Supplemental Material
What IS a Lie Group?
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: A series - unitary transformations in n-dimensional complex space; B series - rotations in odd-dimensional real space; C series - transformations in n-dimensiona quaternion space; and D series - rotations in even-dimensional real space. The B and D are real rotations, the A are complex generalized rotations, the C are quaternion generalized rotations. (I wish they had used the order A B C D instead of B D A C, but they did not.) The only other Lie groups that exist are 5 exceptional ones: (I REALLY do not know why these letters) G2, F4, E6, E7, and E8 You should not be surprised about two facts: they 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. 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.) E6 is in some sense F4 expanded by the complex numbers. E7 is in some sense F4 expanded by the quaternions. E8 is in some sense F4 expanded by the octonions. AND THESE ARE ALL THE LIE GROUPS THAT EXIST.
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). 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. To what Lie group does S7 expand? S7 expands to the twisted product of S7 x S7 x G2, which is the D-series Lie group D4 = Spin(0,8). Spin(8) is the spin covering of the rotations in 8-dimensional space, the space of the octonions. The D4 Lie group Spin(0,8) lives in BOTH: the standard series Lie groups, as D4; and the exceptional octonion Lie groups. Therefore, you would expect Spin(0,8) to be a very special Lie group, and it is. So much so, that it is the basis of my D4-D5-E6 physics model.
What are Lie algebras? 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. 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. 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). 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. 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 physics model by using as building blocks Lie groups, S7, and RP7, I do not discuss here the more abstract H-space structures.
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.
Some references: 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. 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.
