April 8, 2007

This Week's Finds in Mathematical Physics (Week 249)

John Baez

As you may recall, I'm telling a long story about symmetry, geometry, and algebra. Some of this tale is new work done by James Dolan, Todd Trimble and myself. But a lot of it is old work by famous people which deserves a modern explanation.

A great example is Felix Klein's "Erlangen program" - a plan for reducing many sorts of geometry to group theory. Many people tip their hat to the Erlanger program, but few seem to deeply understand it, and even fewer seem to have read what Klein actually wrote about it!

The problem goes back a long ways. In 1871, while at Göttingen, Klein worked on non-Euclidean geometry, and showed that hyperbolic geometry was consistent if and only if Euclidean geometry was. In the process, he must have thought hard about the role of symmetry groups in geometry. When he was appointed professor at Erlangen in 1872, he wrote a lecture outlining his "Erlanger Programm" for reducing geometry to group theory.

But, he didn't actually give this lecture as his inaugural speech! He spoke about something else.

So, nobody ever heard him announce the Erlangen program. And, until recently, the lecture he wrote was a bit hard to find. Luckily, now you can get it online:

1) Felix Klein, Vergleichende Betrachtungen ueber neuere geometrische Forschungen, Verlag von Andreas Deichert, Erlangen, 1872. Also available at the University of Michigan Historical Mathematics Collection, http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABN7632

Even better, Johan Ernst Mebius has recently prepared an HTML version, with links to the above version:

2) Johan Ernst Mebius, Felix Klein's Erlanger Programm, http://www.xs4all.nl/~jemebius/ErlangerProgramm.htm

But what if you have the misfortune of only reading English, not German? Until now the only translation was quite hard to obtain:

3) Felix Klein, A comparative review of recent researches in geometry, trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249.

In case you're wondering, the "New York Mathematical Society" no longer exists! It was founded in 1888, but in 1894 it went national and became the American Mathematical Society.

Luckily, after Thomas Love pointed out the existence of this old translation, Chris Hillman was able to get ahold of it and scan it in! Then Robin Houston created a PDF file of the whole thing, and Lukas-Fabian Moser created a DjVu file. Then Nitin C. Rughoonauth took the marvelous step of putting it into LaTeX! So now, you can read Klein's paper in English here:

4) The Erlangen program, http://math.ucr.edu/home/baez/erlangen/

English-speakers can read more about the Erlangen program here:

5) Felix Klein, Elementary Mathematics from an Advanced Standpoint: Geometry, part 3: Systematic discussion of geometry and its foundations, Dover, New York, 1939.

Luckily Dover keeps its books in print!

For more on the Erlangen program, try these:

6) Garrett Birkhoff and M. K. Bennett, Felix Klein and his "Erlanger Programm", in History and Philosophy of Modern Mathematics, eds. W. Aspray and P. Kitcher, Minnesota Stud. Philos. Sci. XI, University of Minnesota Press, Minneapolis, 1988, pp. 145-176.

7) Hans A. Kastrup, The contributions of Emmy Noether, Felix Klein and Sophus Lie to the modern concept of symmetries in physical systems, in Symmetries in Physics (1600-1980), ed. M. G. Doncel, World Scientific, Singapore, 1987, pp. 113-163.

8) I. M. Yaglom, Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century, trans. S. Sossinsky, Birkhauser, Boston, 1988.

For more about Klein, try "week213" and this little biography:

9) MacTutor History of Mathematics Archive, Felix Klein, http://www-history.mcs.st-andrews.ac.uk/Biographies/Klein.html

But what does the Erlangen program actually amount to, in the language of modern mathematics? This will take a while to explain, so the best thing is to dive right in.

Last week in the Tale of Groupoidification I tried to explain two slogans:

They're a bit vague; they're mainly designed to give you enough intuition to follow the next phase of the Tale, which is all about how:

But before the next phase, I need to say a bit about how groupoids and spans of groupoids fit into Klein's Erlangen program.

Groupoids are a modern way to think about symmetries. A more traditional approach would use a group acting as symmetries of some set. And the most traditional approach of all, going back to Galois and Klein, uses a group acting transitively on a set.

So, let me explain the traditional approach, and then relate it to the modern one.

I hope you know what it means for a group G to "act" on a set X. It means that for any element x of X and any guy g in G, we get a new element gx in X. We demand that

1x = x

and

g(hx) = (gh)x.

More precisely, this is a "left action" of G on X, since we write the group elements to the left of x. We can also define right actions, and someday we may need those too.

We say an action of a group G on a set X is "transitive" if given any two elements of X, there's some guy in G mapping the first element to the second. In this case, we have an isomorphism of sets

X = G/H

for some subgroup H of G.

For example, suppose we're studying a kind of geometry where the symmetry group is G. Then X could be the set of figures of some sort: points, or lines, or something fancier. If G acts transitively on X, then all figures of this sort "look alike": you can get from any one to any other using a symmetry. This is often the case in geometry... but not always.

Suppose G acts transitively on X. Pick any figure x of type X and let H be its "stabilizer": the subgroup consisting of all guys in G that map x to itself. Then we get a one-to-one and onto map

f: X → G/H

sending each figure gx in X to the equivalence class [g] in G/H.

If you haven't seen this fact before, you should definitely prove it - it's one of the big ways people use symmetry!

Here's one kind of thing people do with this fact. The 3d rotation group G = SO(3) acts on the sphere X = S², and the stabilizer of the north pole is the 2d rotation group H = SO(2), so the sphere is isomorphic to G/H = SO(3)/SO(2). The same sort of result holds in any dimension, and we can use it to derive facts about spheres from facts about rotation groups, and vice versa.

A grander use of this fact is to set up a correspondence between sets on which G acts transitively and subgroups of G. This is one of the principles lurking behind Galois theory.

Galois applied this principle to number theory - see "week201" for details. But, it really has nothing particular to do with number theory! In his Erlangen program, Klein applied it to geometry.

Klein's goal was to systematize a bunch of different kinds of non-Euclidean geometry. Each kind of geometry he was interested in had a different group of symmetries. For example:

• n-dimensional spherical geometry has the rotation group SO(n+1) as symmetries. (Or, if you want to include reflections, the bigger group O(n+1).)
• n-dimensional Euclidean geometry has the Euclidean group ISO(n) as symmetries. (This group is built from rotations in SO(n) together with translations in Rⁿ.)
• n-dimensional hyperbolic geometry has the group SO(n,1) as symmetries. (This group also shows up in special relativity under the name of the "Lorentz group": it acts on the "mass hyperboloid", and that's how hyperbolic geometry shows up in special relativity.)
• n-dimensional projective geometry has the group SL(n+1) as symmetries. (This group consists of (n+1)×(n+1) matrices with determinant 1. Scalar multiples of the identity act trivially on projective space, so it's actually better to use the "projective general linear group" PGL(n+1), consisting of invertible matrices mod scalars. But, this has the same Lie algebra as SL(n+1), so people are often a bit slack about which group they use.)

The details here don't matter much yet; the point is that there are lots of interesting kinds of geometry, with interesting symmetry groups!

Klein realized that in any kind of geometry like this, a "type of figure" corresponds to a set on which G acts transitively. Here a "figure" could be a point, a line, a plane, or something much fancier. Regardless of the details, the set of all figures of the same type can be written as G/H, and G acts transitively on this set.

The really cool part is that we can use Klein's idea to define a geometry for any group G. To do this, we just say that every subgroup H of G gives rise to a type of figure. So, we work out all the subgroups of G. Then, we work out all the incidence relations - relations like "a point lies on a line". To do this, we take two sets of figures, say

X = G/H

and

Y = G/K

and find all the invariant relations between them: that is, subsets of X × Y preserved by all the symmetries. I'll say more about how to do this next time - we can use something called "double cosets". In nice cases, like when G is a simple Lie group and H and K are so-called "parabolic" subgroups, these let us express all the invariant relations in terms of finitely many "atomic" ones! So, we can really carry out Klein's program of thoroughly understanding geometry starting from groups - at least in nice cases.

In short, group actions - especially transitive ones - are a traditional and very powerful way of using symmetry to tackle lots of problems.

So, to bridge the gap between the traditional and the new, I should explain how group actions give groupoids. I'll show you that:

It's not very hard to get a groupoid from a group action. Say we have a group G acting on a set X. Then the objects of our groupoid are just elements of X, and a morphism

g: x → y

is just a group element g with

gx = y.

Composing morphisms works the obvious way - it's basically just multiplication in the group G.

Some people call this groupoid an "action groupoid". I often call it the "weak quotient" X//G, since it's like the ordinary quotient X/G, but instead of declaring that x and y are equal when we have a group element g sending x to y, we instead declare they're isomorphic via a specified isomorphism g: x → y.

But for now, let's call X//G the "action groupoid".

So, group actions give action groupoids. But, these groupoids come with extra stuff!

First of all, the action groupoid X//G always comes equipped with a functor

                     X//G 
                      |
                      |p
                      |
                      v
                      G

Second of all, this functor p is always "faithful": given two morphisms from x to y, if p maps them to the same morphism, then they were equal.

And that's all! Any groupoid with a faithful functor to G is equivalent to the action groupoid X//G for some action of G on some set X. This takes a bit of proving... let's not do it now.

So: in my slogan

the "certain extra stuff" was precisely a faithful functor to G.

What if we have a transitive group action? Then something nice happens.

First of all, saying that G acts transitively on X is the same as saying there's a morphism between any two objects of X//G. In other words, all objects of X//G are isomorphic. Or in other words, there's just one isomorphism class of objects.

Just as a groupoid with one object is a group, a groupoid with one isomorphism class of objects is equivalent to a group. Here I'm using the usual notion of "equivalence" of categories, as explained back in "week76".

So, G acts transitively on X precisely when X//G is equivalent to a group!

And what group? Well, what could it possibly be? It's just the stabilizer of some element of X! So, in the case of a transitive group action, our functor

                     X//G 
                      |
                      |p
                      |
                      v
                      G

                      H
                      |
                      |i
                      |
                      v
                      G

So, we see how Klein's old idea of geometrical figures as subgroups of G is being generalized. We can start with any groupoid Y of "figures" and "symmetries between figures", and play with that. It becomes an action groupoid if we equip it with a faithful functor to some group G:

                      Y
                      |
                      |
                      |
                      v
                      G

                      H
                      |
                      |
                      |
                      v
                      G

So, when we blend our ideas on spans of groupoids with Klein's ideas, we'll want to use spans of groupoids "over G" - that is, commutative diamonds of groupoids and functors, like this:

                     S
                    / \
                   /   \
                  /     \
                 /       \
                v         v
               X           Y
                \         /
                 \       /
                  \     /
                   \   /
                    v v
                     G

I'll say one last thing before quitting. It's a bit more technical, but I feel an urge to see it in print.

People often talk about "the" stabilizer group of a transitive action of some group G on some set X. This is a bit dangerous, since every element of X has its own stabilizer, and they're not necessarily all equal!

However, they're all conjugate: if the stabilizer of x is H, then the stabilizer of gx is gHg^-1.

So, when I say above that

                     X//G 
                      |
                      |p
                      |
                      v
                      G

                      H
                      |
                      |i
                      |
                      v
                      G

                      H
                      |
                      |i'
                      |
                      v
                      G

Of course, I'll try to write later issues in a way that doesn't force you to have understood all these nuances!

© 2007 John Baez
baez@math.removethis.ucr.andthis.edu