The split octonions O" have signature (----++++) of a symmetric quadratic form - X0^2 - X1^2 - X2^2 - X3^2 + Y0^2 + Y1^2 + Y2^2 + Y3^2 instead of (++++++++) of a symmetric quadratic form + X0^2 + X1^2 + X2^2 + X3^2 + Y0^2 + Y1^2 + Y2^2 + Y3^2 = = + X0^2 + X1^2 + X2^2 + X3^2 + X4^2 + X5^2 + X6^2 + X7^2 as the octonions O. To build the split octonions and the octonions, use the Cayley-Dickson process: Start with an algebra A, with conjugation * . Define A+ to be the vector space A + A with a new product defined by (x,y)(z,w) = ( x z - w* y , w x + y z* ) and define A- to be the vector space A + A with a new product defined by (x,y)(z,w) = ( x z + w* y , w x + y z* ) Let R denote the real numbers, C denote the complex numbers, H denote quaternions, and O denote octonions. Then: C = R+ H = C+ O = H+ and R- is something different, the Lorentz numbers L C- is not really new and different, but is the 2x2 matrix algebra over R, which can be denoted M2(R) This is because a complex number a + bi can be represented as a 2x2 real matrix a -b b a with complex conjugation defined by the matrix operation 1 0 0 -1 H- is something different, the split octonions O" . Notice that because L, M2(R), and O" have non-trivial "lightcones", they are NOT division algebras, although they ARE normed algebras. The only division algebras over the reals are R, C, H, O and the only normed algebras over the reals are R, C, L, H, M2(R), O, and O". That means that all normed algebras over the reals are subalgebras of either the octonions O or the split octonions O". Now, look at the 7-dimensional imaginary spaces Im(O) and Im(O") Neither O nor O" are associative, but you can find 3 of the 7 imaginaries that, with the real axis, make an associative subalgebra, either quaternions H, in the case of O, or M2(R), in the case of O". Call such a set of 3 an Associative Triple. Call the other 4 imaginaries a Coassociative Square. FOR O THE SIGNATURE OF THE ASSOCIATIVE TRIPLE IS DIFFERENT FROM THE COASSOCIATIVE SQUARE; FOR O" THE SIGNATURE OF THE ASSOCIATIVE TRIPLE IS THE SAME AS THE COASSOCIATIVE SQUARE.
The quadratic form of the Split Octonions is the Symmetric Quadratic Form - X0^2 - X1^2 - X2^2 - X3^2 + Y0^2 + Y1^2 + Y2^2 + Y3^2 of the real Clifford Algebra Cl(4,4). If p = q, the real Clifford Algebra Clf(p,q) is called a Split Clifford Algebra. If the vector space has a mixed signature, in which neither p nor q is zero, the minimum of p and q is the Witt Index. Let, for example, p be less than q and q = p + k. Then the vector space R^(p,q) = R^(p,p+k) of the real Clifford Algebra Cl(p,p+k) can be decomposed into two parts: a positive definite part R^k and a split part R^(p,p). In turn, the split part R^(p,p) can be decomposed into two null subspaces N^p and N'^p each of dimension p. The resulting decomposition of R^(p,p+k) into the direct sum N^p + N'^p + R^k is called a Witt Decomposition. Why are the spaces N^p and N'^p called null subspaces? You can choose a basis { X1, ... , Xp } for N^p and a basis { Y1, ... , Yp } be for N'^p such that the plane spanned by { Xi, Yi } is a Hyperbolic plane for all i from 1 to p. Therefore, the Witt Decomposition decomposes the vector space R^(p,q) = R^(p,p+k) into p copies of the Hyperbolic plane plus a k-dimensional definite space (in the example, positive definite). The number of copies of the Hyperbolic plane is the Witt Index. Some examples of the Witt Index in physics are: For Example, begin with R^2 of U(1) = Spin(2) of Cl(0,2) with quadratic form + Y1^2 + Y2^2 then add one Hyperbolic Plane to get the quadratic form - X0^2 + Y0^2 + Y1^2 + Y2^2 of Spin(1,3) of Cl(1,3) of Witt Index 1 of the Lorentz group of Minkowski space, and then add a second Hyperbolic Plane (and renumber the basis) to get the quadratic form - X0^2 - X1^2 + Y0^2 + Y1^2 + Y2^2 + Y^3^2 of Spin(2,4) of Cl(2,4) of Witt Index 2 of the Conformal Group and Conformal Space. For Another Example, begin with R^24 (the 24-dimensional Euclidean space of the Leech lattice) of Spin(24) of Cl(0,24) with quadratic form + Y1^2 + ... + Y24^2 then add one Hyperbolic Plane to get the quadratic form - X0^2 + Y0^2 + Y1^2 + ... + Y24^2 of Spin(1,25) of Cl(1,25) of Witt Index 1 of the 26-dimensional space of the Lorentz Leech lattice that can be used to construct a representation space for the largest finite simple group, the Monster group. The 26-dimensional space is also used in superstring theory.
begin with the Split real Clifford Algebra Cl(p,p) based on
a Symmetric Inner Product IPsym(X,Y) = IPsym(Y,X) =
- X1 Y1 - ... - Xp Yp + X(p+1) Y(p+1) + ... + X(2p) Y(2p)
which gives the Cl(p,p) quadratic form QF(X,X) =
- X1^2 - ... - Xp^2 + X(p+1)^2 + ... + X(2p)^2
then consider the
Skew-Symmetric Inner Product IPskew(X,Y) = - IPskew(Y,X) =
X1 Y2 - X2 Y1 + ... + X(2p-1) Y(2p) - X(2p) Y(2p-1)
Notice that
the space spanned by the p odd-numbered basis elements
{ X1, X3, ... , X(2p-1) }
is a p-dimensional null space
because IPskew(X,Y) restricted to that space is zero,
and
the space spanned by the p even-numbered basis elements
{ X2, X4, ... , X(2p) }
is an orthogonal p-dimensional null space
because IPskew(X,Y) restricted to that space is zero.
The p pairs of basis elements
{ X1, X2 } , ... , { X(2p-1), X(2p) }
each form a 2-real-dimensional Symplectic Plane
somewhat analogous to
a 2-real-dimensional Hyperbolic Plane
in a Witt Decomposition of the vector space of
a Split Clifford Algebra Cl(p,p).
Any Symplectic Space is just the product of
a number of Symplectic Planes.
Therefore,
any Symplectic Space has even dimension.
One way is to notice that each Symplectic Plane
has a Skew-Symmetric Inner Product
IPskew(X,Y) = - IPskew(Y,X) = X1 Y2 - X2 Y1
that produces a skew Quadratic Form
QFskew(X,X) = X1 X2 - X2 X1
that is the commutator of the basis elements { X1, X2 }
Therefore, if you want
a non-trivial represention of the Symplectic Plane
in terms of a real Division Algebra
and you also want to have an associative representation
so that you can make a matrix representation of any order,
then
you cannot use the commutative Division Algebras
(the Real numbers R or the Complex Numbers C)
and
you cannot use the non-associative Octonions O,
so
you must represent the Symplectic Plane
in terms of the associative non-commutative Quaternions Q.
Since the Quaternions Q are 4-real-dimensional,
they are not a 1-1 representation of
a 2-real-dimensional Symplectic Plane,
so that
the Quaternions represent a pair of Symplectic Planes,
and
the fundamental Quaternionic Symplectic Space
is 4-real-dimensional.
The Quaternionic representation of the Skew Inner Product
is the n-quaternionic dimensional (4n-real-dimensional)
Quaternionic Skew-Hermitian IPskewH(X,Y) =
X1* j Y1 + ... + Xn* j Yn
where * denotes quaternionic conjugation
and j is the element j of the quaternionic basis {1,i,j,k}
that defines the standard isomorphism Q = C + Cj = C^2
relating the Quaternions Q to 2-complex-dimensional C^2.
Generalized rotations in the n-quaternionic-dimensional
space produce the Cn series of Sp(n) Lie groups, which have
the fibration Sp(n+1) / Sp(n)xS3 = QPn, Quaternionic Projective n-space.
Note that the 3-sphere S3 = Sp(1) = SU(2) = Spin(3).
Another way to represent Symplectic Spaces is
to work directly with the real Symplectic Planes,
starting with the Skew-Symmetric Inner Product, usually
with the basis reordered from the ordering used above,
so that IPskew(X,Y) = - IPskew(Y,X) =
X1 Y(p+1) - X2(p+1) Y1 + ... + Xp Y(2p) - X(2p) Yp
which can be written in terms of
the basis { X1, ... , Xp; X(p+1), ... , X(2p) }
and the /\ product of exterior algebra as
X1 /\ Y(p+1) + ... + Xp /\ Y(2p)
Since the p-dimensional exterior algebra has 2^p elements,
with graded structure
0 1 2 k p-2 p-1 p
1 p p(p-1)/2 ... p!/(k!(p-k)!) ... p(p-1)/2 p 1
such that the top line is the grade
and the bottom line is
the dimensionality of the space of elements of that grade,
and since,
if we relable the basis { X1, ... , Xp; X(p+1), ... , X(2p) }
as { U1, ... , Up; V1, ... , Vp }
each set { U1, ... , Up } and { V1, ... , Vp }
spans a p-dimensional null space of the Symplectic Space.
Every p-dimensional null space of the Symplectic Space
is a null space of maximal dimension.
It is transverse to at least one of the 2^p null spaces.
Two subspaces of a vector space are defined to be transverse iff their sum is the whole vector space.
Two p-dimensional spaces of R^2p are transverse iff their intersection is zero.
For example, the null spaces spanned by
{ U1, ... , Up } and { V1, ... , Vp }
are transverse to each other.
Unlike p-dimensional Euclidean space,
in which any k-dimensional space can be rotated
into any other k-dimensional space,
although
any nonzero vector in Symplectic Space can
be symplectically rotated into any other nonzero vector,
it is NOT true
that
any nonzero Symplectic Plane can be symplectically
rotated into any other nonzero Symplectic Plane.
This approach seems more abstract to me
than the Quaternion approach,
but this abstract approach is very useful
in constructing such things as
Poisson brackets of Hamiltonians
based on phase spaces,
or cotangent bundles to configuration manifolds.
Since Symplectic Spaces are always of even dimension 2p,
similar physics applications in odd dimensions (2p-1)
can be constructed as contact structures,
based on the set of contact elements,
or (p-1)-dimensional subspaces of
the p-dimensional tangent spaces
of the p-dimensional configuration manifold.
The space of contact elements has dimensionality
p + -1 = 2p - 1
because p coordinates are needed to specify
the point of contact
and p-1 coordinates to specify which subspace.
Such physics application are described in
Mathematical Methods of Classical Mechanics, 2nd. ed.,
by V. I. Arnold, Springer-Verlag 1989;
and
Classical Mathematical Physics,
Dynamical Systems and Field Theories, 3rd. ed.,
by Walter Thirring and translated by Evans M. Harrell II,
Springer 1997.
The Lie group of rotations
of 7-dimensional real Euclidean space is SO(7),
while the rotation group in 7-dimensional real space
with signature (---++++) is SO(4,3).
SO(4,3) is to SO(7)
as
the split octonions O" are to the octonions O,
Now look at the automorphism groups of O and of O".
The automorphism group of the octonions O
is the 14-dimensional exceptional Lie group G2
G2 is a subgroup of SO(Im(O)) = SO(7)
and
the (also 14-dimensional) automorphism group
of the split octonions O" can be denoted by G"2.
G"2 is a subgroup of SO(Im(O")) = SO(4,3)
In particular,
G2 and G"2 are the subgroups of SO(7) and SO(4,3)
that preserve Associative Triples.
You can also look at SO(3,4) from the point of view
of the Clifford algebra Cl(4,3) of
the real 7 dimensional vector space of signature 4,3.
It gives the covering group Spin(4,3) of SO(4,3).
Signature can make a difference with respect
to Clifford algebras. As matrix algebras:
Cl(0,8) = Cl(8,0) = Cl(4,4) = M16(R),
the 16x16 real matrices,
but
Cl(7,0) = M8(C),
the 8x8 complex matrices,
Cl(0,7) = M8(R) + M8(R),
the sum of two sets of 8x8 real matrices
and
Cl(3,4) = M8(C),
the 8x8 complex matrices,
Cl(4,3) = M8(R) + M8(R),
the sum of two sets of 8x8 real matrices.
However, (see page 146 of
the book Clifford Algebras and the Classical Groups,
by Ian Porteous, Cambridge 1995), when you go from
the Clifford algebra Cl(p,q)
to
the Spin group Spin(p,q)
that is defined by that Clifford algebra,
you find that
Spin(p,q) = Spin(q,p) ,
so that Spin(3,4) = Spin(4,3)
and it makes sense
to talk about Spin(7) = Spin(7,0) = Spin(0,7)
Be careful to realize that,
although Cl(0,7) = Cl(4,3) as matrix algebras,
the difference in their underlying vector space signatures
means that Spin(7) is not the same as Spin(3,4).
Now look at 7-spheres.
For octonions O, define
S7 = { x in O : |x| = 1 }
For split octonions O", define
S7+ = { x in O" : |x| = +1 }
S7- = { x in O" : |x| = -1 }
and
S7+- as the union of S7+ and S7-
Just as there is a fibration
Spin(7) / G2 = S7
there is a fibration
Spin(4,3) / G"2 = S7+
Just as there is a fibration
Spin(8) / Spin(7) = S7
there is a fibration
Spin(4,4) / Spin(4,3) = S7+-
Just as there is a fibration
Spin(9) / Spin(7) = S15
there is a fibration
Spin(4,5) / Spin(4,3) = M15
where
M15 = { x in R(8,8) : |x| = -1 or |x| = -1 }
where R8,8 is real 16-dim space
with signature (8,8).
References:
Spinors and Calibrations, by F. Reese Harvey, Academic Press 1990;
e-mail conversations with Onar Aam;
Clifford Algebras and the Classical Groups,
by Ian Porteous, Cambridge 1995;
e-mail conversation with Scott Chang;
Representation of Compact Lie Groups,
by Theodor Broecker and Tammo tom Dieck, Springer-Verlag 1985;
Mathematical Methods of Classical Mechanics, 2nd. ed.,
by V. I. Arnold, Springer-Verlag 1989;
Classical Mathematical Physics,
Dynamical Systems and Field Theories, 3rd. ed.,
by Walter Thirring and translated by Evans M. Harrell II,
Springer 1997.
Notation here is not quite conventional,
because of ASCII limitations.
I use the signature convention of Porteous,
rather than that of Harvey.
......