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1) A moment frozen in time, NASA Mars Exploration Rover Mission, http://marsrovers.nasa.gov/gallery/press/spirit/20050610a.html
This week I'll talk about Dynkin diagrams, quivers and Hall algebras. But first, some cool identities!
My student Mike Stay did computer science before he came to UCR. When he was applying, he mentioned a result he helped prove, which relates Goedel's theorem to the Heisenberg uncertainty principle:
2) C. S. Calude and M. A. Stay, From Heisenberg to Goedel via Chaitin, International Journal of Theoretical Physics, 44 (2005), 1053-1065. Also available at http://math.ucr.edu/~mike/
Now, this particular combination of topics is classic crackpot fodder. People think "Gee, uncertainty sounds like incompleteness, they're both limitations on knowledge - they must be related!" and go off the deep end. So I got pretty suspicious until I read his paper and saw it was CORRECT... at which point I definitely wanted him around! The connection they establish is not as precise as I'd like, but it's solid math.
So, now Mike is here at UCR working with me on quantum logic and quantum computation using ideas from category theory. In his spare time, he sometimes fools around with math identities and tries to categorify them - see "week184" and "week202" if you don't know what that means. Anyway, maybe that's how he stumbled on this:
3) Jonathan Sondow, A faster product for π and a new integral for ln(π/2), Amer. Math. Monthly 112 (2005), 729-734. Also available as math.NT/0401406.
In this paper, Sondow gives eerily similar formulas for some of our favorite math constants. First, one for e:
2 1/1 22 1/2 23 × 4 1/3 24 × 44 1/4
e = ( - ) ( ----- ) ( ------- ) ( ------------ ) ...
1 1 × 3 1 × 33 1 × 36 × 5
Then, one for π/2:
π 2 1/2 22 1/4 23 × 4 1/8 24 × 44 1/16 - = ( - ) ( ------ ) ( --------- ) ( ------------- ) ... 2 1 1 × 3 1 × 33 1 × 36 × 5
Then one for eγ, where γ is Euler's constant:
γ 2 1/2 22 1/3 23 × 4 1/4 24 × 44 1/5
e = ( - ) ( ----- ) ( ------- ) ( ----------- ) ...
1 1 × 3 1 × 33 1 × 36 × 5
He also points out Wallis' product for π/2 and Pippenger's for e:
π 2 1/1 2×4 1/1 4×6×6×8 1/1 - = ( - ) ( --- ) ( ------- ) ... 2 1 3×3 5×5×7×7
2 1/2 2×4 1/4 4×6×6×8 1/8
e = ( - ) ( --- ) ( ------- ) ...
1 3×3 5×5×7×7
What does it all mean? I haven't a clue! Another mystery thrown
down to us by the math gods, like a bone from on high... we can merely
choose to chew on it or not, as we wish.
Julie Bergner gave a great talk on "derived Hall algebras" at the Mac Lane memorial conference. I just want to explain the very first result she mentioned, due to Ringel - a surprising trick for constructing certain quantum groups from simply-laced Dynkin diagrams. It's very different from the usual method for getting quantum groups from Dynkin diagrams, and it's a miracle that it works.
But, I guess I should start near the beginning!
Way back in 1995, in "week62", "week63", "week64" and "week65", I explained how "Dynkin diagrams" - little gizmos like this:
o
|
o--o--o--o--o--o--o
show up all over mathematics. They have a strange way of tying
together subjects that superficially seem completely unrelated.
In one sense people understand how they work, but in another sense
they're very puzzling - their power keeps growing in unexpected ways.
I love mysterious connections, so as soon I understood enough about Dynkin diagrams to appreciate them, I became fascinated by them, and I've been studying them ever since. I explained their relation to geometry in "week178", "week179", "week180", "week181" and "week182", and their relation to quantum deformation and combinatorics in "week186" and "week187".
You might think that would be enough - but you'd be wrong, way wrong!
I haven't really talked about the most mysterious aspects of Dynkin diagrams, like their relation to singularity theory and representations of quivers. That's because these aspects were too mysterious! I didn't understand them at all. But lately, James Dolan and Todd Trimble and I have been making some progress understanding these aspects.
First, I should remind you how Dynkin diagrams infest so much of mathematics. Let's start with a little puzzle mentioned in "week182".
Draw n dots and connect some of them with edges - at most one edge between any pair of dots, please:
o
/ \
/ \
o o-----o-----o-----o-----o
/
/
o o-----o
Now, try to find a basis of Rn
consisting of one unit vector per dot,
subject to these rules: if two dots are connected by an edge, the angle
between their vectors must be 120 degrees, but otherwise their vectors
must be at right angles.
This sounds like a silly puzzle that only a mathematician could give a hoot about. It takes a while to see its magnificent depth. But anyway, it turns out you can solve this problem only for certain special diagrams called "simply-laced Dynkin diagrams". The basic kinds are called An, Dn, E6, E7, and E8.
The An Dynkin diagram is a line of n dots connected by edges like this:
o---o---o---o---oThe Dn diagram has n dots arranged like this:
o
/
o---o---o---o---o
\
o
A line of them but then a little fishtail at the end! We should
take n to be at least 4, to make the diagram connected and different
from An.
The E6, E7, and E8 diagrams look like this:
o o o
| | |
o--o--o--o--o o--o--o--o--o--o o--o--o--o--o--o---o
You're also allowed to take disjoint unions of the above diagrams.
So, a weird problem with a weird answer! Its depth is revealed only when we see that many different puzzles lead us to the same diagrams. For example:
A) the classification of integral lattices in Rn having a basis of vectors whose length squared equals 2
B) the classification of simply laced semisimple Lie groups
C) the classification of finite subgroups of the 3d rotation group
D) the classification of simple singularities
E) the classification of tame quivers
Let me run through these problems and say a bit about how they're connected:
A) An "integral lattice" in Rn is a lattice where the dot product of any two vectors in the lattice is an integer. There are zillions of these - but if we demand that they have a basis of vectors whose length squared is 2, we can only get them from simply-laced Dynkin diagrams.
It's not very hard to see that finding a lattice like this is equivalent to the puzzle I mentioned earlier. For example, given a solution of that puzzle, you can just multiply all your vectors by √2 and form the lattice of their integer linear combinations.
Here are the lattices you get:
x1 + ... + xn+1 = 0.
For example, A2 is a 2-dimensional hexagonal lattice, the sort you use to pack pennies as densely as possible. Similarly, A3 gives a standard way of packing grapefruit.
x1 + ... + xn
is even. To visualize this, just take an n-dimensional checkerboard, color the cubes alternately red and black, and take the center of each red cube.
x1 + ... + x8
is even.
B) Lie groups are fundamental throughout math and physics: they're groups of continuous symmetries, like rotations. The nicest of the lot are the semisimple Lie groups. Some familiar examples are the group of rotations in n-dimensional space, which is called SO(n), and the group of unitary matrices with determinant 1, which is called SU(n). There are more, but people know what they all are. They're classified by Dynkin diagrams!
Why? The key point is that any semisimple Lie group has a "root lattice". This is an integral lattice spanned by special vectors called "roots". I won't give the details, since I explained this stuff in "week63" and "week64", but it turns out that root lattices, and thus semisimple Lie groups, are classified by Dynkin diagrams.
Not all these Dynkin diagrams look like the A, D and E diagrams listed above. But, it turns out that the length squared of any root must be either 1 or 2. If all the roots have length squared equal to 2, we say our semisimple Lie group is "simply laced". In this case, we're back to problem B), which we already solved! So then our Lie group corresponds to a diagram of type A, D, or E - or a disjoint union of such diagrams.
Here's how it goes:
In short, two regular series and three exotic weirdos.
You may ask where the rotation groups SO(n) with n odd went! Well, these correspond to fancier Dynkin diagrams that aren't simply laced, like this:
o---o---o---o=>=o
The funny arrow here indicates that the last two vectors aren't at a 120-degree angle; they're at a 135-degree angle, and the last vector is shorter than the rest: it has length one instead of √2.
Why are semisimple Lie groups "better" when they're simply laced? What's the big deal? I don't really understand this, but for one, when all the roots have the same length, they're all alike - a certain symmetry group called the Weyl group acts transitively on them.
Anyway, so far our A, D, E Dynkin diagrams have been classifying things that are clearly related to lattices. But now things get downright spooky....
C) Take a Platonic solid and look at its group of rotational symmetries. You get a finite subgroup of the 3d rotation group SO(3). But in general, finite subgroups of SO(3) are classified by ADE Dynkin diagrams!
So, Platonic solids turn out to fit into the game we're playing here!
First I'll say which diagram corresponds to which subgroup of SO(3). Then I'll explain how the correspondence works:
So, the exceptional Lie groups E6, E7 and E8 correspond to Platonic solids in a sneaky way.
To understand what's going on here, first we need to switch from SO(3) to SU(2). The group SU(2) is used to describe rotations in quantum mechanics: it's the double cover of the rotation group SO(3).
It's really finite subgroups of SU(2) that are classified by ADE Dynkin diagrams! It just so happens that these correspond, in a slightly slippery way, to finite subgroups of SO(3).
You'll see how if I list the finite subgroups of SU(2):
Now, how does the correspondence work? For this, I'm afraid I have to raise the sophistication level a bit - I've been trying to keep things simple, but it's getting tough.
In his book on the icosahedron, Felix Klein noticed it was interesting to let the icosahedral group act on the Riemann sphere, and look for rational functions invariant under this group.
It turned out that every such function depends on a single one: Klein's icosahedral function! The explict formula for it is pretty disgusting, but it's a beautiful thing: you can pick it so that it equals 0 at all the vertices of the icosahedron, 1 at the midpoints of the edges, and infinity at the midpoints of the faces. Even better, if you write the function like this:
w = f(z)
then Klein showed that knowing how to solve for z as a function of w lets you solve every quintic equation! The reason is that the Galois group of the general quintic is a close relative of the icosahedral group: the former is S5, the latter is A5.
Anyway, when I said that "every such function depends on a single one", what I really meant was this. Let C(z) be the field of rational functions of one variable; then the icosahedral group acts on this, and the invariant functions form a subfield C(w) where w is Klein's icosahedral function. The Galois group of the little field in the big one is the icosahedral group.
The same kind of thing works for the other finite subgroups of SO(3), except of course for the connection to the quintic equation.
But, it's actually even better to think about finite subgroups of SU(2), since SU(2) acts on C2, and when we projectivize C2 we get SO(3) acting on the Riemann sphere. This viewpoint fits more squarely into the worldview of algebraic geometry.
If we take the quotient of C2 by a finite subgroup G of SU(2), we don't get a smooth manifold: the quotient has a singularity at 0. But we can "resolve" the singularity, finding a smooth complex manifold with a holomorphic map
p: M → C2/G
that has a holomorphic inverse on a dense open set. There may be lots of ways to do this, but in the present case there's just one "minimal" resolution, meaning a resolution that every other resolution factors through.
Then - and here's the magic part! - the inverse image of 0 in M turns out to be the union of a bunch of Riemann spheres. And if we draw a dot for each sphere, and an edge between these dots whenever their spheres intersect, we get a simply laced Dynkin diagram on the above list!!!
Well, almost. We get this diagram with an extra dot thrown in, connected by some extra edges in a specific way. This is called the "extended" Dynkin diagram. It also shows up naturally from the Lie group viewpoint, when we consider central extensions of loop groups.
That's one way the correspondence works. Another way, discovered by McKay, is to draw a dot for each irrep of G. There's always a 2-dimensional representation of G coming from the action of SU(2) on C2. Let's just call this irrep C2. Then, draw an edge from the dot R to the dot S whenever the irrep S shows up in the rep R ⊗ C2. You get the same extended Dynkin diagram as before! The special extra dot in the Dynkin diagram corresponds to the trivial rep of G.
This second way is called the "McKay correspondence". The first way is sometimes called the "geometric McKay correspondence", though I think it was discovered earlier.
Now we're well on the road to the next item...
D) Simply-laced Dynkin diagrams also classify the simple critical points of holomorphic functions
f: C3 → C
A "critical point" is just a place where the gradient of f vanishes. We can try to classify critical points up to a holomorphic change of variables. It's better to classify their "germs", meaning we only look at what's going on right near the critical point. But, even this is hopelessly complicated unless we somehow limit our quest.
To do this, we can restrict attention to "stable" critical points, which are those that don't change type under small perturbations. But we can do better: we can classify "simple" critical points, namely those that change into only finitely many other types under small perturbations.
These correspond to simply-laced Dynkin diagrams!
First I'll say which diagram corresponds to which type of critical point. To do this, I'll give a polynomial f(x,y,z) that has a certain type of critical point at x = y = z = 0. Then I'll explain how the correspondence works:
Here's how the correspondence works. For each of our Dynkin diagrams we have a finite subgroup of SU(2), thanks to item C). This subgroup acts on the ring of polynomials on C2, so we can form the subring of invariant polynomials. This turns out to be generated by three polynomials that we will arbitrarily call x, y, and z. But, they satisfy one relation, given by the polynomial above!
Conversely, we can start with the polynomial
f: C3 → C
The zero set
{f = 0}
has an isolated singularity at the origin. But, we can resolve this singularity, finding a smooth complex manifold N with a holomorphic map
q: N → {f = 0}
that has a holomorphic inverse on a dense open set. There may be lots of ways to do this, but in the present case there's just one "minimal" resolution, meaning one that every other resolution factors through this one.
Then - and here's the magic part! - the inverse image of 0 in N turns out to be the union of a bunch of Riemann spheres. And if we draw a dot for each sphere, and an edge between these dots whenever their spheres intersect, we get back our simply laced Dynkin diagram!!!
This whole section should have given you a feeling of deja vu. It's a lot like section C). If I were smarter, I'd probably see how it's exactly the same stuff, repackaged slightly.
The last item on our list seems different....
E) A quiver is just a category freely generated by some set of morphisms. To specify a quiver we just write down some dots and arrows. The dots are the objects of our category; the arrows are the generating morphisms.
A representation of a quiver Q is just a functor
F: Q → Vect
So, we get a vector space for each dot and a linear map for each arrow, with no extra restrictions. There's an obvious category of representations Rep(Q) of a quiver Q.
A guy named Gabriel proved a divine result about these categories Rep(Q). We say a quiver Q has "finite representation type" if Rep(Q) has finitely many indecomposable objects - objects that aren't direct sums of others. And, it turns out the quivers of finite representation type are just those coming from simply-laced Dynkin diagrams!!
Actually, for this to make sense, you need to take your Dynkin diagram and turn it into a quiver by putting arrows along the edges. If you have an ADE Dynkin diagram, you get a quiver of finite representation type no matter which way you let the arrows point.
There's clearly a lot of mysterious stuff going on here. In particular, this last item sounds completely unrelated to the rest. But it's not! There are cool relationships between quivers and quantum groups, which tie this item to the rest.
I'll just mention one - the one Julie Bergner started her talk with.
For this, you need to know a bit about abelian categories.
Abelian categories are categories like the category of abelian groups, or more generally the category of modules of any ring, where you can talk about chain complexes, exact sequences and stuff like that. You can see the precise definition here:
4) Abelian categories, Wikipedia, http://en.wikipedia.org/wiki/Abelian_category
and learn more here:
5) Peter Freyd, Abelian Categories, Harper and Row, New York, 1964. Also available at http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html
It's really interesting to study the "Grothendieck group" K(A) of an abelian category A. As a set, this consists of formal linear combinations of isomorphism classes of objects of A, where we impose the relations
[a] + [b] = [x]
whenever we have a short exact sequence
0 → a → x → b → 0
It becomes an abelian group in an obvious way.
For example, if A is the category of representations of some group G, it's an abelian category and K(A) is called the "representation ring" of A - it's a ring because we tensor representations. Or, if A is the category of vector bundles over a space X, it's again abelian, and K(A) is called the "K-theory of X".
The Hall ring H(A) of an abelian category is a vaguely similar idea. As a set, this consists of formal linear combinations of isomorphism classes of objects of A. No extra relations! It's an abelian group with the obvious addition. But the cool part is, with a little luck, we can make it into a ring by letting the product [a] [b] be the sum of all isomorphism classes of objects [x] weighted by the number of isomorphism classes of short exact sequences
0 → a → x → b → 0
This only works if the number is always finite.
So far when speaking of "formal linear combinations" I've been implicitly using integer coefficients, but people seem to prefer complex coefficients in the Hall case, and they get something called the "Hall algebra" instead of the "Hall ring".
The fun starts when we take the Hall algebra of Rep(Q), where Q is a quiver. We could look at representations in vector spaces over any field, but let's use a finite field - necessarily a field with q elements, where q is a prime power.
Then, Ringel proved an amazing theorem about the Hall algebra H(Rep(Q)) when Q comes from a Dynkin diagram of type A, D, or E:
5) C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583-592.
He showed this Hall algebra is a quantum group! More precisely, it's isomorphic to the q-deformed universal enveloping algebra of a maximal nilpotent subalgebra of the Lie algebra associated to the given Dynkin diagram.
That's a mouthful, but it's cool. For example, the Lie algebra associated to An is sl(n+1), and the maximal nilpotent subalgebra consists of strictly upper triangular matrices. We're q-deforming the universal enveloping algebra of this. One cool thing is that the "q" of q-deformation gets interpreted as a prime power - something we've already seen in "week185" and subsequent weeks.
So, it seems that all the ways simply-laced Dynkin diagrams show up in math are related. But, I don't think anyone understands what's really going on! It's like black magic.
And, I've just described some of the black magic!
For example, you'll notice I portrayed the Hall algebra H(A) as a kind of evil twin of the more familiar Grothendieck group K(A). They have some funny relations. For example, if you take the minimal resolution of C2/G where G is a finite subgroup of SU(2), you get a variety whose K-theory (as defined above) is isomorphic to the representation ring of G! This was shown here:
6) G. Gonzalez-Springberg and J. L. Verdier, Construction geometrique de la correspondance de McKay, Ann. ENS 16 (1983), 409-449.
For further developments, try this paper, which studies the derived category of coherent sheaves on this minimal resolution of C2/G:
7) Mikhail Kapranov and Eric Vasserot, Kleinian singularities, derived categories and Hall algebras, available as math.AG/9812016.
Now let me give a bunch of references for further study. For a really quick overview of the whole ADE business, try these:
8) Andrei Gabrielov, Coxeter-Dynkin diagrams and singularities, in Selected Papers of E. B. Dynkin with Commentary, eds. A. A. Yushkevich, G. M. Seitz and A. I. Onishchik, AMS, 1999. Also available at http://www.math.purdue.edu/~agabriel/dynkin.pdf
9) John McKay, A rapid introduction to ADE theory, http://math.ucr.edu/home/baez/ADE.html
Here's a more detailed but still highly readable introduction:
10) Joris van Hoboken, Platonic solids, binary polyhedral groups, Kleinian singularities and Lie algebras of type A,D,E, Master's Thesis, University of Amsterdam, 2002, available at http://math.ucr.edu/home/baez/joris_van_hoboken_platonic.pdf
This classic has recently become available online:
11) 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. Also available at http://repos.project.cwi.nl:8888/cwi_repository/docs/I/10/10039A.pdf or http://math.ucr.edu/home/baez/hazewinkel_et_al.pdf
Here's a really nice, elementary introduction to Klein's work on the icosahedron and the quintic:
12) Jerry Shurman, Geometry of the Quintic, Wiley, New York, 1997. Also available at http://people.reed.edu/~jerry/Quintic/quintic.html
I haven't seen this book, but I hear it's good:
13) P. Slodowy, Simple Singularities and Algebraic Groups, Lecture Notes in Mathematics 815, Springer, Berlin, 1980.
Here's a bibliography with links to online references:
14) Miles Reid, Links to papers on McKay correspondence, http://www.maths.warwick.ac.uk/~miles/McKay/
Of those references, I especially like this:
15) Miles Reid, La Correspondence de McKay (in English), Seminaire Bourbaki, 52eme annee, November 1999, no. 867, to appear in Asterisque 2000. Also available as math.AG/9911165.
Here you'll also see some material about generalizations of the McKay correspondence. For example, if we take a finite subgroup G of SU(3), we get a quotient C3/G, which has singularities. If we take a "crepant" resolution of
p: M → C3/G,
which is the right generalization of a minimal resolution, then M is a Calabi-Yau manifold. This gets string theory into the act! Around 1985, Dixon, Harvey, Vafa and Witten used this to guess that the Euler characteristic of M equals the number of irreps of G. A lot of work has been done on this since then, and Reid's article summarizes a bunch.
Apparently a "crepant" resolution is one that induces an isomorphism of canonical bundles; when this fails to happen folks say there's a discrepancy, so a crepant resolution is one with no dis-crepancy. Get it? Since a Calabi-Yau manifold is one whose canonical bundle is trivial, it shouldn't be completely shocking that crepant resolutions yield Calabi-Yaus. This all works in the original 2d McKay correspondence, too - the minimal resolutions we saw there are also crepant.
In fact, string theory also sheds light on the original McKay correspondence. The reason is that the minimal resolution of C2/G is a very nice Riemannian 4-manifold (when viewed as a real manifold). It's an "asymptotically locally Euclidean" manifold, or ALE manifold for short. Doing string theory on this gives a way of seeing how the extended Dynkin diagrams sneak into the McKay correspondence: they're the Dynkin diagrams for central extensions of loop groups, which show up as gauge groups in string theory! I don't really understand this, but it makes a kind of sense.
I guess this is a famous paper about this stuff:
16) Michael R. Douglas and Gregory Moore, D-branes, quivers and ALE instantons, available as hep-th/9603167.
Graham Leuschke wrote: