Can Subgroups Have a Volume in Linear Matrix Spaces?

[raopeng]

Can group have a "volume"?

For example, SL(n, R) is a subgroup in a linear matrix space with det A = 1. So can this equation represent a certain "region" in the n-dimensional linear space and therefore that it has a "volume"?

 

[Erland]

If so, it would be in n^2-space, not n-space, because this is the dimension of the matrix space. But it would be a subset of an n^2-1-dimensional manifold in this space, thus having volume 0.

 

[raopeng]

Thank you. Didn't think it through carefully...

 

[Bacle2]

But your region may have non-zero k-volume for k< n^2 . The topology on subgroups

I'm familiar with turns a matrix [ai] into a point in R^(n^2) by [ai]<-->[a11,a12

,...ann] . This is a region in R^(n^2), and, while, as Erland said, it has zero

n^2-volume, it may have non-zero k-volume for k<n^2 .

 

[Ben Niehoff]

In fact, SL(n,R) is non-compact, so it has infinite volume.

The compact Lie groups all have finite volume, which you can compute without too much difficulty. Constructing a volume form is similar to the construction of a metric (i.e., the Cartan-Killing metric). Then you just integrate this over the group.

Note: As a subset of R^n^2, most groups have measure zero. What I mean is the intrinsic volume of the group manifold, which is k-dimensional (and k < n^2).

 

[A. Bahat]

There is a related notion of what's called a 'Haar measure' on a topological group, which allows one to define an integral over the group.

 

[mathwonk]

excellent point! presumably Haar measure is a translation invariant measure.

integration is about averaging. thus an integration on a group allows one to average

the action of a group on a set. so even if volume is not the main idea, integration is still important.

 

[A. Bahat]

Yes, the key property of Haar measure is translation-invariance. So this generalizes averaging over the elements of a finite group by summing as well as integrating over some familiar topological group like the reals. Plus, the Haar integral allows us to do harmonic analysis on locally compact abelian groups (or even non-abelian groups, for that matter, but this is more complicated) which is really cool. But I barely know anything about this, so I should stop now...

 

[raopeng]

And I am still a bit confused about how to construct a Killing metrics of, say, SU(2) for integration. And also how can we write the defining region of the group for the integration?
I really appreciate all the help, but some of them are quite beyond my current level. So I have to consider SO(n) in an easier way:
Define Killing form as $\langle A, A \rangle = tr(AA^{T})=tr(E)=n$. So for SO(n) the region is a sphere with radius √n? And employing the general formula for n^2-sphere volume I obtain: V = $\frac{\pi^{n^2/2}}{\gamma (n^2/2 + 1)}n^{n^2/2}$. but it looks a bit ridiculous...

 

[raopeng]

I think for SU(n) it is similar, except the dimension is now 2n^2.