Thanks to John Baez and Dave Rusin for pointing out that this page is a non-rigorous, non-technical attempt at answering the question ONLY for compact real forms of complex simple Lie groups, such as groups of rotations acting on spheres, for which a complete classification is known.
There are a lot of Lie groups that are NOT compact real forms of complex simple Lie groups. For instance, the real line with the action of translation is a non-compact Lie group, and solvable Lie groups are certainly not simple groups. An example of a solvable Lie group is the nilpotent Lie group that can be formed from the nilpotent Lie algebra of upper triangular NxN real matrices.
So, when you read this page, be SURE to realize that when I say "Lie group", that is my shorthand for "compact real form of a complex simple Lie group", and similar shorthand is being used when I say "Lie algebra".
As it will turn out that the Lie groups I will discuss are closely related to the division algebras, I will note that you can find a lot about the division algebras on Dave Rusin's division algebra fact page.
A group is just a set of things (numbers, vectors, octonions, whatever) with a multiplication law a times b = ab that is associative (a times b) times c = a times (b times c) (octonions are non-associative, but associative multiplication laws can be defined on sets of octonionic elements to make Lie groups) an identity 1 such that a times 1 = 1 times a = a and an inverse a^(-1) such that a times a^(-1) = 1 (I have written it as multiplication, but I could also have written it as addition: a times b = a + b because both addition and multiplication are group operations.) A manifold is just a continuous geometrical object, such a 2-dim plane (such as the complex numbers), a 4-dim space (such as the quaternions), an 8-dim space (such as the octonions), a circle of radius 1 in the complex plane (1-sphere S1), a 3-dim surface at radius 1 in the quaternions (3-sphere S3), or a 7-dim surface at radius 1 in the octonions (7-sphere S7). By continuous, I mean that for any given point p in the manifold, there are other points in the manifold that are as close to it as you want. (If you tell me you want a point within 1/10,000 (of the unit distance) of p, I can find one and show it to you. Same for 1/1,000,000 or any other number, no matter how small you choose it to be.) As Dave Rusin has commented, you also should require that a manifold be locally isomorphic to a Euclidean space because intersecting lines do not make nice manifolds at the point of intersection, and you should either require a manifold to be Hausdorff or make it clear that you are only dealing with metric spaces, which are automatically Hausdorff.
Now - What is a Lie group? A Lie group is a manifold that is also a group - that is, for any two points a and b in the manifold, there is a multiplication a times b = ab and the group product operation is consistent with the continuous structure of the manifold - that is, if two more points c and d are close to a and b, respectively, then the product cd is close to the product ab. It might not sound like much of a restriction for a manifold to be a Lie group - all you need is a product such that if a is close to c and b is close to d then ab is close to cd However, very few structures (manifold + group) are Lie groups. They were only classified about 100 to 80 years ago, mostly by Sophus Lie (Norwegian) and Wilhelm Killing (German) and Elie Cartan (French). In addition to the Lie groups of translations in n-dimensional space, there are 4 series of Lie groups and they are related to symmetric spaces such as projective spaces and spheres: A series - unitary transformations in n-dimensional complex space CPn = SU(n+1) / S(U(n)xU(1)) sphere S(2n+1) = SU(n+1) / SU(n); B series - rotations in odd-dimensional real space which has sphere of even dimension sphere S(2n) = SO(2n+1) / SO(2n); C series - transformations in n-dimensional quaternion space QPn = Sp(n+1) / Sp(n)xSp(1) sphere S(4n+3) = Sp(n+1) / Sp(n); and D series - rotations in even-dimensional real space which has sphere of odd dimension sphere S(2n+1) = SO(2n+2) / SO(2n+1). The Bn and Dn are real rotations, denoted Spin(2n+1) and Spin(2n), and are called Spin groups, the double covers of special Orthogonal groups; the An are complex generalized rotations, denoted SU(n+1), and are called special Unitary groups; and the Cn are quaternionic generalized rotations, denoted Sp(n), and are called Symplectic groups. (I wish they had used the order A B C D instead of B D A C, but, they did not. See a web page about the origin of the Lie group denotations by Johan E. Mebius.) The only other Lie groups that exist are 5 exceptional ones: G2, F4, E6, E7, and E8 (For history of their notations, see a web page about the origin of the Lie group denotations by Johan E. Mebius.) You should not be surprised about two facts: the exceptional Lie groups are all related to the octonions; and they do not form an infinite series because the non-associativity of the octonions terminates the series. G2 is the automorphism group of the octonions, that is, the group of operations on the octonions that preserve the octonion product. G2 is 14-dimensional and its smallest non-trivial representation is 7-dimensional. F4 is the automorphism group of 3x3 matrices of octonions o11 o12 o13 o21 o22 o23 o31 o32 o33 such that o11, o22, and o33 are real (have no imaginary part), and o12, o13, o23 are the octonion conjugates of o21, o31, o32 respectively. (Such matrices are called Hermitian matrices. In the above example, they form a subspace of a Jordan algebra.) F4 is 52-dimensional and its smallest non-trivial representation is 26-dimensional. E6 is in some sense F4 expanded by the complex numbers. E6 is 78-dimensional and its smallest non-trivial representation is 27-dimensional. E7 is in some sense F4 expanded by the quaternions. E7 is 133-dimensional and its smallest non-trivial representation is 56-dimensional. E8 is in some sense F4 expanded by the octonions. E8 is 248-dimensional and its smallest non-trivial representation is also 248-dimensional. AND THESE ARE ALL THE LIE GROUPS THAT EXIST.
The numbers n of the An, Dn, Bn, and Cn series, and the numbers of E6, E7, E8, and G2 and F4, denote the Rank of the Lie Group, which is the dimension of its Maximal Torus, or, from the point of view of its Lie Algebra, the the dimension of its Maximal Abelian Cartan Sub-Algebra. It is also the dimension of the Euclidean space of its Root Vector Diagram whose symmetries determine its Weyl Group.
Lie, Killing, and Cartan first classified Complex Simple Lie Algebras (and therefore Lie Groups) by looking at their Root Vector Diagrams. In 1999, roughly a century later, J.M. Landsberg and Laurent Manivel classified Complex Simple Lie Algebras (and therefore Lie Groups) using a different technique, by looking at the Projective Geometry of homogeneous varieties. Their proof is constructive: they build a homogeneous space X from a smaller space Y via a rational map P defined using the ideals of the secant and tangential varieties of Y. They first construct a preferred class of homogeneous varities (which they call miniscule varieties), and from them they construct all the fundamental adjoint varieties (for which the adjoint representation of the Lie Algebra is fundamental), and then they prove that there are no more adjoint varieties, except for the two exceptional cases of the An and Cn Lie Algebras (which they also construct). The An and Cn Lie Algebras are exceptional in the Landsberg-Manivel Projective Geometric Classification because the adjoint representation is not a fundamental representation for the An and Cn Lie Algebras.
Luis J. Boya has written a beautiful paper at math-ph/0212067 entitled Problems in Lie Group Theory and here are a few of the interesting things he says:
"... Given a Lie group in a series G(n) ... how is the group G(n+1) constructed?
For the orthogonal series (Bn and Dn) ... given O(n) acting on itself, that is, the adjoint (adj) representation, and the vector representation, n, ... Adj O(n) + Vect O(n) -> Adj O(n+1) ...
For the unitary series SU(n) ... Adj SU(n) + Id + n + n* = Adj SU(n+1) ...
For the symplectic series Sp(n) = Cn ... Adj Sp(n) + Adj Sp(1) + 2( n + n* ) = Adj Sp(n+1) ...
For G2 ... Adj SU(3) + n + n* -> G2 ...[ in addition, I conjecture the existence of an alternate construction: Adj O(4) + Vect O(4) + Spin O(4) = G2 , where Spin O(4) is its Spin representation, a notation that I will continue to use in the rest of this quotation instead of the notation Spin(4) that Boya uses, because I want to reserve the notation Spin(4) for the covering group of SO(4). Note that Spin O(n) for even n is reducible to two copies of mirror image half-spinor representations half-Spin O(n) ]...
For the exceptional groups, the F4 & E series ...
Notice that 8+1 , 8+2 , 8+4 , and 8+8 appear. In this sense the octonions appear as a "second coming " of the reals, completed with the spin, not the vector irrep. ... This confirms that the F4 E6-7-8 corresponds to the octo, octo-complex, octo-quater and octo-octo birings, as the Freudenthal Magic Square confirms. ...
Another ... question ... is the geometry associated to the exceptional groups ... Are we happy with G2 as the automorphism group of the octonions, F4 as the isometry of the [octonion] projective plane, E6 (in a noncompact form) as the collineations of the same, and E7 resp. E8 as examples of symplectic resp. metasymplectic geometries? ... one would like to understand the exceptional groups ... as automorphism groups of some natural geometric objects. ...
The gross topology of Lie groups is well-known. The non-compact case reduces to compact times an euclidean space (Malcev-Iwasawa). The compact case is reduced to a finite factor, a Torus, and a semisimple compact Lie group. H. Hopf determined in 1941 that the real homology of simple compact Lie groups is that of a product of odd spheres .. The exponents of a Lie group are the numbers i such that S(2i+1) is an allowed sphere ...
neither the U-series nor the Sp-series have torsion. The exponents ... for U(n) ... are 0, 1, ... , n-1 ... and jump by two in Sp(n).
But for the orthogonal series one has to consider some Stiefel manifolds instead of spheres, which the same real homology ... It ... introduces (preciesely) 2-torsion: in fact, Spin(n), n>7 and SO(n), n>3, have 2-torsion. The low cases Spin(3,4,5,6) coincide with Sp(1), Sp(1)xSp(1), Sp(2) and SU(4) , and have no torsion.
For ... G2 ... SU(2) -> G2 -> M11 ... where M11 is again a Steifel manifold, with real homology like S11, but with 2-torsion ...
For F4 we do not get the sphere structure from any irrep, and in fact F4 has 2- and 3-torsion. ...
2- and 3-torsion appears in ... E6 and E7 ...
E8 has 2-, 3- and 5-torsion ... The Coxeter number of (dim - rank) of E8 is 30 = 2 x 3 x 5 , in fact a mnemonic for the exponents of E8 is: they are the coprimes up to 30, namely (1,7,11,13,17,19,23,29) ... The first perfect numbers are 6, 28, and 492, associated to the primes 2, 3 and 5 (... Mersenne numbers ...) ... 496 = dim O(32) = dim E(8) x E(8) . Why the square? It also happens in O(4) , dim = 6 (prime 2), as O(4) ...[is like]... O(3) x O(3) ; even O(8) [dim = 28] (prime 3) is like S7 x S7 x G2 ...
The sphere structure of compact simple Lie groups has a curious "capicua" ... Catalan word ( cap i cua 0 = head and tail ) ... form: the exponents are symmetric from each end; for example ...
The real homology algebra of a simple Lie group is a Grassmann algebra, as it is generated by odd (i.e., anticommutative) elements. However, from them we can get, in the enveloping algebra, multilinear symmetric forms, one for each generator; ... in physics they are called Casimir invariants, in mathematics the invariants of the Weyl group. ...".
Since Lie Groups are Manifolds that act (by group multiplication) on themselves and Since rotations take spheres into themselves, We can ask: WHICH SPHERES ARE LIE GROUPS? To answer this, first find out which rotations can be group multiplications. The only ones are rotations of spheres in spaces of the division algebras. The real number sphere is 0-dimensional and discrete, so we don't consider it. That leaves: the complex numbers; the quaternions; and the octonions. The A series contains the complex rotations in the unit circle, S1, and S1 is a Lie group. The B and C series both contain the quaternion rotations on the unit sphere, S3, and S3 is a Lie group. The D series contains the Lorentz group in 4-dim space, consisting of two copies of S3 (3 rotations and 3 boosts).
Note that in some sense all nonablelian Lie groups can be constructed from nonabelian S3. Roughly, you can take as many copies of S3 as the rank of the Lie group, and then add additional root vectors according to the symmetry of the Weyl group.
HOWEVER, the exceptional Lie groups do NOT include S7,
because octonion non-associativity forces S7 to expand,
so that S7 is the only unit sphere in a division algebra that
is not a Lie group,
a fact that is related to the fact that
the only parallelizable spheres are S1, S3, and S7.
To what Lie group does S7 expand?
S7 expands to the twisted product of S7 x S7 x G2,
which is the 4+24 = 28-dimensional D4 Lie group Spin(8).
Spin(8) is the spin covering of the rotations in 8-dimensional space,
the space of the octonions.
The D4 Lie group Spin(8) lives in BOTH:
the standard series Lie groups, as D4;
and
the exceptional octonion Lie groups.
Therefore,
you would expect Spin(8) to be a very special Lie group,
and it is.
So much so,
that it is the basis of my D4-D5-E6-E7-E8 VoDou Physics model.
What are Lie algebras?
Can the generators of S7 form a Lie algebra?
A Lie algebra is a logarithm of a Lie group,
and
a Lie group is an exponential of a Lie algebra.
Lie algebras are flat vector spaces with a
bracket product that takes a times b to (1/2)(ab - ba)
Since ab - ba is a measure of non-commutativity,
define the commutator [a,b] = (1/2)(ab - ba)
The Lie algebra must transform by exponentiation into a Lie group.
It is important to note that the formula
exp(A) exp(B) = exp(A+B)
holds only for a commutative Lie algebra - U(1)
As the Lie algebras get more complicated,
there are correction terms, known as
the the Baker-Campbell-Hausdorff formula.
The first approximation (first-order series term)
for exp(A) exp(B) is
exp(A) exp(B) = exp( A + B )
The second-order approximation for exp(A) exp(B) is
exp(A) exp(B) = exp( A + B + (1/2)[A,B] )
....
(see Varadarajan page 97, Theorem 2.12.4 (i), also)
The Lie algebra must have a basis of invariant vector fields
that is taken by exponentiation into the space of
left-invariant 1-forms on the Lie group.
Such left-invariant 1-forms are called the Maurer-Cartan forms.
Let {Z1,Z2,...,Zn} be the Maurer-Cartan 1-forms
of an n-dimensional Lie group.
The exterior derivative d of the exterior algebra of
forms on the Lie group takes the Maurer-Cartan 1-forms
into 2-forms as follows:
d Zp = -(1/2) SUM(q,r) Fpqr Zq /\ Zr
(where Fpqr are the structure constants that
determine the commutators of the Lie algebra).
Since the exterior derivative d is nilpotent of order 2,
that is, since dd = 0, the identities dd Zp = 0
are true for all p=1,2,...,n.
Since the exterior derivative of the Maurer-Cartan 1-forms {Zp}
is determined by the Lie algebra structure constants Fpqr
the identities dd Zp = 0 can also be expressed in
terms of the Lie algebra structure constants.
So expressed, they give for the Lie algebra
the JACOBI IDENTITY:
J(a,b,c) = [a,[b,c]] + [b,[c,a]] + [c,[a,b]] = 0
J(a,b,c) = a(bc - cb) - (bc - cb)a +
b(ca - ac) - (ca - ac)b +
c(ab - ba) - (ab - ba)c = 0
J(a,b,c) = a(bc) - a(cb) - (bc)a + (cb)a +
b(ca) - b(ac) - (ca)b + (ac)b +
c(ab) - c(ba) - (ab)c + (ba)c = 0
J(a,b,c) = a(bc) - (ab)c + (cb)a - c(ba) +
b(ca) - (bc)a + (ac)b - a(cb) +
c(ab) - (ca)b + (ba)c - b(ac) = 0
Since a(bc)-(ab)c is a measure of non-associativity,
define the associator [a,b,c] = a(bc) - (bc)a
J(a,b,c) = [a,b,c] - [c,b,a] +
[b,c,a] - [a,c,b] +
[c,a,b] - [b,a,c] = 0
An algebra is called an Alternative algebra
if its associator [a,b,c] is alternating function of a,b,c
that is, if [a,b,c] = -[c,b,a] = -[a,c,b] = -[b,a,c]
For Alternative algebras, we have that
J(a,b,c) = 2[a,b,c] + 2[b,c,a] + 2[c,a,b] = 6[a,b,c]
For all associative Alternative algebras,
the commutator algebra is a Lie algebra.
The octonion algebra is an Alternative algebra,
but since it is non-associative the imaginary octonions
do not form a Lie algebra because J(a,b,c) = 6[a,b,c] =/= 0
CAN THE IMAGINARY OCTONION ALGEBRA BE EMBEDDED IN A LIE ALGEBRA?
To define a Lie algebra that includes the imaginary octonions,
start with the imaginary octonions {i,j,k,E,I,J,K},
and, for definiteness, use the following multiplication table
(out of the 480 multiplications):
i j k E I J K
i -1 k -j I -E -K J
j -k -1 i J K -E -I
k j -i -1 K -J I -E
E -I -J -K -1 i j k
I E -K J -i -1 -k j
J K E -I -j k -1 -i
K -J I E -k -j i -1
An example of octonion non-associativity is
[i,j,E] = (1/2)(i(jE) - (ij)E) =
= (1/2)(iJ - kE) =
= (1/2)(-K -K) = -K =/= 0
The corresponding example of violation of the Jacobi identity is
J(i,j,E) = 6[i,j,E] = -12K
To construct a larger Lie algebra that can be projected
into the imaginary octonion commutator algebra,
start with the commutator [i,j] = (1/2)(ij - ji) = k
Then add to it a new independent term [ij] (without the comma)
such that the projection of [ij] into
the 7-dimensional space of imaginary octonions is k
to get, in the larger Lie algebra,
[i,j] = [ij]
Do the same thing for all 7x7=49 commutators of {i,j,k,E,I,J,K}
with the rule that [ab] = -[ba]
so that there are only (7x6)/2 = 21 independent new elements:
[ij] [ik] [iE] [iI] [iJ] [iK]
[jk] [jE] [jI] [jJ] [jK]
[kE] [kI] [kJ] [kK]
[EI] [EJ] [EK]
[IJ] [IK]
[JK]
As is suggested by this upper triangular arrangement,
the 21 new elements can be given commutator product rules
for commutators of the form [[ab],[cd]]
that are the commutators of 7x7 antisymmetric real matrices,
which form the 21-dimensional Lie algebra of Spin(0,7),
the covering group of the rotation group in 7-dim space.
Spin(0,7) can be decomposed by a fibration
into a 7-sphere S7 and the exceptional Lie group G2.
To define the commutator product rules
for commutators of the form [a,[bc]]
with one term from the 7 imaginary octonions and
the other term from the 21 independent new terms,
write all 7+21=28 of them together as
i [ij] [ik] [iE] [iI] [iJ] [iK]
j [jk] [jE] [jI] [jJ] [jK]
k [kE] [kI] [kJ] [kK]
E [EI] [EJ] [EK]
I [IJ] [IK]
J [JK]
K
and give them commutator product rules
that are commutators of 8x8 antisymmetric real matrices,
which form the 28-dimensional Lie algebra of Spin(0,8),
the covering group of the rotation group in 8-dim space.
Spin(0,8) can be decomposed by two fibrations
into two 7-spheres and the exceptional Lie group G2,
so that Spin(0,8) = S7 x S7 x G2
(where x = twisted fibre product).
Therefore:
the Spin(8) Lie algebra is
the Lie algebra expansion of
the imaginary octonion commutator algebra.
The structure Spin(0,8) = S7 x S7 x G2 can be seen
in the sedenions and in the design of the Temple of Luxor.
NOW WE CAN LOOK AT THE COMMUTATOR ALGEBRAS
OF THE SPHERES S1, S3, and S7:
Complex S1 [S1,S1] = 0 S1 COLLAPSES!
is a Lie algebra.
Quaternion S3 [S3,S3] = S3 S3 IS STABLE!
is a Lie algebra.
Octonion S7 [S7,S7] = S7xS7xG2 = Spin(0,8) S7 EXPANDS!
is (x=twisted fibration product)
NOT a Lie algebra
because
it does NOT satisfy
the Jacobi identity.
We have seen that
the 7-dim imaginary octonion commutator algebra
lives inside the 28-dim Lie algebra of Spin(0,8)
and that
it is not a Lie algebra
It belongs to the class of algebras called Malcev algebras. For more about Malcev algebras and related S7 structures, see, for example, this page and the following papers by Martin Cederwall et al:
Now we can ask, what kind of group is
formed by the corresponding 7-dim subspace
of the 28-dim Lie group Spin(0,8)?
That subspace is a 7-sphere S7,
the unit sphere in the 8-dim space of octonions.
S7 is not a 7-dim Lie group,
because the corresponding 7-dim algebra is not a Lie algebra.
However:
S7 IS a 7-dim manifold; and
S7 HAS a multiplication taking a times b into ab
such that for all a,X,Y in S7,
a(X(aY)) = ((aX)a)Y
a(X(YX)) = ((aX)Y)X
(aX)(Ya) = a((XY)a) = (a(XY))a = a(XY)a
These identities are the Moufang identities,
so that S7 can be called a Moufang loop.
From the identity (aX)(Ya) = a(XY)a
it is clear that
if we take Y = X^(-1)
we get (aX)(X^(-1)a) = a(X X^(-1))a = aa
However,
it is NOT true that (aX)(Yb) = a(XY)b
or that (aX)(X^(-1)b) = a(X X^(-1))b = ab
This is the basis of the definitions of
the S7 X-product by Cederwall et al
and the S7 XY-product by Dixon.
This construction of Spin(8) from S7 is related to the fact that the structure constants of a Lie algebra correspond to the Torsion of its Lie group manifold.
Intuitively, you can see that the S7 Moufang loop product is expanded by the X-product to include a 7-dim "spherical loop" S7 parameter space for the parameter X. For many years (see Kane, The Homology of Hopf Spaces, North-Holland 1988) S7 (and the real projective space RP7) were known to be interesting loop spaces that were not compact Lie groups. In 1951, Serre (Ann. Math. 54 (1951) 425-505) developed the concept of H-spaces to have an abstract structure that could be used to study compact Lie groups, S7, and RP7 together. However, such discoveries as the Hilton-Roitberg criminal (Hilton and Roitberg, Ann. Math. 90 (1969) 91-107; Stasheff, Bull. Amer. Math. Soc. 75 (1969) 998-1000; and Zabrodsky, Invent. Math. 16 (1972) 260-266) of different types of H-spaces showed that H-spaces included other things as well. Since I can construct the D4-D5-E6-E7 physics model by using as building blocks Lie groups, S7, and RP7, I do not say much about the more abstract H-space structures.
S7 has two homogeneous Einstein Riemannian metrics, one canonical one and one related to Sp(2)xSp(1) and the squashed 7-sphere. According to Einstein Manifolds (Arthur L. Besse, Springer-Verlag 1987, at page 259, "... Theorem (W. Ziller) ... The homogeneous Einstein Riemannian metrics on spheres and projective spaces ... (up to a scaling factor) ... S15 has 3 homogeneous Einstein Riemannian metrics, CP(2q+1) and S(4q+3) (q=/=3) have 2 homogeneous Einstein Riemannian metrics and the other ones [ S(2m), S(4q+1), CP(2m), QPq, OP2 ] ... have only one homogeneous Einstein Riemannian metric. ...".All S7 structures are not H-spaces. As Jim Lin (jimlin@euclid.ucsd.edu) said in 1995: "... the 7 sphere bundle over the 15 sphere could not be an h-space. the proof uses secondary cohomology operations and the factorization of Sq(16) thru these operations. ...".
Even though, as (asimov@nas.nasa.gov) Daniel A. Asimov wrote in 1993 on sci.math.research, "... If n is of the form n = 8*k - 1, then S^n admits a continuous field of tangent 7-planes. (See N. Steenrod, Topology of Fibre Bundles, sections 20 and 27.), so that, in particular, S^23 admits a continuous field of tangent 7-planes ...", S^23 is NOT the total space of a fibre bundle whose fibre is S^7. (See sci.math.research article <1993Feb16.205507.26378@nas.nasa.gov>, (asimov@nas.nasa.gov) Daniel A. Asimov wrote: "... is S^23 the total space of a fibre bundle whose fibre is S^7 ? ...", to which Geoffrey Mess (geoff@math.ucla.edu) replied: "... No. For then attaching a 24-ball by the bundle projection to the base of the bundle would yield a space with integer cohomology ring Z[x]/(x^4) having a single generator in dimension 8. Adams showed that Z[x]/(x^4) with x of degree 8 cannot occur as an integer cohomology ring. The proof is (reputedly-I haven't read it) very difficult, using secondary or tertiary cohomology operations. ...".).
As stated in Manifolds All of Whose Geodesics Are Closed (Arthur L. Besse, Springer-Verlag 1978, at page 5): "... The basic reason for ... [... the nonexistence of a Hopf fibration S23 -> OP2 with fiber S7 which would allow us to define OP2 as a suitable quotient of S23 ...] is the fact that the system of Cayley numbers [Octonions O] ... is not associative. ...".
The only Hopf fibrations that exist are
- S0 -> Sn -> RPn,
- S1 -> S(2n+1) -> CPn,
- S3 -> S(4n+3) -> QPn, and
- S7 -> S15 -> S8 = OP1
There are no fibrations S7 -> S23 -> OP2 or S15 -> S31 -> SedenionP1
and
there is no OP3.
Another way to describe Spin(8) is based on Clifford Algebras: Spin(8) is the Lie Group whose Lie Algebra is the commutator algebra of bivectors of the real Clifford Algebra Cl(8) with basis elements: G0 G1 G2 G3 G4 G5 G6 G7 These 8 real basis elements form an 8-real-dimensional representation space for Spin(8). Denote the basis of the complex numbers {1,i}. The 8-real-dimensional basis of Cl(8) can be rewritten as a 4-complex dimensional basis: G0-iG1 G2-iG3 G4-iG5 G6-iG7 These 4 basis elements form a 4-complex-dimensional representation space for the SU(4) subgroup of Spin(8). Denote the basis of the quaternions {1,i,j,k}. The 8-real-dimensional basis of Cl(8) can be rewritten as a 2-quaternionic dimensional basis: G0-iG1-jG2-kG3 G4-iG5-jG6-kG7 These 2 basis elements form a 2-quaternionic-dimensional representation space for the Sp(2) subgroup of Spin(8). Denote the basis of the octonions {1,i,j,k,E,I,J,K}. The 8-real-dimensional basis of Cl(8) can be rewritten as a 1-octonionic dimensional basis: G0-iG1-jG2-kG3-EG4-IG5-JG6-KG7 This 1 basis element forms a 1-octonionic-dimensional representation space for an S7 subset of Spin(8). The S7 subset of Spin(8) is acted upon by the G2 subgroup of Spin(8). Notice that the 7-sphere S7 is not a Lie algebra, but if you extend it to make a Lie algebra, you get Spin(8), which has an 8-real-dimensional representation space, that corresponds to the 1-octonionic-dimensional space. This construction generalizes as follows: Spin(2n) has a SU(n) subgroup; Spin(4n) has a Sp(n) subgroup; by Periodicity of Clifford Algebras (similar to Periodicity of Homotopy Groups) Spin(8n) corresponds to the n-fold tensor product Spin(8) x...x Spin(8); By the Lie algebra magic square constructions, the exceptional Lie algebras F4, E6, E7, and E8 are constructed from Spin(8) and Spin(2x8) = Spin(16). Therefore, in a sense all Lie algebras can be constructed from the fundamental Spin(n) Lie algebras, which in turn can be constructed from Clifford Algebras, which in turn can be constructed from Set Theory, thus showing that the D4-D5-E6-E7-Ei VoDou Physics model can be constructed from Set Theory.
OCTONION FRACTALS show two kinds of fractal structure: ordinary z to zz + c additive structure; and non-associative octonion X-product and XY-product multiplicative structure. It seems to me that: octonion X- and XY-product structure is a logarithm of z to zz + c structure; and z to zz + c structure is an exponential of octonion X- and XY-product structure.
Kac and Smilga, in hep-th/9908096, say:
"... According to Dynkin ... one has a bijective correspondence
between conjugacy classes of distinguished nilpotent elements of g
and distinguished markings of the Dynkin diagram of g. ...
... The full classification of distinguished markings for all algebras
was done in ... E.B. Dynkin, Mat. Sbornik 30 (1952) 349 ...[and]
... P. Bala and R.W. Carter, Math. Proc. Camb. Phil. Soc. 79 (1976) 3
... Translating it into our physical language gives immediately
the following result:
Theorem 6. The number of inequivalent by conjugation solutions ...
n[G] ... with gauge group G is
(a) n[SU(n)] = 1 ...
(b) n[Sp(2r)] coincides with
the number of partitions of r into distinct parts ...
(c) n[SO(n)] coincides with
the number of partitions of n into distinct odd parts ...
(d) n[G2] = 2,
n[F4] = 4,
n[E6] = 3,
n[E7] = 6, and
n[E8] = 11. ...".
Here, I have used the notation n[G] instead of the
notation #vac[G} used by Kac and Smilga (whose notation
#vac is motivated by their interest in the physics of
supersymmetric vacua, as opposed to pure mathematics).
Here is a table of all n[G] through n[G] = 2 through 11:
--------------------------------------------------------
n[G] Gauge group G
2 SO(8), SO(9), SO(10), SO(11), Sp(6), Sp(8), G2
3 SO(12), SO(13), SO(14), Sp(10), E6
4 SO(15), Sp(12), F4
5 SO(16), SO(17), SO(18), Sp(14)
6 SO(19), Sp(16), E7
7 SO(20)
8 SO(21), SO(22), Sp(18)
9 SO(23)
10 Sp(20)
11 SO(24), E8
--------------------------------------------------------
As I see it, n[G] is the number of
distinct marked Dynkin diagrams of a given Lie algebra G.
For example,
in ascii with x = marked o = unmarked,
there are two D4 marked diagrams
x x | | x-x-x x-o-x
I think that it is interesting to look at Figure 4 on page 19 of the paper of Kac and Smilga hep-th/9908096:
which shows the distinguished marked Dynkin diagrams of all the exceptional groups G2, F4, E6, E7, and E8. I particularly like the way that you can see how the E8 diagrams (11 of them) are the 6 E7 diagrams with one marked vertex added at the end plus 5 "new for E8" diagrams. the E7 diagrams (6 of them) are the 3 E6 diagrams with one marked vertex added at the end plus 3 "new for E7" diagrams You could also see (not shown on Figure 4) that the E6 diagrams (3 of them) are the 2 D5 diagrams with one marked vertex added at the end plus 1 "new for E6" diagram the D5 diagrams (2 of them) are just the 2 D4 diagrams with one marked vertex added at the end Here, in ascii with x = marked o = unmarked, are the two D4 marked diagrams (see pages 16 and 17 of Kac and Smilga):
x x | | x-x-x x-o-x
I like all this because those groups make up the D4-D5-E6-E7-E8 chain in the VoDou Physics model.
Here is some material about SYMMETRIC SPACES.
Here is a page about how LIE GROUPS come from FINITE REFLECTION (WEYL) GROUPS.
Here is a web page about Graded Lie Algebras.
Some references and acknowledgements: Thanks to Ben Bullock (ben@theory.kek.jp) for pointing out that a group needs an inverse, and without it you just have a monoid; Differential Geometry, Gauge Theories, and Gravity, by Gockeler and Schucker, Cambridge 1987; Nonassociative Algebras in Physics, by Lohmus, Paal, and Sorgsepp, Hadronic Press 1994; Topological Geometry, 2nd ed, (new edition to be titled Clifford Algebras and the Classical Groups) by Porteous, Cambridge 1981. J. Math. Phys. 14 (1973) 1651-1667, by Gunaydin and Gursey. Lie Groups, Lie Algebras, and Their Representations, by V. S. Varadarajan, Springer Grad. Text Math. No. 102, 1984. Reflection Groups and Coxeter Groups, by Humphreys, Cambridge 1990. Introduction to Lie Algebras and Representation Theory, by Humphreys, Springer-Verlag 1972. Groupes et Algebres de Lie Chapitres 1; 2 et 3; 4, 5 et 6; 7 et 8; 9 by Bourbaki Edward Dunne has some nice WWW pages about some Lie Groups and related structures, such as Hermitian Symmetric Spaces, E8, F4, and SU(2,2). HERE IS AN EXCELLENT FREE BOOK: Semi-Simple Lie Algebras and their Representations by Robert N. Cahn. The original publisher of the 1984 book, Benjamin-Cummings, gave him permission to put the entire book on his WWW site in the form of freely downloadable postscript files. Thanks to both Robert Cahn and Benjamin-Cummings for making such good material freely available to everybody. The Lie groups G2, F4, E6, E7, and E8 are described in the book Lectures on Exceptional Lie Groups by J. F. Adams, published posthumously by Un. of Chicago Press in 1996, edited by Zafer Mahmud and Mamoru Mimura.
Predrag Cvitanovic has written a web book on Group Theory. In its introduction, he says: "... This monograph offers a derivation of all classical and exceptional semi-simple Lie algebras through a classification of "primitive invariants" ... the invariant tensors are represented by diagrams ... many groups and group representations are mutually related by interchanges of symmetrizations and anti-symmetrizations and replacement of the dimension parameter n by -n. I call this phenomenon "negative dimensions". ...". He has also written a web book on Chaos and a web book on Field Theory. He says in the Epilogue of his web book on Group Theory: "... once I learned that chaos is generic for generic Hamiltonian flows, I lost faith in doing field theory by pretending that it is a bunch of harmonic oscillators, with interactions accounted for as perturbative corrections. This picture is simply wrong - strongly coupled field theories (hydrodynamics, QCD, gravity) are nothing like that ... So they excommunicated me from the ranks of high energy theorists, and now ... we are working out quantum chaos ... this is ... replace the path integral with a fractal set of semi-classical orbits. ...".
OY! Barry Simon has written YABOGR! The official title is: Representations of Finite and Compact Groups (AMS 1996) What is YABOGR? Read the Book!
Tony Smith's Home Page ......