 0 and a^3 =/= 0 in 256 dimensions, are related to the appearance of spinors that are not pure. The 15 bivectors of the 64-dimensional Clifford algebra form the Lie algebra Spin(6) = SU(4). The 64-dimensional algebra is related to 64 of the 72 = 64+8 (64+8 = dimensionality of Cl(6) and its full spinor space) root vectors of the 78-dimensional Lie algebra E6. Also, the 64-dimensional algebra is related to the full spinors of the Clifford algebra Cl(13), whose Lie algebra Spin(13) is 78-dimensional, the same dimensionality as E6. Also, the 64-dimensional algebra is related to E7/(Spin(12)xSpin(3)) which is Rosenfeld's elliptic projective plane (QxO)P2. The 21 bivectors of the 128-dimensional Clifford algebra form the Lie algebra Spin(7). The 128-dimensional algebra is related to the 126 = 28+70+28 (128-2=7+21+35+35+21+7 = dim of Cl(7) - scalar and pseudoscalar) root vectors of the 133-dimensional Lie algebra E7. Also, the 128-dimensional algebra is related to the half-spinors of the Clifford algebra Cl(16), whose Lie algebra Spin(16) is 120-dimensional, and whose adjoint and half-spinor representations make up the 120+128 = 248-dimensional E8. Also, the 128-dimensional algebra is related to E8/Spin(16) which is Rosenfeld's elliptic projective plane (OxO)P2. The 28 bivectors of the 256-dimensional Clifford algebra form the Lie algebra Spin(8). The 256-dimensional algebra is related to the 240 = 256-16 (256-16 = dim of Cl(8) - full spinor space) root vectors of the 248-dimensional Lie algebra E8. There is an alternative way to look at ZeroDivisor Algebras of dimensions 64, 128, and 256.
John Baez uses homotopy periodicity to notice that dimension 7 corresponds to dimension -1: n Cl(n) PI(n)(O) PI(n)(Sp) Clc(n) PI(n)(U) 7=-1 R+R Z Z C+C Z 0=8 R Z/2 0 C 0 1 C Z/2 0 C+C Z 2 Q 0 0 C 0 3 Q+Q Z Z C+C Z 4 Q 0 Z/2 C 0 5 C 0 Z/2 C+C Z 6 R 0 0 C 0 He also notices that dimension -1 corresponds to the concept of dimensionality, in that homotopy in dimension -1 counts the difference in dimension between the two fibres of a vector bundle over the 0-sphere S0. Charles Muses notes that n-dimensional Dn lattices are related to n! (the Dn Weyl group is of order 2^(n-1) n!) and that (by the Riemann Zeta Function) (-1)! = 1 + 1/2 + 1/3 + 1/4 + 1/5 + .... (Harmonic Series), so that dimension -1 should be related to Harmonic structures.
Many thanks to Douglas Welty for telling me about the works of Charles Muses, and for sending me some of them (they are hard to find). References: Charles Muses, The First Nondistributive Algebra, with Relations to Optimization and Control Theory, in Functional Analysis and Optimization, ed. by E. R. Caianiello, Academic Press 1966. Charles Muses, The Amazing 24th Dimension, Journal for the Study of Consciousness, Research Notes. Charles Muses, Hypernumbers and Quantum Field Theory with a Summary of Physically Applicable Hypernumber Arithmetics and their Geometries, Applied Mathematics and Computation 6 (1960) 63-94. Kevin Carmody, Circular and Hyperbolic Quaternions, Octonions, and Sedenions, Applied Mathematics and Computation 28 (1988) 47-72. Guillermo Moreno, The zero divisors of the Cayley-Dickson algebras over the real numbers, q-alg/9710013. R. Penrose and W. Rindler, Spinors and Space-Time, vol. 2, Cambridge 1986, particularly Appendix: spinors in n dimensions. Boris Rosenfeld, Geometry of Lie Groups, Kluwer 1997. Arthur L. Besse, Einstein Manifolds, Springer-Verlag 1987.
......