A path-connected compact abelian matrix group is a torus (i.e. it's isomorphic to for some ).
Compact connected abelian groups are tori
This video is optional.
Since is abelian, we have Therefore the exponential map is a group homomorphism where we think of as with addition.
Here's how the proof will go:
We'll prove that exp is surjective.
The first isomorphism theorem for groups will then show that .
Then we will identify with .
Why is exp surjective? Because is path-connected, we know from an earlier exercise that is generated by any neighbourhood of the identity matrix. In particular, is generated by the image of an exponential chart. We can always add generators to a generating set to get a bigger generating set, so is generated by . But is a subgroup, as we saw in the last video, so the subgroup generated by is itself, therefore , and the exponential map is surjective.
We now identify the kernel of the exponential map.
linearly independent vectors
and take all possible integer linear combinations of them to get a subgroup
A subgroup of this form is called a
is a lattice, taking to be the th standard basis vector.
is a lattice of rank (where ).
Once we have established this, we are done. To see why, consider the basis of given by the s. This identifies with in such a way that is identified with . Therefore is isomorphic to .
Note that because we have a surjective homomorphism , , with kernel , so the first isomorphism theorem tells us .
The fact that the rank must be is easy to see once you know that is a lattice. If the rank were strictly less than then we would get , so would fail to be compact because is not compact unless .
We will need to assume the following lemma:
A discrete subgroup of is a lattice.
The proof of this lemma would take us too far away from the main business of the course. You can find it in Arnold's book "Mathematical Methods of Classical Mechanics" (on page 276 in my version of the book, where he is proving the (Arnold)-Liouville theorem on integrable systems).
We will just define discrete subgroups and prove that is discrete.
In other words, the points of are sufficiently spaced out that you can separate them by balls.
An example of something that is not discrete would be : any ball (interval) contains infinitely many rational numbers.
By constrast, the integers are discrete: an interval of length containing an integer contains only one integer.
is a discrete subgroup of .
The origin is in , so let's start by finding a ball such that . We can take to be the domain of an exponential chart.
Remember that we have the exponential map and that we can find neighbourhoods of the origin and of the identity such that is bijective. In particular, is injective, so (i.e. is the only possible preimage of the identity in this neighbourhood).
For a general , we will take . This works because but because is abelian and , so we deduce that . Since , this implies . Therefore
This final lemma shows that is a discrete subgroup of , so by the previous (unproven) lemma, it is a lattice (which we saw has rank ). This implies that is isomorphic to as required.