Let me continue the tale of "ADE classifications". Last week I described an "ABDEFGHI classification" of all finite reflection groups - that is, finite symmetry groups of Euclidean space, every element of which is a product of reflections. Now we'll build on that to get other related classifications.
Every element of a finite reflection group is a product of reflections through certain special vectors, which people call "roots". These roots are all at angles π/n from each other, where n > 1 is an integer. To describe the group, we draw a diagram with one dot for each root. If two roots are perpendicular we don't draw a line between them, but otherwise, if they are at an angle π/n from each other, we draw a line and label it with the integer n. Actually, the integer n = 3 comes up so often that we don't bother labelling the line in this case.
Now, not all of these diagrams correspond to finite reflection groups. The following ones, together with disjoint unions of them, are all the possibilities.
An, which has n dots like this: o---o---o---o Bn, which has n dots, where n > 1: 4 o---o---o---o Dn, which has n dots, where n > 3: o / o---o---o---o \ o 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: m o---o
Recall that Im is the symmetry group of the of regular m-gon, while others of these are the symmetry groups of Platonic solids, and still others are symmetry groups of regular polytopes in n-dimensional space. For example, the symmetry group of the dodecahedron is H3, while that of its 4-dimensional relative is H4.
Now you may know that there are no perfect crystals in the shape of a regular dodecahedron. However, iron pyrite comes close. In his wonderful book:
1) Hermann Weyl, Symmetry, Princeton University Press, Princeton, New Jersey, 1989.
Weyl suggests that this is how people discovered the regular dodecahedron:
...the discovery of the last two [Platonic solids] is certainly one of the most beautiful and singular discoveries made in the whole history of mathematics. With a fair amount of certainty, it can be traced to the colonial Greeks in southern Italy. The suggestion has been made that they abstracted the regular dodecahedron from the crystals of pyrite, a sulfurous mineral abundant in Sicily.
Thus while iron pyrite is nothing but "fool's gold" to the miner, it may have done a good deed by fooling the Greeks into discovering the regular dodecahedron. Could this be why the ratio of the diagonal to the side of a regular pentagon, (√5 + 1)/2, is called the golden ratio? Or am I just getting carried away? One is tempted to call the shape of pyrite crystals the "fool's dodecahedron," but in fact it's called a "pyritohedron". (All this information on pyrite, as well as the puns, I owe to Michael Weiss.)
More recently, I think people have discovered "quasicrystals" (related to Penrose tiles) having true dodecahedral symmetry. But no perfectly repetitive crystals form dodecahedra! And the reason is that there is no lattice having H3 as its symmetries.
Recall that we get a "lattice" by taking n linearly independent vectors in n-dimensional Euclidean space and forming all linear combinations with integer coefficients. If someone hands us a finite reflection group, we can look for a lattice having it as symmetries. If one exists, we say the group satisfies the "crystallographic condition". The only ones that do are
An, Bn, Dn, E6, E7, E8, F4, and G2
(and those corresponding to disjoint unions of these diagrams). In other words, the symmetry groups of the pentagon (I5), the heptagon and so on (Im with m > 6), and the dodecahedron and its 4-dimensional relative (H3 and H4) are ruled out.
Now let us turn to the theory of Lie groups. Lie groups are the most important "continuous" (as opposed to discrete) symmetry groups. Examples include the real line (with addition as the group operation), the circle (with addition mod 2π), and the groups SO(n) and SU(n) discussed in "week61". These groups are incredibly important in both physics and mathematics. Thus it is wonderful, and charmingly ironic, that the same diagrams that classify the oh-so-discrete finite reflection groups also classify some of the most beautiful of Lie groups: the "simple" Lie groups. It turns out that the simple Lie groups correspond to the diagrams of forms A,B,D,E,F, and G. Oh yes, and C. I have to tell you what happend to C.
There is a vast amount known about semisimple Lie groups, and everyone really serious about mathematics winds up needing to learn some of this stuff. I took courses on Lie groups and their Lie algebras in grad school, but it was only later that I really came to appreciate the beauty of the simple Lie groups. Back then I found it mystifying because the work involved in the classification was so algebraic, and I preferred the more geometrical aspects of Lie groups. Part of the reason is that the treatment I learned emphasized the Lie algebras and downplayed the groups. A nice treatment that emphasizes the groups is:
2) John Frank Adams, Lectures on Lie groups, Benjamin, New York, 1969.
So what's the basic idea? Let me summarize two semesters of grad school, and tell you the basic stuff about Lie groups and the classification of simple Lie groups. Forgive me if it's a bit rushed, sketchy, and even mildly inaccurate: hopefully the main ideas will shine through the murk this way.
A Lie group is a group that's also a manifold, for which the group operations (multiplication and taking inverses) are smooth functions. This lets you form the tangent space to any point in the group, and the tangent space at the identity plays a special role. It's called the Lie algebra of the group. If we have any element x in the Lie algebra, we can exponentiate it to get an element exp(x) in the group, and we can keep track of the noncommutativity of the group by forming the "bracket" of elements x and y in the Lie algebra:
[x,y] = (d/dt)(d/ds) exp(sx) exp(ty) exp(-sx) exp(-ty)
where s and t are real numbers, and we evaluate the derivative at s,t = 0. Note that [x,y] = 0 if the group is commutative. This bracket operation satisfies some axioms, and algebraists call anything a Lie algebra that satisfies those axioms. For example, you could take n x n matrices and let [x,y] = xy - yx.
Now a Lie algebra is called "semisimple" if for any z, there are x and y with z = [x,y]. This is sort of the opposite of an abelian, or commutative, Lie algebra, where [x,y] = 0 for all x and y. It turns out that we can take direct sums of Lie algebras by defining operations componentwise, and it turns out that if you have a compact Lie group, its Lie algebra is always the direct sum of a semsimple Lie algebra and an abelian one. The abelian ones are pretty trivial, so all the hard works lies in understanding the semisimple ones. Any semisimple one is the direct sum of a bunch of semisimple ones that aren't sums of anything else, and these basic building blocks are called the "simple" ones. They are like the prime numbers of Lie algebra theory. Unlike the prime numbers, though, we can completely classify all of them!
Now how does one classify the simple Lie algebras? Basically, it goes like this. We'll assume our simple Lie algebra is the Lie algebra of a compact Lie group G - it turns out that they all are. Now, sitting inside G there is a maximal commutative subgroup T that's a torus: a product of a bunch of circles. Let Lie(T) stand for the Lie algebra of this torus T. Now, sitting inside Lie(T) there is a lattice, consisting of all elements x with exp(x) = 1. This is how lattices sneak into the picture!
Moreover, for some elements g in G, if we "conjugate" T by g, that is, form the set of all elements gtg-1 where t is in T, we get T back. In other words, these elements of g act as symmetries of the torus T. Now, if something acts as symmetries of something else, it also acts as symmetries of everything naturally cooked up from that something else. (Roughly speaking, "naturally" means "without dependence on arbitrary choices.) For this reason, these special elements of G also act as symmetries of Lie(T) and of the lattice sitting inside Lie(T). So we have a lattice together with a group of symmetries, which by the way is called the Weyl group of G. Now the cool part is that the Weyl group is actually a finite reflection group, so it must correspond to one of the diagrams in the list above! Even better, it turns out that the Lie algebra of G is determined by the lattice together with its Weyl group.
The upshot is that to classify semisimple Lie algebras, all we need is the classification of finite reflection groups satisfying the crystallographic condition - which we've done already using diagrams - together with a classification of lattices having such finite reflection groups as symmetries. It turns out that the operation of taking direct sums of semisimple Lie algebras corresponds to taking disjoint unions of diagrams, so to get the "building blocks" - the simple Lie algebras - we only need to worry about the diagrams we've drawn above, not disjoint unions of them. Now it turns out that for every type except B, there is (up to isomorphism) only one lattice having that group of symmetries, but for B there are two. Recall the diagram Bn looks like:
with n dots. And recall that the dots correspond to "roots", which in the present context are vectors in Lie(T). Now it turns out that whenever we have a finite reflection group satisfying the crystallographic condition, we can get a lattice having it as symmetries by taking integer linear combinations of the roots, but not necessarily roots that are unit vectors; the lengths of the roots matter. In all cases except B, there is basically just one way to get the lengths right, but for B there are two. We can keep track of the root lengths with some extra markings on our diagrams, and then we get two diagrams, which we call Bn and Cn. One of them has the root at the right of the diagram being longer, and one has the root right next to it being longer. This makes no difference when n = 2, since then we just have
which is perfectly symmetrical. So folks usually consider Cn only for n > 2, to avoid double counting.
In other words, all the simple Lie algebras are of the form:
Okay, so what are these things, really? What do they mean, and what are the implications of the fact that the symmetries of the forces of nature are given by the some of the corresponding Lie groups? Why are 4 infinite series of them and 5 "exceptional" Lie algebras? What's so special about A, D, and E, that makes people keep talking about "ADE classifications"? What do the exceptional Lie algebras (and their corresponding Lie groups) have to do with octonions? Why do some string theorists think the symmetry group of nature is E8, the largest exceptional Lie group???
Well, I'm afraid that I'm going camping in a couple of hours, so I'll have to leave you hanging, even though I do know the answers to some of these questions. I'll try to finish talking about ADE classifications in the next couple of issues.
© 1995 John Baez