## Towards integration on matrix Lie groups

You may have heard (perhaps in a representation-theoretic context) of something called the Haar measure, which gives a way of assigning a “size” to subsets of a locally compact group. In this series of articles we will eventually discuss something known as the Haar integral on a matrix Lie group (henceforth “Lie group”), without concentrating so much on the Haar measure itself. This discussion is meant as an accompaniment to the corresponding section of the PMATH 763 notes (indeed this post was mainly written so I could figure out what this stuff means). Of course, the Riesz representation theorem tells us that there exists a unique regular Borel measure which implements what we will call the Haar integral. This measure is precisely the Haar measure.

Since a Lie group is a manifold with a group structure, it is in particular a (very nice) topological group, certainly nice enough to satisfy all the hypotheses required to obtain a well-behaved, left-invariant integral. Although we will see that there are Lie groups for which this integral is not right-invariant, we will also discuss a property known as unimodularity, a situation where we do get right-invariance as well. There are enough examples of such Lie groups to make this worthwhile.

It turns out that, given a Lie group $G \leq \mathrm{GL}_n(\mathbb{F})$, its associated Lie algebra $\mathfrak{g} \leq \mathfrak{gl}_n(\mathbb{F}) = \mathrm{M}_n(\mathbb{F})$ is nothing more than $T_I G$, the tangent space at the identity (denoted $I$) of the smooth manifold $G$. Moreover, for any $g \in G$, the tangent space at $g$ is just given by $g \cdot \mathfrak{g} = \mathfrak{g} \cdot g$.

We want to be able to integrate functions $G \to \mathbb{C}$. Since $G$ is equipped with smooth manifold structure, there is an obvious strategy: use charts (“local coordinates”) to reduce our problem to integrating functions $\mathbb{R}^n \to \mathbb{C}$. This is exactly what we shall do, but figuring out how to do it properly necessitates a discussion of smooth functions and vector fields on $G$.

### Smooth functions and vector fields

The Lie algebra $\mathfrak{g}$ controls infinitesimal phenomena on $G$ in a crucial way, via the exponential map $\mathfrak{g} \to G$ which serves as a local diffeomorphism (since $\mathfrak{g}$ is a finite-dimensional vector space, it has a natural smooth structure). We fix neighbourhoods $U$ of $0$ in $\mathfrak{g}$ and $V$ of $I$ in $G$ such that $\exp : U \to V$ is a diffeomorphism. Since it is convenient, we define a smooth ($\mathcal{C}^\infty$) function on $G$ as one which is smooth with respect to that structure: that is,

$\displaystyle \mathcal{C}^\infty(G) = \{ f : G \to \mathbb{R} \mid f(g \cdot \exp(-)) \in \mathcal{C}^\infty(U) \text{ for all } g \in G \}.$

Having done this, we can introduce the notion of a vector field: these are functions $\xi : G \to \mathrm{M}_n(\mathbb{F})$ such that $\xi(g)$ is a tangent vector to $g$ for all $g \in G$. Since these $\xi$ are matrix-valued, we obtain $n^2$ coordinate functions $\xi_{ij} : G \to \mathbb{R}$ (or in the complex case, $2n^2$ such functions $\mathrm{Re} \; \xi_{ij} : G \to \mathbb{R}$ and $\mathrm{Im} \; \xi_{ij} : G \to \mathbb{R}$), so that

$\displaystyle \xi = \sum_{i,j=1}^n \xi_{ij} E_{ij} \qquad \text{or, if } \mathbb{F} = \mathbb{C}, \qquad \xi = \sum_{i,j=1}^n (\mathrm{Re} \; \xi_{ij} + i \mathrm{Im} \; \xi_{ij})E_{ij}.$

We say $\xi$ is smooth (or $\mathcal{C}^\infty$) if all the coordinate functions are in $\mathcal{C}^\infty(G)$. The set of all smooth vector fields on $G$ will be denoted $\Xi(G)$. Note that it carries the structure of a $\mathcal{C}^\infty(G)$-module: given $\xi, \eta \in \Xi(G)$ and $f \in \mathcal{C}^\infty(G)$ we define

$(f \cdot \xi)(g) = f(g) \xi(g), \qquad (\xi + \eta)(g) = \xi(g) + \eta(g)$

for all $g \in G$.

### Volume on G

Notice that since $T_g G = g \mathfrak{g}$ for any $g \in G$, all the tangent spaces have the same dimension $d := \dim \mathfrak{g}$. Now I am going to introduce something bizarre, which aims at the heart of why differential geometers use exterior algebra to define what are called “differential forms” (don’t worry, we won’t talk about these here, at least not unreservedly, heh heh…) to achieve the feat of integration on manifolds. We will discuss afterwards why anyone would ever do something like this, and hopefully it will become clear. We define

$\displaystyle \mathrm{Alt}^d(G) = \{ \omega : \Xi(G)^d \to \mathcal{C}^\infty(G) \mid \omega \text{ an alternating } \mathcal{C}^\infty\text{-multilinear map} \}$.

When I read the definition above, a myriad of questions filled my head, not all of which I have completely figured out (one that still remains is when do we take absolute values of volume forms, and when do we not?). I will now attempt to address them, as carefully as possible. Here, we want (note that I haven’t told you what $\omega_g$ are yet)

$\omega(\xi_1,\ldots,\xi_d)(g) = \omega_g(\xi_1(g), \ldots, \xi_d(g)).$

Also, by “multilinear map” we mean that in any given argument, $\omega$ behaves like a $\mathcal{C}^\infty$-module homomorphism, which we prefer to describe as “$\mathcal{C}^\infty$-linearity” by analogy with vector spaces. By “alternating” we mean that if you simply swap two of the inputs to $\omega$, then its value gets negated. We also require smoothness i.e. that for each $(\xi_1, \ldots, \xi_d) \in \Xi(G)$ and $g \in G$, the function $G \to \mathbb{R}$ given by

$g \mapsto \omega_g(\xi_1(g),\ldots,\xi_d(g))$

is smooth. Parsing this definition, we apparently want to talk about those things $\omega$, which consume $d$ vector fields on $G$, and spit out a smooth function on $G$, satisfying some properties. What is even going on? Note that giving an ordered list of $d$ vector fields on $G$ is the same as giving an “ordered parallelotope” at each point $g \in G$ in a smoothly varying way. Hmmm…

First note that such an $\omega$ (usually referred to as a density) is really a global object. It is defined on all of $G$. In contrast, if we fix a $g \in G$ and consider the tangent space $T_g G$, we note that (up to scaling) there is a unique way $\omega_g$ of assigning (unsigned) volumes to “ordered parallelotopes” in $T_g G$. By an ordered parallelotope, I mean an element of $(T_g G)^d$. That is, $\omega_g$ is the absolute value of an alternating $d$-linear form on $T_g G$, or phrased differently, $\omega_g$ is obtained from an element of the top exterior power $\bigwedge^d(T^*_g G)$ of the cotangent space. The reason for this is that the top exterior power of a vector space is one-dimensional (if you are interested, I may perhaps elaborate on this later). Roughly speaking, each $\omega_g$ gives us a local way of measuring volume, and an $\omega$ is the global object that results from smoothly piecing all these together.

### Final comments… because I have class tomorrow and it’s 2 am

I think this post is sufficiently long for the moment, but I am certainly not finished reflecting on these matters. In the next post, I will explore more geometric intuition about these “densities” (also, I may have said some false things about them; if I realize this is the case, I will point it out in the next post, but this one will not be updated). I will then proceed to talk about how we actually define the Haar integral by using partitions of unity, its invariance behaviour, and the concept of unimodularity. In the mean time, you should think about how the integration of a function $\mathbb{R}^n \to \mathbb{R}$ familiar from multivariable calculus can be cast as a special case of all this ridiculous machinery: the tangent spaces are spanned by directional derivatives like $\frac{\partial}{\partial x^1}, \ldots, \frac{\partial}{\partial x^d}$, so if we form the dual basis in the cotangent space we get some guys we might denote by $dx^1, \ldots, dx^n$. Then the top exterior power is spanned by $dx^1 \wedge \ldots \wedge dx^n$

just another guy trying to make the diagrams commute.
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### One Response to Towards integration on matrix Lie groups

1. Erik Crevier says:

So here is really the reason the absolute values come into the picture. In calc 1, you sometimes want things to work out so that $\int_0^1 dx=1$ while $\int_1^0 dx=-1$. On the other hand, sometimes you to just be able to write $\int_{[0,1]} 1 dm=1$, where $dm$ is a proto-lesbeque measure. If you want things to work consistently, then it really doesn’t make any sense to write $\int_{[0,1]}dx=1$

Generalizing $\int_0^1dx$ gives you integration of differential forms. If you actually look up in a differential geometry textbook precisely how this is defined, you’ll see that you are really only allowed to integrate a $k$-form $\omega$ on $M$ against a smooth map $C:[0,1]^k\to M$ (something a topologist might call a “smooth singular $k$-cube, which I mention because of a related notion that will probably come up soon in 467). And this makes sense when you consider what $\omega$ does on an infinitesimal level: It takes a $k$-tuple of tangent vectors (ie an “infinitesimal $k$-cube), and spits out a scalar. The integral $\int_C\omega$ is just a globablized version of this.

But if you want to be able to integrate functions, as opposed to differential forms, then you basically need to have some measure in the background (because of the Riesz representation theorem). In particular, you should be able to make sense of the measure of a set $E$ as being $\int_{E}1dm$. Because this doesn’t have any way of tracking orientations, it should act locally by sending an infinitesimal $k$-cubes to its unsigned volume, as opposed to a signed volume. This is why we need to work with $\int_G f |\omega|$ instead of $\int_G g\omega$.

By the way here is a rather lovely little expository article which helped me enormously at understanding this stuff.