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A popular way to visualise relationships between some finite number of sets is via Venn diagrams, or more generally Euler diagrams. In these diagrams, a set is depicted as a two-dimensional shape such as a disk or a rectangle, and the various Boolean relationships between these sets (e.g., that one set is contained in another, or that the intersection of two of the sets is equal to a third) is represented by the Boolean algebra of these shapes; Venn diagrams correspond to the case where the sets are in “general position” in the sense that all non-trivial Boolean combinations of the sets are non-empty. For instance to depict the general situation of two sets ${A,B}$ together with their intersection ${A \cap B}$ and ${A \cup B}$ one might use a Venn diagram such as

(where we have given each region depicted a different color, and moved the edges of each region a little away from each other in order to make them all visible separately), but if one wanted to instead depict a situation in which the intersection ${A \cap B}$ was empty, one could use an Euler diagram such as

One can use the area of various regions in a Venn or Euler diagram as a heuristic proxy for the cardinality ${|A|}$ (or measure ${\mu(A)}$) of the set ${A}$ corresponding to such a region. For instance, the above Venn diagram can be used to intuitively justify the inclusion-exclusion formula

$\displaystyle |A \cup B| = |A| + |B| - |A \cap B|$

for finite sets ${A,B}$, while the above Euler diagram similarly justifies the special case

$\displaystyle |A \cup B| = |A| + |B|$

for finite disjoint sets ${A,B}$.

While Venn and Euler diagrams are traditionally two-dimensional in nature, there is nothing preventing one from using one-dimensional diagrams such as

or even three-dimensional diagrams such as this one from Wikipedia:

Of course, in such cases one would use length or volume as a heuristic proxy for cardinality or measure, rather than area.

With the addition of arrows, Venn and Euler diagrams can also accommodate (to some extent) functions between sets. Here for instance is a depiction of a function ${f: A \rightarrow B}$, the image ${f(A)}$ of that function, and the image ${f(A')}$ of some subset ${A'}$ of ${A}$:

Here one can illustrate surjectivity of ${f: A \rightarrow B}$ by having ${f(A)}$ fill out all of ${B}$; one can similarly illustrate injectivity of ${f}$ by giving ${f(A)}$ exactly the same shape (or at least the same area) as ${A}$. So here for instance might be how one would illustrate an injective function ${f: A \rightarrow B}$:

Cartesian product operations can be incorporated into these diagrams by appropriate combinations of one-dimensional and two-dimensional diagrams. Here for instance is a diagram that illustrates the identity ${(A \cup B) \times C = (A \times C) \cup (B \times C)}$:

In this blog post I would like to propose a similar family of diagrams to illustrate relationships between vector spaces (over a fixed base field ${k}$, such as the reals) or abelian groups, rather than sets. The categories of (${k}$-)vector spaces and abelian groups are quite similar in many ways; the former consists of modules over a base field ${k}$, while the latter consists of modules over the integers ${{\bf Z}}$; also, both categories are basic examples of abelian categories. The notion of a dimension in a vector space is analogous in many ways to that of cardinality of a set; see this previous post for an instance of this analogy (in the context of Shannon entropy). (UPDATE: I have learned that an essentially identical notation has also been proposed in an unpublished manuscript of Ravi Vakil.)

In the last few notes, we have been steadily reducing the amount of regularity needed on a topological group in order to be able to show that it is in fact a Lie group, in the spirit of Hilbert’s fifth problem. Now, we will work on Hilbert’s fifth problem from the other end, starting with the minimal assumption of local compactness on a topological group ${G}$, and seeing what kind of structures one can build using this assumption. (For simplicity we shall mostly confine our discussion to global groups rather than local groups for now.) In view of the preceding notes, we would like to see two types of structures emerge in particular:

• representations of ${G}$ into some more structured group, such as a matrix group ${GL_n({\bf C})}$; and
• metrics on ${G}$ that capture the escape and commutator structure of ${G}$ (i.e. Gleason metrics).

To build either of these structures, a fundamentally useful tool is that of (left-) Haar measure – a left-invariant Radon measure ${\mu}$ on ${G}$. (One can of course also consider right-Haar measures; in many cases (such as for compact or abelian groups), the two concepts are the same, but this is not always the case.) This concept generalises the concept of Lebesgue measure on Euclidean spaces ${{\bf R}^d}$, which is of course fundamental in analysis on those spaces.

Haar measures will help us build useful representations and useful metrics on locally compact groups ${G}$. For instance, a Haar measure ${\mu}$ gives rise to the regular representation ${\tau: G \rightarrow U(L^2(G,d\mu))}$ that maps each element ${g \in G}$ of ${G}$ to the unitary translation operator ${\rho(g): L^2(G,d\mu) \rightarrow L^2(G,d\mu)}$ on the Hilbert space ${L^2(G,d\mu)}$ of square-integrable measurable functions on ${G}$ with respect to this Haar measure by the formula

$\displaystyle \tau(g) f(x) := f(g^{-1} x).$

(The presence of the inverse ${g^{-1}}$ is convenient in order to obtain the homomorphism property ${\tau(gh) = \tau(g)\tau(h)}$ without a reversal in the group multiplication.) In general, this is an infinite-dimensional representation; but in many cases (and in particular, in the case when ${G}$ is compact) we can decompose this representation into a useful collection of finite-dimensional representations, leading to the Peter-Weyl theorem, which is a fundamental tool for understanding the structure of compact groups. This theorem is particularly simple in the compact abelian case, where it turns out that the representations can be decomposed into one-dimensional representations ${\chi: G \rightarrow U({\bf C}) \equiv S^1}$, better known as characters, leading to the theory of Fourier analysis on general compact abelian groups. With this and some additional (largely combinatorial) arguments, we will also be able to obtain satisfactory structural control on locally compact abelian groups as well.

The link between Haar measure and useful metrics on ${G}$ is a little more complicated. Firstly, once one has the regular representation ${\tau: G\rightarrow U(L^2(G,d\mu))}$, and given a suitable “test” function ${\psi: G \rightarrow {\bf C}}$, one can then embed ${G}$ into ${L^2(G,d\mu)}$ (or into other function spaces on ${G}$, such as ${C_c(G)}$ or ${L^\infty(G)}$) by mapping a group element ${g \in G}$ to the translate ${\tau(g) \psi}$ of ${\psi}$ in that function space. (This map might not actually be an embedding if ${\psi}$ enjoys a non-trivial translation symmetry ${\tau(g)\psi=\psi}$, but let us ignore this possibility for now.) One can then pull the metric structure on the function space back to a metric on ${G}$, for instance defining an ${L^2(G,d\mu)}$-based metric

$\displaystyle d(g,h) := \| \tau(g) \psi - \tau(h) \psi \|_{L^2(G,d\mu)}$

if ${\psi}$ is square-integrable, or perhaps a ${C_c(G)}$-based metric

$\displaystyle d(g,h) := \| \tau(g) \psi - \tau(h) \psi \|_{C_c(G)} \ \ \ \ \ (1)$

if ${\psi}$ is continuous and compactly supported (with ${\|f \|_{C_c(G)} := \sup_{x \in G} |f(x)|}$ denoting the supremum norm). These metrics tend to have several nice properties (for instance, they are automatically left-invariant), particularly if the test function is chosen to be sufficiently “smooth”. For instance, if we introduce the differentiation (or more precisely, finite difference) operators

$\displaystyle \partial_g := 1-\tau(g)$

(so that ${\partial_g f(x) = f(x) - f(g^{-1} x)}$) and use the metric (1), then a short computation (relying on the translation-invariance of the ${C_c(G)}$ norm) shows that

$\displaystyle d([g,h], \hbox{id}) = \| \partial_g \partial_h \psi - \partial_h \partial_g \psi \|_{C_c(G)}$

for all ${g,h \in G}$. This suggests that commutator estimates, such as those appearing in the definition of a Gleason metric in Notes 2, might be available if one can control “second derivatives” of ${\psi}$; informally, we would like our test functions ${\psi}$ to have a “${C^{1,1}}$” type regularity.

If ${G}$ was already a Lie group (or something similar, such as a ${C^{1,1}}$ local group) then it would not be too difficult to concoct such a function ${\psi}$ by using local coordinates. But of course the whole point of Hilbert’s fifth problem is to do without such regularity hypotheses, and so we need to build ${C^{1,1}}$ test functions ${\psi}$ by other means. And here is where the Haar measure comes in: it provides the fundamental tool of convolution

$\displaystyle \phi * \psi(x) := \int_G \phi(y) \psi(y^{-1}x) d\mu(y)$

between two suitable functions ${\phi, \psi: G \rightarrow {\bf C}}$, which can be used to build smoother functions out of rougher ones. For instance:

Exercise 1 Let ${\phi, \psi: {\bf R}^d \rightarrow {\bf C}}$ be continuous, compactly supported functions which are Lipschitz continuous. Show that the convolution ${\phi * \psi}$ using Lebesgue measure on ${{\bf R}^d}$ obeys the ${C^{1,1}}$-type commutator estimate

$\displaystyle \| \partial_g \partial_h (\phi * \psi) \|_{C_c({\bf R}^d)} \leq C \|g\| \|h\|$

for all ${g,h \in {\bf R}^d}$ and some finite quantity ${C}$ depending only on ${\phi, \psi}$.

This exercise suggests a strategy to build Gleason metrics by convolving together some “Lipschitz” test functions and then using the resulting convolution as a test function to define a metric. This strategy may seem somewhat circular because one needs a notion of metric in order to define Lipschitz continuity in the first place, but it turns out that the properties required on that metric are weaker than those that the Gleason metric will satisfy, and so one will be able to break the circularity by using a “bootstrap” or “induction” argument.

We will discuss this strategy – which is due to Gleason, and is fundamental to all currently known solutions to Hilbert’s fifth problem – in later posts. In this post, we will construct Haar measure on general locally compact groups, and then establish the Peter-Weyl theorem, which in turn can be used to obtain a reasonably satisfactory structural classification of both compact groups and locally compact abelian groups.