Processing math: 100%

Tuesday, February 16, 2010

Orbit-Stabilizer and Covering Maps

Last week, in our quantum Mechanics class, we were going over symplectic spaces and symplectic transformations. A symplectic space is just a manifold together with a skew-symmetric non-degenerate bilinear form J defined on it, and a symplectic transformation S is a transformation of the manifold into itself such that it preserves J. One of the most common examples is when we take our manifold to be M=\mathbb{F}^{2n} and the symplectic form

J=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}

where F is a field and I_n is the identity matrix. This is a symplectic manifold, and the set of symplectic transformations is know as Sp(n,F). This is a well known Lie Group acting by multiplication on M, and one of its goodness is that this action is transitive, that is, for any non-zero x,y\in M, there is S\in Sp(n,F) such that y=Sx.

This statement was actually part of our homework, to prove that the action is transitive, and I wanted to find a nicer way to prove it and not to do a proof that I had seen before in my previous courses, so I started thinking a bit of many different ways of saying that this action was transitive.

One way of seen this is by turning around the problem saying what would happen if we let S to run over Sp(n,F) and look at Sx for x fixed? Well, that is saying something like the orbit of x is M/\{0\} and that started to sound a bit familiar.

I was trying then to use some kind of orbit-stabilizer theorem and then use some cardinality argument and kill the problem. Although, I only did remember the finite version of this powerful theorem, which obviously, wouldn't help me at all, but in essence, that was what I was looking for. A cardinality argument would not help me in this situation, because I could have some proper subspace of the same cardinality of M and this wouldn't lead me to the conclusion I was going after. Instead, a dimensionality argument was needed.

While searching for this and thinking what actually was going on behind the scenes in this group action, I saw how helpful is the notion of representation for understanding a strange object.

If G is a Lie Group, we call a representation of it, a vector space V in which G acts on. We can think as G be some sort of subgroup of Gl(V), the set of linear transformations of V into itself. For an element x of V, we can talk about the G orbit through x, O_x as the set of all g.x for g\in G. In some sense, O_x is a copy of the shape of G. Also, from the geometrical point of view, a Lie Group is a manifold, endowed with superpowers (group structure) and hence, we can think of these orbits into V as coordinate maps of G given by \phi(g)=g.x, so really O_x is how G looks locally.

For example, take O(2), which is the group of all 2\times 2 matrices O such that OO^T=I. This group is quite odd to picture, since it is a 1 dimensional manifold living in a 4 dimensional space, but by means of orbits, one can have a pretty good idea of how this group looks like. By picking a nonzero vector x and looking at its orbit in \mathbb{R}^2, one can find that O(2) looks locally like a circle.



In the general case, one can think as G being a covering for O_x and the degree of the cover is the number of connected components of G, for instance, in the above case, O(2) has 2 connected components, the set of matrices with determinant equal to 1 and those of determinant equal to -1, and that fact is reflected in O_x as the vector g.x rotates counter clockwise for O(2)_e (the identity component) and rotates clockwise in the other component, so each circle is drawn twice, and that means that O(2) is a 2-fold cover for each O_x.

In this language, we can say that the stabilizer G_X of an element x, is the fiber \phi^{-1}(x) whose cardinality gives us the degree of the covering map.

Actually, from this point of view, \phi defines a quotient map, which is very suitable for an orbit-stabilizer type argument. Since the stabilizer G_x is a normal subgroup, one can think of G as a principal G_x-bundle as G/G_x\times G_x and making the identification G/G_x\sim O_x and G_x\sim \phi^{-1}(x) we have that G\sim O_x\times\phi^{-1}(x).

Going away from counting arguments and going more into dimensionality, I found the so called Orbit-Stabilizer Theorem for Lie Groups which have the same feeling as the covering map approach. It states that

dim(G)=dim(O_x)+dim(G_s)

where dim is regarded as manifolds.

In the O(2) case, we have that dim(G)=1, dim(O_x)=1 and dim(G_x)=0 as any of the other cases when G_x is a finite group, and hence, we have that \phi is a quotient map and dim(G)=dim(O_x) as expected from a covering map.

At the end, I didn't use any of these arguments for my proof, but I found quite enjoyable doing this diversion from my first thought.


No comments:

Post a Comment