In the last post I discussed some preliminaries from differential geometry which we’ll need to define the integral of a function , where is a matrix Lie group. Naturally, we want to do this in a left-invariant way, namely if is the map (left multiplication by ), then we want . This is analogous to the case of (say) the group where “translation invariance” of an integral would translate (no pun intended) into a condition like . The group is abelian of course, so left and right invariance are the same concept.
Let us now return to our friend , the set of volume forms on . In the language of differential geometry, the elements of are smooth sections of , or in other words, top-degree differential forms on . One might thus prefer to write rather than . Recall that . There is no reason, a priori, that should be nonempty. Yes, we know has dimension 1 for every , but why should there exist a smooth section of this bundle? Well, it turns out that if you fix some basis for the Lie algebra and simply define, for each , to be the thing that assigns a “volume” of 1 to the parallelotope formed by the “-translated basis” , and then piece all these together by declaring
then you actually find that .
Let me now correct some errata from last time. I said that an is usually referred to as a density, and also claimed that is a way of assigning “unsigned volumes” to ordered parallelotopes in . This is slightly incorrect. The actual situation is that these guys in really are (as mentioned above) legit top-degree differential forms on . The objects that we would call densities are the absolute values of some such . Notice that eats vector fields on and produces a smooth function on . After we compose this with the absolute value, I believe we merely get a map ; note the modesty of the codomain. Indeed, is a smooth function, but is not even continuously differentiable, although it is continuous. However, we don’t care, since any continuous function is surely still good enough to integrate.
It should be noted that unless one chooses a basis for the Lie algebra ahead of time, the Haar integral (which we will soon define) is only going to make sense up to scaling by a positive factor. This is because in order to fully determine a left-invariant form as desired, we have to say “which” parallelotopes in the Lie algebra we want to take as fundamental, that is, which ones we want to assign a volume of 1.
As an aside, one of the nice things about (oriented) Riemannian manifolds is that their tangent spaces carry inner products, at which point Hodge duality phenomena, which yield natural isomorphisms , would give us a distinguished element of (); merely pick the one corresponding to the constant function 1. In less high-brow parlance, we could use the orientation and inner product on each tangent space to establish some isometric isomorphism to and then invoke the determinant over there. However we’re not so lucky to have such Riemannian structure here.
Integrating a function supported on one chart
Having now discussed the geometric intuition behind the machinery, we are ready to define the integral. In general, to do this, we will choose some coordinate system on our Lie group . The actual definition of the integral turns out to be independent of which coordinate system we use; it depends only on the density . The simplest case, of course, is when the function we are integrating actually vanishes outside of a single . In this case we write and put
As disgusting as the above may look on first sight, I claim this is the natural thing to do. Parsing the right-hand side, we see that we are now integrating over a region . The integrand, of course, is merely the function that takes points in to their corresponding points in , and thereafter feeds them to . Well… almost. The huge expression involving that we’re multiplying by actually turns out to be very important. Intuitively, this should not be surprising: what if we chose charts such that were larger and larger subsets of ? Our integral could be made arbitrarily large. However, this is where the second factor comes in, and keeps things under control. Let’s look at it a little more closely:
I guess you can probably interpret this scaling factor as some kind of Radon-Nikodym derivative. If we think again about the signature of , it wants to take in elements of the tangent space at . Here is perhaps where the PMATH 365 point of view comes in handy again: can also be interpreted as consisting of derivations at of the algebra of smooth functions. The tangent space at in admits the natural coordinate basis
Now is a map from to , so the things
which we might write
which appear above are nothing more than the images of the coordinate basis elements under the pushforward map . Notice that if was mapping onto some HUGE set , then it seems reasonable to expect that would have very small partial derivatives, which would cause the scaling factor given by to be a very small positive number.
That’s all I have time to write for now. Next time we will discuss partitions of unity, which will permit the integration of functions even when they are not supported on a single chart, and finish by mentioning the modular function and what unimodularity has to do with right-invariance criteria.