September 23, 1995

This Week's Finds in Mathematical Physics (Week 64)

John Baez

I have been talking about different "ADE classifications". This time I'll start by continuing the theme of last Week, namely simple Lie algebras, and then move on to discuss affine Lie algebras and quantum groups. These are algebraic structures that describe the symmetries appearing in quantum field theory in 2 and 3 dimensions. They are very important in string theory and topological quantum field theory, and it's largely physics that has gotten people interested in them.

Remember, we began by classifying finite reflection groups. A finite reflection group is simply a finite group of linear transformations of Rn, every element of which is a product of reflections. Finite reflection groups are in 1-1 correspondence with the following "Coxeter diagrams", together with disjoint unions of such diagrams:

An, which has n dots like this:


Bn, which has n dots, where n > 1:


Dn, which has n dots, where n > 3:


E6, E7, and E8:

      o               o                   o
      |               |                   |
o--o--o--o--o   o--o--o--o--o--o    o--o--o--o--o--o---o

F4:                   G2:               H3 and H4:

      4                6                5                5
o---o---o---o        o---o            o---o---o        o---o---o---o

Im, where m = 5 or m > 6:


Not all of these finite reflection groups satisfy the "crystallographic condition", namely that they act as symmetries of some lattice. The ones that do are of types A,B,D,E,F, and G, and disjoint unions thereof --- but I'm going to stop reminding you about disjoint unions all the time!

Now, if we have a finite reflection group that's the symmetries of some lattice, we can take the dimension of the lattice to be the number of dots in the Coxeter diagram. In fact, the dots correspond to a basis of the lattice, and the lines between them (and their numberings) keep track of the angles between the basis vectors. These basis vectors are called "roots". To describe the lattice completely, in principle we need to know the lengths of the roots as well as the angles between them. But it turns out that except for type B, there is just one choice of lengths that works (up to overall scale). For type B there are two choices, which people call Bn and Cn, respectively. People keep track of the lengths with a "Dynkin diagram" like this:

Bn has n dots, where n > 1:       Cn has n dots, where n > 2:

          4                                  4
o---o---o->-o                      o---o---o-<-o

The arrow points to the shorter root; for Bn all the roots except the last one are sqrt(2) times as long as the last one, while for Cn all the roots except the last one are 1 / √2 as long. (In fact, the lattices corresponding to Bn and Cn are "dual", in the hopefully obvious sense.) The only reason why we require n > 2 for Cn is that B2 is basically the same as C2!

Now last Week, I also sketched how the Lie algebras of the compact simple Lie groups were also classified by the same Dynkin diagrams of types A, B, C, D, E, F, and G. These were real Lie algebras; we can also switch viewpoint and work with complex Lie algebras if we like, in which case we simply say we're studying the complex simple Lie algebras, as opposed to their "compact real forms".

Unfortunately, I didn't say much about what these Lie algebras actually are! Basically, they go like this:

An - The Lie algebra An is just sln+1(C), the (n+1) × (n+1) complex matrices with vanishing trace, which form a Lie algebra with the usual bracket [x,y] = xy - yx. The compact real form of sln(C) is sun, and the corresponding compact Lie group is SU(n), the n × n unitary matrices with determinant 1. The symmetry group of the electroweak force is U(1) x SU(2), where U(1) is the 1 × 1 unitary matrices. The symmetry group of the strong force is SU(3). The study of An is thus a big deal in particle physics. People have also considered "grand unified theories" with symmetry groups like SU(5).

Bn - The Lie algebra Bn is so2n+1(C), the (2n+1) × (2n+1) skew- symmetric complex matrices with vanishing trace. The compact real form of son(C) is son, and the corresponding compact Lie group is SO(n), the n × n real orthogonal matrices with determinant 1, that is, the rotation group in Euclidean n-space. For some basic cool facts about SO(n), check out "week61".

Cn - The Lie algebra Cn is spn(C), the 2n × 2n complex matrices of the form

     A  B
     C  D

where B and C are symmetric, and D is minus the transpose of A. The compact real form of spn(C) is spn, and the corresponding compact Lie group is called Sp(n). This is the group of n × n quaternionic matrices which preserve the usual inner product on the space Hn of n-tuples of quaternions.

Dn - The Lie algebra Dn is so2n(C), the 2n × 2n skew-symmetric complex matrices with vanishing trace. See Bn above for more about this. It may seem weird that SO(n) acts so differently depending on whether n is even or odd, but it's true: for example, there are "left-handed" and "right-handed" spinors in even dimensions, but not in odd dimensions. Some clues as to why are given in "week61".

Those are the "classical" Lie algebras, and they are things that are pretty easy to reinvent for yourself, and to get interested in for all sorts of reasons. As you can see, they are all about "rotations" in real, complex, and quaternionic vector spaces.

The remaining ones are called "exceptional", and they are much more mysterious. They were only discovered when people figured out the classification of simple Lie algebras. As it turns out, they tend to be related to the octonions! Some other week I will say more about them, but for now, let me just say:

F4 - This is a 52-dimensional Lie algebra. Its smallest representation is 26-dimensional, consisting of the traceless 3×3 hermitian matrices over the octonions, on which it preserves a trilinear form.

G2 - This is a 14-dimensional Lie algebra, and the compact Lie group corresponding to its compact real form is also often called G2. This group is just the group of symmetries (automorphisms) of the octonions! In fact, the smallest representation of this Lie algebra is 7-dimensional, corresponding to the purely imaginary octonions.

E6 - This is a 78-dimensional Lie algebra. Its smallest representation is 27-dimensional, consisting of all the 3x3 hermitian matrices over the octonions this time, on which it preserves the anticommutator.

E7 - This is a 133-dimensional Lie algebra. Its smallest representation is 56-dimensional, on which it preserves a tetralinear form.

E8 - This is a 248-dimensional Lie algebra, the biggest of the exceptional Lie algebras. Its smallest representation is 248-dimensional, the so-called "adjoint" representation, in which it acts on itself. Thus in some vague sense, the simplest way to understand the Lie group corresponding to E8 is as the symmetries of itself! (Thanks go to Dick Gross for this charming information; I think he said all the other exceptional Lie algebras have representations smaller than themselves, but I forget the sizes.) In "week20" I described a way to get its root lattice (the 8-dimensional lattice spanned by the roots) by playing around with the icosahedron and the quaternions.

People have studied simple Lie algebras a lot this century, basically studied the hell out of them, and in fact some people were getting a teeny bit sick of it recently, when along came some new physics that put a lot of new life into the subject. A lot of this new physics is related to string theory and quantum gravity. Some of this physics is "conformal field theory", the study of quantum fields in 2 dimensional spacetime that are invariant under all conformal (angle-preserving) transformations. This is important in string theory because the string worldsheet is 2-dimensional. Some other hunks of this physics go by the name of "topological quantum field theory", which is the study of quantum fields, usually in 3 dimensions so far, that are invariant under all transformations (or more precisely, all diffeomorphisms).

Every simple Lie algebra gives rise to an "affine Lie algebra" and a "quantum group". The symmetries of conformal field theories are closely related to affine Lie algebras, and the symmetries of topological quantum field theories are quantum groups. I won't tell you what affine Lie algebras and quantum groups ARE, since it would take quite a while. Instead I'll refer you to a good good introduction to this stuff:

1) Juergen Fuchs, Affine Lie Algebras and Quantum Groups, Cambridge Monographs on Mathematical Physics, Cambridge U. Press, Cambridge 1992.

Let me whiz through his table of contents and very roughly sketch what it's all about.

1. Semisimple Lie algebras

This is a nice summary of the theory of semisimple Lie algebras (remember, those are just direct sums of simple Lie algebras) and their representations. Especially if you are a physicist, a slick summary like this might be a better way to start learning the subject than a big fat textbook on the subject. He covers the Dynkin diagram stuff and much, much more.

2. Affine Lie algebras

This starts by describing Kac-Moody algebras, which are certain infinite-dimensional analogs of the simple Lie algebras. Fuchs concentrates on a special class of these, the affine Lie algebras, and describes the classification of these using Dynkin diagrams. The main nice thing about the affine Lie algebras is that their corresponding infinite-dimensional Lie groups are very nice: they are almost "loop groups". If we have a Lie group G, the loop group LG is just the set of all smooth functions from the circle to G, which we make into a group by pointwise multiplication. If you're a physicist, this is obviously relevant to string theory, because at each time, a string is just a circle (or bunch of circles), and if you are doing gauge theory on the string, with symmetry group G, the gauge group is then just the loop group LG. So you'd expect the representation theory of loop groups and their Lie algebras to be really important.

You'd almost be right, but there is a slight catch. In quantum theory, what counts are the "projective" representations of a group, that is, representations that satisfy the rule g(h(v)) = (gh)(v) up to a phase. (This is because "phases are unobservable in quantum theory" - one of those mottoes that needs to be carefully interpreted to be correct.) The projective representations of the loop group LG correspond to the honest-to-goodness representations of a "central extensions" of LG, a slightly fancier group than LG itself. And the Lie algebra of this group is called an affine Lie algebra.

So, people who like gauge theory and string theory need to know a lot about affine Lie algebras and their representations, and that's what this chapter covers. A real heavy-duty string theorist will need to know more about Kac-Moody algebras, so if you're planning on becoming one of those, you'd better also try:

2) Victor Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.

You'll also need to know more about loop groups, so try:

3) Loop groups, by Andrew Pressley and Graeme Segal, Oxford University Press, Oxford, 1986.

3. WZW theories

Well, I just said that physicists liked affine Lie algebras because they were the symmetries of conformal field theories that were also gauge theories. Guess what: a Wess-Zumino-Witten, or WZW, theory, is a conformal field theory that's also a gauge theory! You can think of it as the natural generalization of the wave equation in 2 dimension (which is conformally invariant, btw) from the case of real-valued fields, to general G-valued fields, where G is our favorite Lie group.

4. Quantum groups

When you quantize a WZW theory whose symmetry group G is some simple Lie group, something funny happens. In a sense, the group itself also gets quantized! In other words, the algebraic structure of the group, or its Lie algebra, gets "deformed" in a way that depends on the parameter ħ (Planck's constant). I have muttered much about quantum groups on This Week's Finds, especially concerning their relevance to topological quantum field theory, and I will not try to explain them any better here! Eventually I will discuss a bunch of books that have come out on quantum groups, and I hope to give a mini-introduction to the subject in the process.

5. Duality, fusion rules, and modular invariance

The previous chapter described quantum groups as abstract algebraic structures, showing how you can get one from any simple Lie algebra. Here Fuchs really shows how you get them from quantizing a WZW theory. WZW theories are invariant under conformal transformations, and quantum groups inherit lots of cool properties from this fact. For example, suppose you form a torus by taking the complex plane and identifying two points if they differ by any number of the form n z1 + m z2, where z1 and z2 are fixed complex numbers and n, m are arbitrary integers. For example, we might identify all these points:

                x      x      x      x      x

                x      x      x      x      x

                x      x      x      x      x

The resulting torus is a "Riemann surface" and it has lots of transformations, called "modular transformations". The group of modular transformations is the discrete group SL(2,Z) of 2×2 integer matrices with determinant 1; I leave it as an easy exercise to guess how these give transformations of the torus. (This is an example of a "mapping class group" as discussed in "week28".) In any event, the way the the WZW theory transforms under modular transformations translates into some cool properties of the corresponding quantum group, which Fuchs discusses. That's roughly what "modular invariance" means.

Similarly, "fusion rules" have to do with the thrice-punctured sphere, or "trinion", which is another Riemann surface. And "duality" has to do with the sphere with four punctures, which can be viewed as built up from trinions in either of two "dual" ways:

    0\          /0
     //        \\
    0/          \0


      O    O
      \\  //
      //  \\
      O    O

This is one of the reasons string theory was first discovered - we can think of the above pictures as two Feynman diagrams for interacting strings, and the fact that they are really just distorted versions of each other gives rise to identities among Feynman diagrams. Similarly, this fact gives rise to identities satisfied by the fusion rules of quantum groups.

So - Fuchs' book should make clear how, starting from the austere beauty of the Dynkin diagrams, we get not only simple Lie groups, but also a wealth of more complicated structures that people find important in modern theoretical physics.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. - Bertrand Russell.

© 1995 John Baez