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This time I'll finish up talking about "ADE classifications" for a while, although there is certainly more to say. Recall where we were: the following diagrams correspond to the simple Lie algebras, and they also define certain lattices, the "root lattices" of those Lie algebras:
An, which has n dots like this:
o---o---o---o
Bn, which has n dots, where n > 1:
4
o---o---o->-o
Cn, which has n dots, where n > 2:
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:
4 6
o---o->-o---o o-<-o
The dots in one of these "Dynkin diagrams" correspond to certain set of basis vectors, or "roots", of the lattice. The lines, with their decorative numbers and arrows, give enough information to recover the lattice from the diagram. In particular, two dots that are not connected by a line correspond to roots that are at a 90 degree angle from each other, while two dots connected by an unnumbered line correspond to roots that are at a 60 degree angle from each other. Numbered lines mean the angle between roots is something else, and the arrows point from the longer to the shorter root in this case, as partially explained in "week63".
However, we will now concentrate on the cases A, D, and E, where all the roots are 90 or 60 degrees from each other, and they are all the same length - usually taken to be length 2. These are the "simply laced" Dynkin diagrams. I want to explain what's so special about them! But first, I should describe the corresponding lattices more explicitly, to make it clear how simple they really are.
The following information, and more, can be found in Chapter 4 of:
1) Sphere Packings, Lattices and Groups, J. H. Conway and N. J. A. Sloane, second edition, Grundlehren der mathematischen Wissenschaften 290, Springer, Berlin, 1993.
which I described in more detail in "week20".
So, what are A, D, and E like?
A: We can describe the lattice An as the set of all (n+1)-tuples of integers (x1,...,xn+1) such that
x1 + ... + xn+1 = 0.
It's a fun exercise to show that A2 is a 2-dimensional hexagonal lattice, the sort of lattice you use to pack pennies as densely as possible. Similarly, A3 gives a standard way of packing grapefruit, which is in fact the densest lattice packing of spheres in 3 dimensions. (Hsiang has claimed to have shown it's the densest packing, lattice or not, but this remains controversial.)
D: We can describe Dn as the set of all n-tuples of integers (x1,...,xn) such that
x1 + ... + xn is even.
Or, if you like, you can imagine taking an n-dimensional checkerboard, coloring the cubes alternately red and black, and taking the center of each red cube. In four dimensions, D4 gives a denser packing of spheres than A4; in fact, it gives the densest lattice packing possible. Moreover, D5 gives the densest lattice packing of in dimension 5. However, in dimensions 6, 7, and 8, the En lattices are the best!
E: We can describe E8 as the set of 8-tuples (x1,...,x8) such that the xi are either all integers or all half-integers - a half-integer being an integer plus 1/2 - and
x1 + ... + x8 is even.
Each point has 240 closest neighbors. Alternatively, as described in "week20", we can describe E8 in a slick way in terms of the quaternions. Another neat way to think of E8 is in terms of the octonions! Not too surprising, perhaps, since the octonions and E8 are both 8-dimensional, but it's still sorta neat. For the details, check out
2) Geoffrey Dixon, Octonion X-product and E8 lattices, preprint available as hep-th/9411063.
Briefly, this goes as follows. In "week59" we described a multiplication table for the "seven dwarves" - a basis of the imaginary octonions - but there are lots of other multiplication tables that would also give an algebra isomorphic to the octonions. Given any unit octonion a, we can define an "octonion X-product" as follows:
b X c = (b a)(a* c)
where a* is the conjugate of a (as defined in "week59") and the product on the right-hand side is the usual octonion product, parenthesized because it ain't associative. For exactly 480 choices of the unit octonion a, the X-product gives us a new multiplication table for the seven dwarves, such that we get an algebra isomorphic to the octonions again! 240 of these choices have all rational coordinates (in terms of the seven dwarves), and these are precisely the 240 closest neighbors of the origin in a copy of the E8 lattice! The other 240 have all irrational coordinates, and these are the closest neighbors to the origin of a different copy of the E8 lattice. (Here we've rescaled the E8 lattice so the nearest neighbors have distance 1 from the origin, instead of sqrt(2) as above.)
Once we have E8 in hand, we can get its little pals E7 and E6 as follows. To get E7, just take all the vectors in E8 that are perpendicular to some closest neighbor of the origin. To get E6, find a copy of the lattice A2 in E8 (there are lots) and then take all the vectors in E8 perpendicular to everything in that copy of A2.
So, now that we have a nodding acquaintance with A, D, and E, let me describe some of the many places they show up. First, what's so great about these lattices, apart from the fact that they're the root lattices of simple Lie algebras, with a special "simply-laced" property? I don't think I really understand the answer to this in a deep way, but I know various things to say!
First, Witt's theorem says that the A, D, and E lattices and their direct sums are the only integral lattices having a basis consisting of vectors v with ||v||2 = 2. Here a lattice is "integral" if the dot product of any two vectors in it is an integer. In fact, any integral lattice having a basis consisting of vectors with ||v||2 equal to 1 or 2 is a direct sum of copies of A, D, and E lattices and the integers, thought of as a 1-dimensional lattice.
This makes ADE classifications pop up in various places in math and physics. For example, there is a cool relationship between the ADE diagrams and the symmetry groups of the Platonic solids, called the McKay correspondence. Briefly, this is what you do to get it. First, learn about SO(3) and SU(2) from "week61" or somewhere. Then, take the symmetry group of a Platonic solid, or more generally any finite subgroup G of SO(3). Since SO(3) has SU(2) as a double cover, you can get a double cover of G, say G~, sitting inside SU(2). For example, if G was the symmetry group of the icosahedron, G~ would be the icosians (see "week24").
Since G~ is finite, it has finitely many irreducible representations. Draw a dot for each of the irreducible representations. One of these will be 2-dimensional, coming from the spin-1/2 representation of SU(2). Now, when you tensor this 2d rep with any other irreducible rep R, you get a direct sum of irreducible reps; draw one line from the dot for R to each other dot for each time that other irreducible rep appears in this direct sum. What do you get? Well, you get an "affine Dynkin diagram" of type A, D, or E, which is like the usual Dynkin diagram but with an extra dot thrown in (corresponding to the trivial rep of G~). And, you get all of them this way!
In fact, playing around with this stuff some more, you can get the affine Dynkin diagrams of the other simple Lie algebras. There is a lot more to this... you should probably look at:
3) John McKay, Graphs, singularities and finite groups, in Proc. Symp. Pure Math. vol 37, Amer. Math. Soc. (1980), pages 183- and 265-.
John McKay, Representations and Coxeter Graphs, in "The Geometric Vein" Coxeter Festschrift (1982), Springer-Verlag, Berlin, pages 549-.
John McKay, A rapid introduction to ADE theory, http://math.ucr.edu/home/baez/ADE.html
4) Pavel Etinghof and Mikhail Khovanov, Representations of tensor categories and Dynkin diagrams, preprint available as hep-th/9408078.
McKay does lots of other mindblowing group theory. He's clearly in tune with the symmetries of the universe... and occaisionally he deigns to post to the net! A beautiful way of thinking about the McKay correspondence in terms of category theory appears in the paper by Etinghof and Khovanov; what we are really doing, it turns out, is classifying the representations of the tensor category of unitary representations of SU(2). This tensor category is generated by one object, the spin-1/2 representation, meaning that every other representation sits inside some tensor power of that one. This way of thinking of it is important in
5) Jurg Froehlich and Thomas Kerler, Quantum Groups, Quantum Categories, and Quantum Field Theory, Springer Lecture Notes in Mathematics 1542, Springer-Verlag, Berlin, 1991.
Here Froehlich and Kerler give a classification of certain "quantum categories" that show up in conformal field theory and topological quantum field theory. These are certain braided tensor categories with properties like those of the categories of representations of quantum groups at roots of unity. In such categories, every object has a "quantum dimension", which need not be integral, and Froehlich and Kerler concentrate on those categories which are generated by a single object of quantum dimension less than 2, getting an ADE-type classification of them. The category of representations of SU(2), on the other hand, is generated by a single object of dimension equal to 2 - the spin-1/2 representation - so Froehlich and Kerler are basically generalizing the McKay stuff to certain quantum groups related to SU(2).
Where else do ADE diagrams show up? Well, here I won't try to say anything about their role in the representation theory of "quivers", or in singularity theory; these are covered pretty well in
6) M. Hazewinkel, W. Hesselink, D. Siermsa, and F. D. Veldkamp, The ubiquity of Coxeter-Dynkin diagrams (an introduction to the ADE problem), Niew. Arch. Wisk., 25 (1977), 257-307.
Instead, I'll mention something more recent. In string theory, there is a Lie algebra called the Virasoro algebra that plays a crucial role; its almost just the Lie algebra of the group of diffeomorphisms of the circle, but it's actually just one dimension bigger, being a "central extension" thereof; projective representations of the Lie algebra of the group of diffeomorphisms of the circle correspond to honest representations of the Virasoro algebra. An important task in string theory was to classify the unitary representations of the Virasoro algebra having a given "central charge" c (this describes the action of that one extra dimension) and "conformal weight" h (this describes the action of dilations). It turns out that to get unitary reps one needs c and h to be nonnegative. The representations with c between 0 and 1 are especially nice, for reasons I don't really understand, and they are called "minimal models". An ADE classification of these was conjectured by Capelli and Zuber, and proved by
7) Capelli and Zuber, Comm. Math. Phys. 113 (1987) 1.
8) Kato, Mod. Phys. Lett. A2 (1987) 585.
Friedan, Qiu, and Shenker also played a big role in this, in part by figuring out the allowed values of c. For a good introduction to this stuff and what it has to do with honest physics (which I admit I've been slacking off on here), try:
9) Claude Itzykson and Jean-Michel Drouffe, Statistical Field Theory, 1: From Brownian Motion to Renormalization and Lattice Gauge Theory, and 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems. Cambridge U. Press, 1989.
I will probably come back to this ADE stuff as time goes by, since I'm sort of fascinated by it, and hopefully folks can refer back to the last few weeks when I do, so they'll remember what I'm talking about. But in the next few Weeks I want to catch up with some new developments in math and physics that have happened in the last few months...
© 1995 John Baez
baez@math.removethis.ucr.andthis.edu
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