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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-dimensional quaternion space; and D series - rotations in even-dimensional real space.   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.)   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.  


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.


  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.  
 
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-E7 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.   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).   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 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.  


Here is some material about SYMMETRIC SPACES.


Here is a page about how LIE GROUPS come from FINITE REFLECTION (WEYL) GROUPS.


 
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. 
 
OY!  Barry Simon has written YABOGR!
The official title is: 
Representations of Finite and Compact Groups (AMS 1996)
What is YABOGR? Read the Book!
 

   

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