
This week I'd like to talk about exceptional Lie algebras and the Standard Model, Witten's new paper on the Monster group and black holes in 3d gravity, and Connes and Marcolli's new book! Then I want to continue the Tale of Groupoidification.
However, I don't have the energy to do this all now. And even if I did, you wouldn't have the energy to read it.
So, I'll just point you towards Connes and Marcolli's new book, which you can download for free:
1) Alain Connes and Mathilde Marcolli, Noncommutative Geometry, Quantum Fields and Motives, available at http://www.alainconnes.org/downloads.html
I hope to discuss it sometime, especially since it tackles a question I've been mulling lately: is there a good "explanation" for the Standard Model of particle physics?
For now, I'll start by discussing Witten's latest paper:
2) Edward Witten, Threedimensional gravity revisited, available as arXiv:0706.3359.
This is a bold piece of work, which seeks to relate the entropy of black holes in 3d quantum gravity to representations of the Monster group  the largest sporadic finite simple group, with about 10^{54} elements.
If the main idea is right, this gives a whole new view of "Monstrous Moonshine"  the bizarre connection between the Monster and fundamental concepts in complex analysis like the jfunction. (See "week66" for a quick intro to Monstrous Moonshine.)
As the title hints, Witten had already tackled quantum gravity in 3 spacetime dimensions. In this earlier work, he argued it was an exactly soluble problem: a topological field theory called ChernSimons theory. However, this theory is really an extension of gravity to the case of "degenerate" metrics: roughly speaking, geometries of spacetime where certain regions get squashed down to zero size. Degenerate metrics are weird. So, what happens if we try to quantize 3d gravity while insisting that the metric be nondegenerate?
It's hard to say. So, Witten takes a few clues and cleverly fits them together to make a surprising guess. He considers 3d general relativity with negative cosmological constant. This has 3d antiDeSitter space as a solution. AntiDeSitter space has a "boundary at infinity": a 2d cylinder with a conformal structure. The "AdSCFT" idea, also known as "holography", suggests that in this sort of situation, 3d quantum gravity should be completely described by a field theory living on this boundary at infinity  a field theory theory with all conformal transformations as symmetries.
Which conformal field theory should correspond to 3d quantum gravity with negative cosmological constant? It depends on the value of the cosmological constant! Some topological arguments suggest that the ChernSimons description of 3d quantum gravity is only gaugeinvariant when the cosmological constant Λ takes certain special values, namely
Λ = 1/(16 k)^{2}
where k is an integer, known as the "level" in ChernSimons theory.
By the way: I'm working in Planck units here, and I'm assuming our ChernSimons theory is leftright symmetric, just to keep things simple. I may also be making some small numerical errors.
This quantization of the cosmological constant must seem strange if you've never seen it before, but it's not really so weird. What's weird is that Witten is using ChernSimons theory to determine the allowed values of the cosmological constant even though he wants to study what happens if gravity is not described by ChernSimons theory!
Witten knows this is weird: later he says "we used the gauge theory approach to get some hints about the right values of the cosmological constant simply because it was the only tool available."
Indeed, the whole paper seems designed to refute the notion that mathematicians get less daring as they get older. He writes: "We make at each stage the most optimistic possible assumption." Perhaps he has some evidence for his guesses that he's not revealing yet. Or perhaps he's decided it takes courage verging on recklessness to track the Monster to its lair.
Anyway: next Witten relates the level k to something called the "central charge" of the conformal field theory living at the boundary at infinity.
What's the "central charge"? This is a standard concept in conformal field theory. Perhaps the simplest explanation is that in a conformal field theory, the total energy of the vacuum state is c/24, where c is the central charge. So, naively you'd expect c = 0, but quantum effects make nonzero values of the vacuum energy possible, and even typical. A closely related cool fact is that the partition function of a conformal field theory is only a welldefined number up to multiples of
exp(2 π i c / 24)
This means the partition function is a welldefined number when c is a multiple of 24. This happens in certain especially nice conformal field theories which are said to have "holomorphic factorization".
The appearance of the magic number 24 here is the first taste of Monstrous Moonshine! For more on the importance of this number in string theory, see "week124", "week125" and "week126".
As you can see, there are lots of subtleties here, which I really don't want to get into, but feel guilty about glossing over.
Here's another. There are really two conformal field theories in this game: one that describes ripples of the gravitational field moving clockwise around the boundary at infinity, and another for ripples moving counterclockwise. Our simplifying assumption about leftright symmetry lets us describe these "rightmovers" and "leftmovers" with the same theory. So, both have the same central charge.
In this case, the relation between central charge and level is simple:
c = 24 k
Next, Witten considers the situation where k takes its smallest interesting value: k = 1, so c = 24. It just so happens that c = 24 conformal field theories with holomorphic factorization have been classified, at least modulo a certain conjecture:
2) A. N. Schellekens, Meromorphic c = 24 conformal field theories, Comm. Math. Phys. 153 (1993) 159196. Also available as hepth/9205072.
It's believed there are 71 of them. Which one could describe 3d quantum gravity?
Of these 71, all but one have gauge symmetries! Now, Witten is assuming 3d quantum gravity is not described by ChernSimons theory, which is a gauge theory. So, he guesses that the one exceptional theory is the right one!
And this is a very famous conformal field theory. It's a theory of a bosonic string wiggling around in a 26dimensional spacetime curled up in clever way with the help of a 24dimensional lattice called the Leech lattice. This theory is famous because its symmetry group is the Monster group! It is, in fact, the simplest thing we know that has the Monster group as symmetries.
For more details, try these  in rough order of increasing thoroughness:
3) Terry Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, available as arXiv:math/0109067.
Terry Gannon, Monstrous Moonshine: the first twentyfive years, available as arXiv:math/0402345.
4) Richard Borcherds, online papers, available at http://math.berkeley.edu/~reb/papers/
5) Igor Frenkel, James Lepowsky, Arne Meurman, Vertex operator algebras and the Monster, Academic Press, New York, 1988.
Now, if this monstrous conformal field theory turned out to be 3d quantum gravity in disguise  viewed from infinity, so to speak  it might someday give a much better understanding of Monstrous Moonshine. However, Witten gives no explanation as to why this theory should be 3d gravity, except for the indirect argument I just sketched. The precise relation between 3d quantum gravity and the bosonic string wiggling around in 26 dimensions remains obscure.
However, while Witten leaves this mysterious, he does offer a tantalizing extra tidbit of evidence that the relation is real!
The partition function of the monstrous conformal field theory I just mentioned is the jfunction, or more precisely:
J(q) = q^{1} + 196884 q + 21493760 q^{2} + ...
As I mentioned, this function shows up naturally in complex analysis. More precisely, it parametrizes the moduli space of elliptic curves (see "week125"). But, its bizarre coefficients turn out to be dimensions of interesting representations of the Monster group. For example, the smallest nontrivial representation of the Monster has dimension 196883; adding the trivial representation, we get 196884. This was one of several strange clues leading to the discovery of Monstrous Moonshine.
What Witten does is assume that the monstrous conformal field theory describes 3d quantum gravity for k = 1, and then use properties of the jfunction to compute the entropy of black holes!
I won't attempt to explain the calculation. Suffice it to say that the lightest possible black hole turns out to have 196883 quantum states  its space of states is the smallest nontrivial representation of the Monster group. So, its entropy is:
ln(196883) ≈ 12.19
On the other hand, Hawking's semiclassical calculation gives
4 π ≈ 12.57
The match is not perfect  but it doesn't need to be, since we expect quantum corrections to Hawking's formula for small black holes.
What's more impressive is that Witten can guess the entropy of the lightest possible black hole for other values of k  meaning, other values of the cosmological constant. The space of states of these black holes are always representations of the Monster group, so we get logarithms of weirdlooking integers. For example, for k = 4 the entropy is
ln(81026609426) ≈ 25.12
while Hawking's formula gives
8 π ≈ 25.13
Much better! And, using a formula of Petersson and Rademacher for asymptotics of coefficients of the jfunction, together with some facts about Hecke operators, he shows that as k → ∞, the agreement becomes perfect!
In short, there are some fascinating hints of a relation between the Monster group and black hole entropies in 3d gravity, but the details of Witten's hopedfor "AdSCFT correspondence" between 3d gravity and the monstrous conformal field theory remain obscure. Indeed, there are lots of problems with Witten's proposal:
6) Jacques Distler, Witten on 2+1 gravity, http://golem.ph.utexas.edu/~distler/blog/archives/001335.html
But, time will tell. In fact, if history is any guide, we can expect to see armies of string theorists marching into this territory any day now. So, I'll just pose one question.
There's a wellknown route from 2d rational conformal field theories (or "RCFTs") to 3d topological quantum field theories (or "TQFTs"), which passes through modular tensor categories. For example, an RCFT called the WessZuminoWitten model gives the TQFT called ChernSimons theory.
But, now Witten is saying "3d quantum gravity isn't ChernSimons theory; instead, it's something related to the monstrous CFT".
So: is the monstrous conformal field theory known to be an RCFT? If so, what 3d TQFT does it give? Could this TQFT be the 3d quantum gravity theory Witten is seeking?
Even though Witten is now claiming 3d quantum gravity can't be a TQFT, I think this is an interesting question. At the very least, I'd like to know more about this "Monster TQFT"  if it exists.
Now let's move from 3d quantum gravity to realworld particle physics...
Last week I described some mathematical relations between the Standard Model of particle physics, the most famous grand unified theories, and some "exceptional" structures in mathematics: the exceptional Lie group E_{6}, the complexified octonionic projective plane, and the exceptional Jordan algebra.
This week I want to go a bit further, and talk about the work of Kac, Larsson and others on the exceptional Lie superalgebras E(36), E(38) and E(510).
As before, my goal is to point out some curious relations between the messy pack of particles we see in nature and the "exceptional" structures we find in mathematics. By this, I mean structures that show up when you classify algebraic gadgets, but don't fit into nice systematic infinite families. Right now the Monster is the king of all exceptional structures, the biggest of the 26 sporadic finite simple groups. But, there are lots of other such structures, and they all seem to be related.
As mentioned back in "week66", Edward Witten once suggested that the correct theory of our universe could be an exceptional structure of some sort. There's even a fun handwavy argument for this idea. It goes like this: the theory of our universe must be incredibly special, since out of all the theories we can write down, only one describes the universe that actually exists!
In particular, lots of very simple theories do not describe our universe. So there must be some principle besides simplicity that picks out the theory of our universe.
Unfortunately, when we try to think about these issues seriously, we're quickly led into very deep waters. In practice, people quickly muddy these waters and create a quagmire. It's very hard to discuss this stuff without uttering nonsense. If you want to see my try, look at "week146".
But right now, I prefer to act like a sober, serious mathematical physicist. So, I'll tell you a bit about exceptional Lie superalgebras and how they could be related to the Standard Model.
First, some history. In 1887, Wilhelm Killing sent a letter to Friedrich Engel saying he'd classified the simple Lie algebras. Besides the "classical" ones  namely the infinite series sl(n,C), so(n,C) and sp(n,C)  he found 6 exceptions: a 14dimensional one, two 52dimensional ones, a 78dimensional one, a 133dimensional one and a 248dimensional one.
In 1894, Eli Cartan finished a doctoral thesis in which he cleaned up Engel's work. In the process, he noticed that Engel's two 52dimensional Lie algebras were actually the same. Whoops!
So, we now have just 5 "exceptional" simple Lie algebras. In order of increasing size, they're called g_{2}, f_{4}, e_{6}, e_{7} and e_{8}.
In 1914, Cartan realized that the smallest exceptional Lie algebra g_{2}, comes from the symmetry group of the octonions! Later it was realized that all 5 are connected to the octonions. I've written a lot about this in previous Weeks, but most of that material can be found here:
6) John Baez, Exceptional Lie algebras, http://math.ucr.edu/home/baez/octonions/node13.html
Now, whenever mathematicians do something fun, they want to keep doing it, which means generalizing it.
One way to generalize Cartan's work is to study "symmetric spaces", which I defined last week. Briefly, a symmetric space is a manifold equipped with a geometrical structure that's very symmetrical: so much so that every point is just like every other, and the view in any direction is the same as the view in the opposite direction.
In fact, it was Cartan himself who invented the concept of symmetric space, and after he classified the simple Lie algebras he went ahead and classified these.
More precisely, I think he classified the "compact Riemannian" symmetric spaces. Every simple Lie algebra gives one of these, namely a compact simple group. But, there are others too. So, compact Riemannian symmetric spaces are a nice generalization of simple Lie algebras  and I believe Cartan succeeded in classifying them all.
Again, there are some infinite series, but also some exceptions coming from the octonions. I talked about one of these last week, namely EIII, the complexified octonionic projective plane. You can see a list here:
7) Wikipedia, Riemannian symmetric space, http://en.wikipedia.org/wiki/Riemannian_symmetric_space
For a quick intro to the classification of simple Lie algebras and compact Riemannian symmetric spaces, try this great book:
8) Daniel Bump, Lie Groups, Springer, Berlin, 2004.
For a slower, more thorough introduction, try the book by Helgason mentioned last Week.
A second way to generalize Cartan's work is to consider simple Lie superalgebras.
Lie superalgebras are just like Lie algebras, except they're split into an "even" or bosonic and "odd" or fermionic part. The idea is that we stick minus signs in the usual Lie algebra formulas whenever we switch two "odd" elements.
This is very natural from a physics viewpoint, since whenever you switch two identical fermions, the wavefunction of the universe gets multiplied by 1. (Take my word for it  I've seen it happen!)
It's also very natural from a math viewpoint, since "super vector spaces" form a symmetric monoidal category with almost all the nice properties of plain old vector spaces. This lets crazed mathematicians and physicists systematically generalize pretty much all of linear algebra to the "super" world. So, why not Lie algebras?
The simple Lie superalgebras were classified by Victor Kac in 1977:
9) Victor Kac, Lie superalgebras, Adv. Math. 26 (1977), 896.
Not counting the ordinary simple Lie algebras, there are 8 series of simple Lie superalgebras and a few exceptional ones. At least some of these exceptions come from the octonions:
10) Anthony Sudbery, Octonionic description of exceptional Lie superalgebras, Jour. Math. Phys. 24 (1983), 19861988.
Do they all? I don't know! Someone please tell me!
A third way to generalize Cartan's work is to classify infinitedimensional simple Lie algebras  or for that matter, Lie superalgebras.
So far I've implicitly assumed all our algebraic gadgets are finitedimensional, but we can lift that restriction. If you try to classify infinitedimensional gadgets without any restrictions, it can get really hairy. It turns out the nice thing is to classify "linearly compact" infinitedimensional simple Lie algebras. I won't define the quoted phrase, since it's technical and it's explained near the beginning of this paper:
11) Victor Kac, Classification of infinitedimensional simple linearly compact Lie superalgebras, Erwin Schrödinger Institut preprint, 1998. Available at http://www.esi.ac.at/Preprintshadows/esi605.html
Anyway, back in 1880 Lie himself made a guess about infinitedimensional Lie algebras that would solve the problem I'm talking about now, though he didn't phrase it in the modern way. And, Cartan proved Lie's guess in 1909! Actually, there was a hole in Cartan's proof, which was only noticed much later. It was filled by Guillemin, Quillen and Sternberg in 1966.
So, here's the answer: there are 4 families of linearly compact infinitedimensional simple Lie algebras, and no exceptions. Ignoring an important nuance I'll explain later, these are:
div v = 0
L_{v} ω = 0
where ω is the usual symplectic structure on C^{2n}, and L means "Lie derivative"
L_{v} α = f α
for some function f depending on v, where α is the usual contact structure on C^{2n+1}.
If you don't know about symplectic structures or contact structures, don't worry  we won't need them now. The main point is that they're differential forms that show up throughout classical mechanics. So, this classification theorem is surprisingly nice.
Notice: no exceptions! That's a kind of exception in its own right.
In 1998, Victor Kac proved the "super" version of this result. In other words, he classified linearly compact infinitedimensional Lie superalgebras! This result is Theorem 6.3 of his paper above. There turn out to be 10 families and 6 exceptions, which are called E(16), E(22), E(36), E(38), E(44) and E(510).
Many of the families are straightforward "super" generalizations of the 4 families I just showed you. Some are stranger. Most important for us today are the exceptions discovered by Irina Shchepochkina in 1983:
12) Irena Shchepochkina, New exceptional simple Lie superalgebras, C. R. Bul. Sci. 36 (1983), 313314.
The easiest to explain is E(510). And, you'll soon see that the number 5 here is related to the math of the SU(5) grand unified theory, which I explained last Week!
The even part of E(510) is the Lie algebra of divergencefree complex vector fields on C^{5}.
The odd part of E(510) consists of closed complex 2forms on C^{5}.
The bracket of two even guys is the usual Lie bracket of vector fields.
The bracket of an even guy and an odd guy is the usual "Lie derivative" of a differential form with respect to a vector field.
The only tricky bit is the bracket of two odd guys! So, suppose μ and ν are closed complex 2forms on C^{5}. Their wedge product is a 4form μ ^ ν. But, we can identify this with a vector field v by demanding:
i_{v} vol = μ ^ ν
Here vol is the volume form:
vol = dx_{1} ^ dx_{2} ^ dx_{3} ^ dx_{4} ^ dx_{5}
and i_{v} vol is the "interior product", which feeds v into vol and leaves us with a 4form. You can check that this vector field v is divergencefree. So, we define the bracket of μ and ν to be v.
sl(5,C) sits inside the even part of E(510) in a nice way, as the divergencefree vector fields whose coefficients are linear functions on C^{5}. So, since su(5) sits inside sl(5,C), we get a tempting relation to SU(5).
(Now I'll come clean now and explain the "important nuance" I ignored earlier. For the classification theorems I mentioned earlier, we must use vector fields and differential forms with formal power series as coefficients. But for the purposes of mathematical physics, we should keep a more flexible attitude.)
Next, what about E(36)? This is contained in E(510). To define it, we give E(510) a clever grading where x_{1}, x_{2}, x_{3} are treated differently from the other two variables. Then we take the subalgebra of degreezero guys. The details are explained in the above papers  or more simply, here:
13) Victor Kac, Classification of infinitedimensional simple groups of supersymmetries and quantum field theory, available as math.QA/9912235.
All this is reminiscent of how SU(5) contains the gauge group of the Standard Model, namely S(U(3) × U(2)). In particular, the even part of E(36) contains the Lie algebra
sl(3,C) ⊕ sl(2,C) ⊕ gl(1,C)
in a canonical way. So, any representation of E(36) automatically gives a representation of the Standard Model Lie algebra
su(3) ⊕ su(2) ⊕ u(1)
And in the above paper Kac goes even further! He defines a fairly natural class of representations of E(36), and proves something remarkable: these restrict to representations of
su(3) ⊕ su(2) ⊕ u(1)
that correspond precisely to the gluon, the photon and the W and Z bosons, and the quarks and leptons in one generation...
... together with one other particle, which is not the Higgs boson, but instead acts like a gluon with electric charge ±1.
Darn.
One nice thing is how these Lie superalgebras get both bosons and fermions into the game in a natural way without forcing the existence of a bunch of unseen "superpartners". One unfortunate thing is that the above result gives no hint as to why there should be three generations of quarks and leptons. However, Kac and Rudakov develop some mathematics to address that question here:
14) Victor Kac and Alexi Rudakov, Representations of the exceptional Lie superalgebra E(3,6): I. Degeneracy conditions. Available as mathph/0012049.
Representations of the exceptional Lie superalgebra E(3,6): II. Four series of degenerate modules. Available as mathph/0012050.
Representations of the exceptional Lie superalgebra E(3,6) III: Classification of singular vectors. Available as mathph/0310045.
Their results are summarize at the end of this review article:
15) Victor Kac, Classification of supersymmetries, Proceedings of the ICM, Beijing, 2002, vol. 1, 319344. Available as mathph/0302016
Here Kac writes that "three generations of leptons occur in the stable region [whatever that means], but the situation with quarks is more complicated: this model predicts a complete fourth generation of quarks and an incomplete fifth generation (with missing down type triplets)."
So, while I don't understand "this model", it seems tantalizingly close to capturing the algebraic patterns in the Standard Model... without quite doing so.
Some more nice explanations and references can be found here:
16) Irina Shchepochkina, The five exceptional simple Lie superalgebras of vector fields. Available as hepth/9702121.
17) Pavel Grozman, Dimitry Leites and Irina Shchepochkina, Defining relations for the exceptional Lie superalgebras of vector fields pertaining to The Standard Model, available as mathph/0202025.
18) Pavel Grozman, Dimitry Leites and Irina Shchepochkina, Invariant operators on supermanifolds and Standard Models, available as math.RT/0202193.
Thomas Larsson has been working on similar ideas, mainly using E(38) instead. This also contains
sl(3,C) ⊕ sl(2,C) ⊕ gl(1,C)
in a canonical way.
19) Thomas A. Larsson, Symmetries of everything, available as math.RT/0202193.
Exceptional Lie superalgebras, invariant morphisms, and a secondgauged Standard Model, available as mathph/020202.
Thomas A. Larsson, Maximal depth implies su(3)+su(2)+u(1), available as hepth/0208185.
Alas, E(38) gets the hypercharges of some fermions wrong. Larsson seems to say this problem also occurs for E(36), which would appear to contradict what Kac claims  but I could be misunderstanding.
I'll end with few questions. First, is there any relation between the exceptional Lie superalgebras E(510), E(36) or E(510) and the exceptional Lie algebra e6? Last week I explained some relations between e6 and the Standard Model; are those secretly connected to what I'm discussing this week?
Second, has anyone tried to unify all three generalizations of Cartan's classification of simple Lie algebras? Starting from simple Lie algebras, we've seen three ways to generalize:
So: has anyone tried to classify infinitedimensional super versions of symmetric spaces? Or even finitedimensional ones?
(Maybe the super version of a symmetric space should be called a "supersymmetric space", just for the sake of a nice pun.)
Suppose you're trying to do a CAT scan. You want to obtain a 3d image of someone's innards. Unfortunately, all you do is take lots of 2d Xray photos of them. How can you assemble all this information into the picture you want?
Who better to help you out than a guy named after a radioactive gas: Radon!
In 1917, the Viennese mathematician Johann Radon tackled a related problem one dimension down. You could call it a "CAT scan for flatlanders".
Suppose you want to obtain a complete image of the insides of a 2dimensional person, but all you can do is shine beams of Xrays through them and see how much each beam is attenuated.
So, mathematically: you have a realvalued function on the plane  roughly speaking, the density of your flatlander. You're trying to recover this function from its integrals along all possible lines. Someone hands you this function on the space of lines, and you're trying to figure out the original function on the space of points.
(Points lying on lines! If you've been following the Tale of Groupoidification, you'll know this "incidence relation" is connected to Klein's approach to geometry, and ultimately to spans of groupoids. But pretend you don't notice, yet.)
Now, it's premature to worry about this tricky "inverse problem" before we ponder what it's the inverse of: the "Radon transform". This takes our original function on the space of points and gives a function on the space of lines.
Let's call the Radon transform T. It takes a function f on the space of points and gives a function Tf on the space of lines, as follows. Given a line y, (Tf)(y) is the integral of f(x) over the set of all points x lying on y.
What Radon did is figure out a nice formula for the inverse of this transform. But that's not what I'm mainly interested in now. It's the Radon transform itself that's a kind of Hecke operator!
Next, look at another example.
This is an obvious generalization to higher dimensions of what I just described. Before we had a space
X = {points in the plane}
and a space
Y = {lines in the plane}
and an incidence relation
S = {(x,y): x is a point lying on the line y}
If we go to n dimensions, we can replace all this with
X = {points in R^{n}}
Y = {lines in R^{n}}
S = {(x,y): x is a point lying on the line y}
Again, the Xray transform takes a function f on the space of points and gives a function Tf on the space of lines. Given a line y, (Tf)(y) is the integral of f(x) over the set of all x with (x,y) in S.
Next, yet another example!
Radon actually considered a different generalization of the 2d Radon transform, using hyperplanes instead of lines. Using hyperplanes is nicer, because it gives a very simple relationship between the Radon transform and the Fourier transform. But never mind  that's not the point here! The point is how similar everything is. Now we take:
X = {points in R^{n}}
Y = {hyperplanes in R^{n}}
S = {(x,y): x is a point lying on the hyperplane y}
And again, the Radon transform takes a function f on X and gives a function Tf on Y. Given y in Y, (Tf)(y) is the integral of f(x) over the set of all x with (x,y) in S.
We're always doing the same thing here. Now I'll describe the general pattern a bit more abstractly.
We've always got three spaces, and maps that look like this:
S / \ / \ P/ \Q / \ v v X YIn our examples so far these maps are given by
P(x,y) = x
Q(x,y) = y
But, they don't need to be.
Now, how do we get a linear operator in this situation?
Easy! We start with a realvalued function on our space X:
f: X → R
Then we take f and "pull it back along P" to get a function on S. "Pulling back along P" is just impressive jargon for composing with P:
f o P: S → R
Next, we take f o P and "push it forwards along Q" to get a function on Y. The result is our final answer, some function
Tf: Y → R
"Pushing forwards along Q" is just impressive jargon for integrating: Tf(y) is the integral over all s in S with Q(s) = y. For this we need a suitable measure, and we need the integral to converge.
This is the basic idea: we define an operator T by pulling back and then pushing forward functions along a "span", meaning a diagram shaped like a bridge:
S / \ / \ P/ \Q / \ v v X YBut, the reason this operator counts as a "Hecke operator" is that it gets along with a symmetry group G that's acting on everything in sight. In the examples so far, this is the group of Euclidean symmetries of R^{n}. But, it could be anything.
This group G acts on all 3 spaces: X, Y, and S. This makes the space of functions on X into a representation of G! And, ditto for the space of function on Y and S.
Furthermore, the maps P and Q are "equivariant", meaning
P(gs) = gP(s)
and
Q(gs) = gQ(s)
This makes "pulling back along P" into an intertwining operator between representations of G. "Pushing forwards along Q" will also be an intertwining operator if the measure we use is Ginvariant in a suitable sense. In this case, our transform T becomes an intertwining operator between group representations! Let's call an intertwining operator constructed this way a "Hecke operator".
If you're a nitpicky person, e.g. a mathematician, you may wonder what I mean by "a suitable sense". Well, each "fiber" Q^{1}(y) of the map
Q: S → Y
needs a measure on it, so we can take a function on S and integrate it over these fibers to get a function on Y. We need this family of measures to be invariant under the action of G, for pushing forwards along Q be an intertwining operator.
This stuff about invariant families of measures is mildly annoying, and so is the analysis involved in making precise which class of functions on X we can pull back to S and then push forward to Y  we need to make sure the integrals converge, and so on. When I really get rolling on this Hecke operator business, I'll often focus on cases where X, Y, and S are finite sets... and then these issues go away!
Hmm. I'm getting tired, but I can't quit until I say one more thing. If you try to read about Hecke operators, you won't see anything about the examples I just mentioned! You're most likely to see examples where X and Y are spaces of lattices in the complex plane. This is the classic example, which we're trying to generalize. But, this example is more sophisticated than the ones I've mentioned, in that the "functions" on X and Y become "sections of vector bundles" over X and Y. The same sort of twist happens when we go from the Radon transform to the more general "Penrose transform".
Anyway, next time I'll talk about some really easy examples, where X, Y, and S are finite sets and G is a finite group. These give certain algebras of Hecke operators, called "Hecke algebras".
In the meantime, see if you can find any reference in the literature which admits that "Hecke algebras" are related to "Hecke operators". It ain't easy!
It's a great example of a mathematical coverup  one we're gonna bust wide open.
Addendum: David Corfield notes that Helgason has a good textbook on the Radon transform which is free online. Snap it up while you can!
20) Sigurdur Helgason, Radon Transform, second edition, Birkhäuser, New York, 1999. Also available at http://wwwmath.mit.edu/~helgason/Radonbook.pdf.
For more discussion, go to the nCategory Café.
© 2007 John Baez
baez@math.removethis.ucr.andthis.edu
