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In the last few months, I have been working my way through the theory behind the solution to Hilbert’s fifth problem, as I (together with Emmanuel Breuillard, Ben Green, and Tom Sanders) have found this theory to be useful in obtaining noncommutative inverse sumset theorems in arbitrary groups; I hope to be able to report on this connection at some later point on this blog. Among other things, this theory achieves the remarkable feat of creating a smooth Lie group structure out of what is ostensibly a much weaker structure, namely the structure of a locally compact group. The ability of algebraic structure (in this case, group structure) to upgrade weak regularity (in this case, continuous structure) to strong regularity (in this case, smooth and even analytic structure) seems to be a recurring theme in mathematics, and an important part of what I like to call the “dichotomy between structure and randomness”.

The theory of Hilbert’s fifth problem sprawls across many subfields of mathematics: Lie theory, representation theory, group theory, nonabelian Fourier analysis, point-set topology, and even a little bit of group cohomology. The latter aspect of this theory is what I want to focus on today. The general question that comes into play here is the extension problem: given two (topological or Lie) groups {H} and {K}, what is the structure of the possible groups {G} that are formed by extending {H} by {K}. In other words, given a short exact sequence

\displaystyle  0 \rightarrow K \rightarrow G \rightarrow H \rightarrow 0,

to what extent is the structure of {G} determined by that of {H} and {K}?

As an example of why understanding the extension problem would help in structural theory, let us consider the task of classifying the structure of a Lie group {G}. Firstly, we factor out the connected component {G^\circ} of the identity as

\displaystyle  0 \rightarrow G^\circ \rightarrow G \rightarrow G/G^\circ \rightarrow 0;

as Lie groups are locally connected, {G/G^\circ} is discrete. Thus, to understand general Lie groups, it suffices to understand the extensions of discrete groups by connected Lie groups.

Next, to study a connected Lie group {G}, we can consider the conjugation action {g: X \mapsto gXg^{-1}} on the Lie algebra {{\mathfrak g}}, which gives the adjoint representation {\hbox{Ad}: G \rightarrow GL({\mathfrak g})}. The kernel of this representation consists of all the group elements {g} that commute with all elements of the Lie algebra, and thus (by connectedness) is the center {Z(G)} of {G}. The adjoint representation is then faithful on the quotient {G/Z(G)}. The short exact sequence

\displaystyle  0 \rightarrow Z(G) \rightarrow G \rightarrow G/Z(G) \rightarrow 0

then describes {G} as a central extension (by the abelian Lie group {Z(G)}) of {G/Z(G)}, which is a connected Lie group with a faithful finite-dimensional linear representation.

This suggests a route to Hilbert’s fifth problem, at least in the case of connected groups {G}. Let {G} be a connected locally compact group that we hope to demonstrate is isomorphic to a Lie group. As discussed in a previous post, we first form the space {L(G)} of one-parameter subgroups of {G} (which should, eventually, become the Lie algebra of {G}). Hopefully, {L(G)} has the structure of a vector space. The group {G} acts on {L(G)} by conjugation; this action should be both continuous and linear, giving an “adjoint representation” {\hbox{Ad}: G \rightarrow GL(L(G))}. The kernel of this representation should then be the center {Z(G)} of {G}. The quotient {G/Z(G)} is locally compact and has a faithful linear representation, and is thus a Lie group by von Neumann’s version of Cartan’s theorem (discussed in this previous post). The group {Z(G)} is locally compact abelian, and so it should be a relatively easy task to establish that it is also a Lie group. To finish the job, one needs the following result:

Theorem 1 (Central extensions of Lie are Lie) Let {G} be a locally compact group which is a central extension of a Lie group {H} by an abelian Lie group {K}. Then {G} is also isomorphic to a Lie group.

This result can be obtained by combining a result of Kuranishi with a result of Gleason; I am recording this argument below the fold. The point here is that while {G} is initially only a topological group, the smooth structures of {H} and {K} can be combined (after a little bit of cohomology) to create the smooth structure on {G} required to upgrade {G} from a topological group to a Lie group. One of the main ideas here is to improve the behaviour of a cocycle by averaging it; this basic trick is helpful elsewhere in the theory, resolving a number of cohomological issues in topological group theory. The result can be generalised to show in fact that arbitrary (topological) extensions of Lie groups by Lie groups remain Lie; this was shown by Gleason. However, the above special case of this result is already sufficient (in conjunction with the rest of the theory, of course) to resolve Hilbert’s fifth problem.

Remark 1 We have shown in the above discussion that every connected Lie group is a central extension (by an abelian Lie group) of a Lie group with a faithful continuous linear representation. It is natural to ask whether this central extension is necessary. Unfortunately, not every connected Lie group admits a faithful continuous linear representation. An example (due to Birkhoff) is the Heisenberg-Weyl group

\displaystyle  G := \begin{pmatrix} 1 & {\bf R} & {{\bf R}/{\bf Z}} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix} / \begin{pmatrix} 1 & 0 & {\bf Z} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.

Indeed, if we consider the group elements

\displaystyle  A := \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

and

\displaystyle  B := \begin{pmatrix} 1 & 0 & 1/p \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

for some prime {p}, then one easily verifies that {B} has order {p} and is central, and that {AB} is conjugate to {A}. If we have a faithful linear representation {\rho: G \rightarrow GL_n({\bf C})} of {G}, then {\rho(B)} must have at least one eigenvalue {\alpha} that is a primitive {p^{th}} root of unity. If {V} is the eigenspace associated to {\alpha}, then {\rho(A)} must preserve {V}, and be conjugate to {\alpha \rho(A)} on this space. This forces {\rho(A)} to have at least {p} distinct eigenvalues on {V}, and hence {V} (and thus {{\bf C}^n}) must have dimension at least {p}. Letting {p \rightarrow \infty} we obtain a contradiction. (On the other hand, {G} is certainly isomorphic to the extension of the linear group {{\bf R}^2} by the abelian group {{\bf R}/{\bf Z}}.)

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This is yet another post in a series on basic ingredients in the structural theory of locally compact groups, which is closely related to Hilbert’s fifth problem.

In order to understand the structure of a topological group {G}, a basic strategy is to try to split {G} into two smaller factor groups {H, K} by exhibiting a short exact sequence

\displaystyle  0 \rightarrow K \rightarrow G \rightarrow H \rightarrow 0.

If one has such a sequence, then {G} is an extension of {H} by {K} (which includes direct products {H \times K} and semidirect products {H \ltimes K} as examples, but can be more general than these situations, as discussed in this previous blog post). In principle, the problem of understanding the structure of {G} then splits into three simpler problems:

  1. (Horizontal structure) Understanding the structure of the “horizontal” group {H}.
  2. (Vertical structure) Understanding the structure of the “vertical” group {K}.
  3. (Cohomology) Understanding the ways in which one can extend {H} by {K}.

The “cohomological” aspect to this program can be nontrivial. However, in principle at least, this strategy reduces the study of the large group {G} to the study of the smaller groups {H, K}. (This type of splitting strategy is not restricted to topological groups, but can also be adapted to many other categories, particularly those of groups or group-like objects.) Typically, splitting alone does not fully kill off a structural classification problem, but it can reduce matters to studying those objects which are somehow “simple” or “irreducible”. For instance, this strategy can often be used to reduce questions about arbitrary finite groups to finite simple groups.

A simple example of splitting is as follows. Given any topological group {G}, one can form the connected component {G^\circ} of the identity – the maximal connected set containing the identity. It is not difficult to show that {G^\circ} is a closed (and thus also locally compact) normal subgroup of {G}, whose quotient {G/G^\circ} is another locally compact group. Furthermore, due to the maximal connected nature of {G^\circ}, {G/G^\circ} is totally disconnected – the only connected sets are the singletons. In particular, {G/G^\circ} is Hausdorff (the identity element is closed). Thus we have obtained a splitting

\displaystyle  0 \rightarrow G^\circ \rightarrow G \rightarrow G/G^\circ \rightarrow 0

of an arbitrary locally compact group into a connected locally compact group {G^\circ}, and a totally disconnected locally compact group {G/G^\circ}. In principle at least, the study of locally compact groups thus splits into the study of connected locally compact groups, and the study of totally disconnected locally compact groups (though the cohomological issues are not always trivial).

In the structural theory of totally disconnected locally compact groups, the first basic theorem in the subject is van Dantzig’s theorem (which we prove below the fold):

Theorem 1 (Van Danztig’s theorem) Every totally disconnected locally compact group {G} contains a compact open subgroup {H} (which will of course still be totally disconnected).

Example 1 Let {p} be a prime. Then the {p}-adic field {{\bf Q}_p} (with the usual {p}-adic valuation) is totally disconnected locally compact, and the {p}-adic integers {{\bf Z}_p} are a compact open subgroup.

Of course, this situation is the polar opposite of what occurs in the connected case, in which the only open subgroup is the whole group.

In view of van Dantzig’s theorem, we see that the “local” behaviour of totally disconnected locally compact groups can be modeled by the compact totally disconnected groups, which are better understood (for instance, one can start analysing them using the Peter-Weyl theorem, as discussed in this previous post). The global behaviour however remains more complicated, in part because the compact open subgroup given by van Dantzig’s theorem need not be normal, and so does not necessarily induce a splitting of {G} into compact and discrete factors.

Example 2 Let {p} be a prime, and let {G} be the semi-direct product {{\bf Z} \ltimes {\bf Q}_p}, where the integers {{\bf Z}} act on {{\bf Q}_p} by the map {m: x \mapsto p^m x}, and we give {G} the product of the discrete topology of {{\bf Z}} and the {p}-adic topology on {{\bf Q}_p}. One easily verifies that {G} is a totally disconnected locally compact group. It certainly has compact open subgroups, such as {\{0\} \times {\bf Z}_p}. However, it is easy to show that {G} has no non-trivial compact normal subgroups (the problem is that the conjugation action of {{\bf Z}} on {{\bf Q}_p} has all non-trivial orbits unbounded).

Returning to more general locally compact groups, we obtain an immediate corollary:

Corollary 2 Every locally compact group {G} contains an open subgroup {H} which is “compact-by-connected” in the sense that {H/H^\circ} is compact.

Indeed, one applies van Dantzig’s theorem to the totally disconnected group {G/G^\circ}, and then pulls back the resulting compact open subgroup.

Now we mention another application of van Dantzig’s theorem, of more direct relevance to Hilbert’s fifth problem. Define a generalised Lie group to be a topological group {G} with the property that given any open neighbourhood {U} of the identity, there exists an open subgroup {G'} of {G} and a compact normal subgroup {N} of {G'} in {U} such that {G'/N} is isomorphic to a Lie group. It is easy to see that such groups are locally compact. The deep Gleason-Yamabe theorem, which among other things establishes a satisfactory solution to Hilbert’s fifth problem (and which we will not prove here), asserts the converse:

Theorem 3 (Gleason-Yamabe theorem) Every locally compact group is a generalised Lie group.

Example 3 We consider the locally compact group {G = {\bf Z} \ltimes {\bf Q}_p} from Example 2. This is of course not a Lie group. However, any open neighbourhood {U} of the identity in {G} will contain the compact subgroup {N := \{0\} \times p^j {\bf Z}_p} for some integer {j}. The open subgroup {G' := \{0\} \times {\bf Z}_p} then has {G'/N} isomorphic to the discrete finite group {{\bf Z}/p^j{\bf Z}}, which is certainly a Lie group. Thus {G} is a generalised Lie group.

One important example of generalised Lie groups are those locally compact groups which are an inverse limit (or projective limit) of Lie groups. Indeed, suppose we have a family {(G_i)_{i\in I}} of Lie groups {G_i} indexed by partially ordered set {I} which is directed in the sense that every finite subset of {I} has an upper bound, together with continuous homomorphisms {\pi_{i \rightarrow j}: G_i \rightarrow G_j} for all {i > j} which form a category in the sense that {\pi_{j \rightarrow k} \circ \pi_{i \rightarrow j} = \pi_{i \rightarrow k}} for all {i>j>k}. Then we can form the inverse limit

\displaystyle G := \lim_{\stackrel{\leftarrow}{i \in I}} G_i,

which is the subgroup of {\prod_{i \in I} G_i} consisting of all tuples {(g_i)_{i \in I} \in \prod_{i \in I} G_i} which are compatible with the {\pi_{i \rightarrow j}} in the sense that {\pi_{i \rightarrow j}(g_i) = g_j} for all {i>j}. If we endow {\prod_{i \in I} G_i} with the product topology, then {G} is a closed subgroup of {\prod_{i \in I} G_i}, and thus has the structure of a topological group, with continuous homomorphisms {\pi_i: G \rightarrow G_i} which are compatible with the {\pi_{i \rightarrow j}} in the sense that {\pi_{i \rightarrow j} \circ \pi_i = \pi_j} for all {i>j}. Such an inverse limit need not be locally compact; for instance, the inverse limit

\displaystyle \lim_{\stackrel{\leftarrow}{n \in {\bf N}}} {\bf R}^n

of Euclidean spaces with the usual coordinate projection maps is isomorphic to the infinite product space {{\bf R}^{\bf N}} with the product topology, which is not locally compact. However, if an inverse limit

\displaystyle G = \lim_{\stackrel{\leftarrow}{i \in I}} G_i

of Lie groups is locally compact, it can be easily seen to be a generalised Lie group. Indeed, by local compactness, any open neighbourhood {G} of the identity will contain an open precompact neighbourhood of the identity; by construction of the product topology (and the directed nature of {I}), this smaller neighbourhood will in turn will contain the kernel of one of the {\pi_i}, which will be compact since the preceding neighbourhood was precompact. Quotienting out by this {\pi_i} we obtain a locally compact subgroup of the Lie group {G_i}, which is necessarily again a Lie group by Cartan’s theorem, and the claim follows.

In the converse direction, it is possible to use Corollary 2 to obtain the following observation of Gleason:

Theorem 4 Every Hausdorff generalised Lie group contains an open subgroup that is an inverse limit of Lie groups.

We show Theorem 4 below the fold. Combining this with the (substantially more difficult) Gleason-Yamabe theorem, we obtain quite a satisfactory description of the local structure of locally compact groups. (The situation is particularly simple for connected groups, which have no non-trivial open subgroups; we then conclude that every connected locally compact Hausdorff group is the inverse limit of Lie groups.)

Example 4 The locally compact group {G := {\bf Z} \ltimes {\bf Q}_p} is not an inverse limit of Lie groups because (as noted earlier) it has no non-trivial compact normal subgroups, which would contradict the preceding analysis that showed that all locally compact inverse limits of Lie groups were generalised Lie groups. On the other hand, {G} contains the open subgroup {\{0\} \times {\bf Q}_p}, which is the inverse limit of the discrete (and thus Lie) groups {\{0\} \times {\bf Q}_p/p^j {\bf Z}_p} for {j \in {\bf Z}} (where we give {{\bf Z}} the usual ordering, and use the obvious projection maps).

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This is another post in a series on various components to the solution of Hilbert’s fifth problem. One interpretation of this problem is to ask for a purely topological classification of the topological groups which are isomorphic to Lie groups. (Here we require Lie groups to be finite-dimensional, but allow them to be disconnected.)

There are some obvious necessary conditions on a topological group in order for it to be isomorphic to a Lie group; for instance, it must be Hausdorff and locally compact. These two conditions, by themselves, are not quite enough to force a Lie group structure; consider for instance a {p}-adic field {{\mathbf Q}_p} for some prime {p}, which is a locally compact Hausdorff topological group which is not a Lie group (the topology is locally that of a Cantor set). Nevertheless, it turns out that by adding some key additional assumptions on the topological group, one can recover Lie structure. One such result, which is a key component of the full solution to Hilbert’s fifth problem, is the following result of von Neumann:

Theorem 1 Let {G} be a locally compact Hausdorff topological group that has a faithful finite-dimensional linear representation, i.e. an injective continuous homomorphism {\rho: G \rightarrow GL_d({\bf C})} into some linear group. Then {G} can be given the structure of a Lie group. Furthermore, after giving {G} this Lie structure, {\rho} becomes smooth (and even analytic) and non-degenerate (the Jacobian always has full rank).

This result is closely related to a theorem of Cartan:

Theorem 2 (Cartan’s theorem) Any closed subgroup {H} of a Lie group {G}, is again a Lie group (in particular, {H} is an analytic submanifold of {G}, with the induced analytic structure).

Indeed, Theorem 1 immediately implies Theorem 2 in the important special case when the ambient Lie group is a linear group, and in any event it is not difficult to modify the proof of Theorem 1 to give a proof of Theorem 2. However, Theorem 1 is more general than Theorem 2 in some ways. For instance, let {G} be the real line {{\bf R}}, which we faithfully represent in the {2}-torus {({\bf R}/{\bf Z})^2} using an irrational embedding {t \mapsto (t,\alpha t) \hbox{ mod } {\bf Z}^2} for some fixed irrational {\alpha}. The {2}-torus can in turn be embedded in a linear group (e.g. by identifying it with {U(1) \times U(1)}, or {SO(2) \times SO(2)}), thus giving a faithful linear representation {\rho} of {{\bf R}}. However, the image is not closed (it is a dense subgroup of a {2}-torus), and so Cartan’s theorem does not directly apply ({\rho({\bf R})} fails to be a Lie group). Nevertheless, Theorem 1 still applies and guarantees that the original group {{\bf R}} is a Lie group.

(On the other hand, the image of any compact subset of {G} under a faithful representation {\rho} must be closed, and so Theorem 1 is very close to the version of Theorem 2 for local groups.)

The key to building the Lie group structure on a topological group is to first build the associated Lie algebra structure, by means of one-parameter subgroups.

Definition 3 A one-parameter subgroup of a topological group {G} is a continuous homomorphism {\phi: {\bf R} \rightarrow G} from the real line (with the additive group structure) to {G}.

Remark 1 Technically, {\phi} is a parameterisation of a subgroup {\phi({\bf R})}, rather than a subgroup itself, but we will abuse notation and refer to {\phi} as the subgroup.

In a Lie group {G}, the one-parameter subgroups are in one-to-one correspondence with the Lie algebra {{\mathfrak g}}, with each element {X \in {\mathfrak g}} giving rise to a one-parameter subgroup {\phi(t) := \exp(tX)}, and conversely each one-parameter subgroup {\phi} giving rise to an element {\phi'(0)} of the Lie algebra; we will establish these basic facts in the special case of linear groups below the fold. On the other hand, the notion of a one-parameter subgroup can be defined in an arbitrary topological group. So this suggests the following strategy if one is to try to represent a topological group {G} as a Lie group:

  1. First, form the space {L(G)} of one-parameter subgroups of {G}.
  2. Show that {L(G)} has the structure of a (finite-dimensional) Lie algebra.
  3. Show that {L(G)} “behaves like” the tangent space of {G} at the identity (in particular, the one-parameter subgroups in {L(G)} should cover a neighbourhood of the identity in {G}).
  4. Conclude that {G} has the structure of a Lie group.

It turns out that this strategy indeed works to give Theorem 1 (and variants of this strategy are ubiquitious in the rest of the theory surrounding Hilbert’s fifth problem).

Below the fold, I record the proof of Theorem 1 (based on the exposition of Montgomery and Zippin). I plan to organise these disparate posts surrounding Hilbert’s fifth problem (and its application to related topics, such as Gromov’s theorem or to the classification of approximate groups) at a later date.

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Recall that a (real) topological vector space is a real vector space {V = (V, 0, +, \cdot)} equipped with a topology {{\mathcal F}} that makes the vector space operations {+: V \times V \rightarrow V} and {\cdot: {\bf R} \times V \rightarrow V} continuous. One often restricts attention to Hausdorff topological vector spaces; in practice, this is not a severe restriction because it turns out that any topological vector space can be made Hausdorff by quotienting out the closure {\overline{\{0\}}} of the origin {\{0\}}. One can also discuss complex topological vector spaces, and the theory is not significantly different; but for sake of exposition we shall restrict attention here to the real case.

An obvious example of a topological vector space is a finite-dimensional vector space such as {{\bf R}^n} with the usual topology. Of course, there are plenty of infinite-dimensional topological vector spaces also, such as infinite-dimensional normed vector spaces (with the strong, weak, or weak-* topologies) or Frechet spaces.

One way to distinguish the finite and infinite dimensional topological vector spaces is via local compactness. Recall that a topological space is locally compact if every point in that space has a compact neighbourhood. From the Heine-Borel theorem, all finite-dimensional vector spaces (with the usual topology) are locally compact. In infinite dimensions, one can trivially make a vector space locally compact by giving it a trivial topology, but once one restricts to the Hausdorff case, it seems impossible to make a space locally compact. For instance, in an infinite-dimensional normed vector space {V} with the strong topology, an iteration of the Riesz lemma shows that the closed unit ball {B} in that space contains an infinite sequence with no convergent subsequence, which (by the Heine-Borel theorem) implies that {V} cannot be locally compact. If one gives {V} the weak-* topology instead, then {B} is now compact by the Banach-Alaoglu theorem, but is no longer a neighbourhood of the identity in this topology. In fact, we have the following result:

Theorem 1 Every locally compact Hausdorff topological vector space is finite-dimensional.

The first proof of this theorem that I am aware of is by André Weil. There is also a related result:

Theorem 2 Every finite-dimensional Hausdorff topological vector space has the usual topology.

As a corollary, every locally compact Hausdorff topological vector space is in fact isomorphic to {{\bf R}^n} with the usual topology for some {n}. This can be viewed as a very special case of the theorem of Gleason, which is a key component of the solution to Hilbert’s fifth problem, that a locally compact group {G} with no small subgroups (in the sense that there is a neighbourhood of the identity that contains no non-trivial subgroups) is necessarily isomorphic to a Lie group. Indeed, Theorem 1 is in fact used in the proof of Gleason’s theorem (the rough idea being to first locate a “tangent space” to {G} at the origin, with the tangent vectors described by “one-parameter subgroups” of {G}, and show that this space is a locally compact Hausdorff topological space, and hence finite dimensional by Theorem 1).

Theorem 2 may seem devoid of content, but it does contain some subtleties, as it hinges crucially on the joint continuity of the vector space operations {+: V \times V \rightarrow V} and {\cdot: {\bf R} \times V \rightarrow V}, and not just on the separate continuity in each coordinate. Consider for instance the one-dimensional vector space {{\bf R}} with the co-compact topology (a non-empty set is open iff its complement is compact in the usual topology). In this topology, the space is {T_1} (though not Hausdorff), the scalar multiplication map {\cdot: {\bf R} \times {\bf R} \rightarrow {\bf R}} is jointly continuous as long as the scalar is not zero, and the addition map {+: {\bf R} \times {\bf R} \rightarrow {\bf R}} is continuous in each coordinate (i.e. translations are continuous), but not jointly continuous; for instance, the set {\{ (x,y) \in {\bf R}: x+y \not \in [0,1]\}} does not contain a non-trivial Cartesian product of two sets that are open in the co-compact topology. So this is not a counterexample to Theorem 2. Similarly for the cocountable or cofinite topologies on {{\bf R}} (the latter topology, incidentally, is the same as the Zariski topology on {{\bf R}}).

Another near-counterexample comes from the topology of {{\bf R}} inherited by pulling back the usual topology on the unit circle {{\bf R}/{\bf Z}}. Admittedly, this pullback topology is not quite Hausdorff, but the addition map {+: {\bf R} \times {\bf R} \rightarrow {\bf R}} is jointly continuous. On the other hand, the scalar multiplication map {\cdot: {\bf R} \times {\bf R} \rightarrow {\bf R}} is not continuous at all. A slight variant of this topology comes from pulling back the usual topology on the torus {({\bf R}/{\bf Z})^2} under the map {x \mapsto (x,\alpha x)} for some irrational {\alpha}; this restores the Hausdorff property, and addition is still jointly continuous, but multiplication remains discontinuous.

As some final examples, consider {{\bf R}} with the discrete topology; here, the topology is Hausdorff, addition is jointly continuous, and every dilation is continuous, but multiplication is not jointly continuous. If one instead gives {{\bf R}} the half-open topology, then again the topology is Hausdorff and addition is jointly continuous, but scalar multiplication is only jointly continuous once one restricts the scalar to be non-negative.

Below the fold, I record the textbook proof of Theorem 2 and Theorem 1. There is nothing particularly original in this presentation, but I wanted to record it here for my own future reference, and perhaps these results will also be of interest to some other readers.

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A topological space {X} is said to be metrisable if one can find a metric {d: X \times X \rightarrow [0,+\infty)} on it whose open balls {B(x,r) := \{ y \in X: d(x,y) < r \}} generate the topology.

There are some obvious necessary conditions on the space {X} in order for it to be metrisable. For instance, it must be Hausdorff, since all metric spaces are Hausdorff. It must also be first countable, because every point {x} in a metric space has a countable neighbourhood base of balls {B(x,1/n)}, {n=1,2,\ldots}.

In the converse direction, being Hausdorff and first countable is not always enough to guarantee metrisability, for a variety of reasons. For instance the long line is not metrisable despite being both Hausdorff and first countable, due to a failure of paracompactness, which prevents one from gluing together the local metric structures on this line into a global one. Even after adding in paracompactness, this is still not enough; the real line with the lower limit topology (also known as the Sorgenfrey line) is Hausdorff, first countable, and paracompact, but still not metrisable (because of a failure of second countability despite being separable).

However, there is one important setting in which the Hausdorff and first countability axioms do suffice to give metrisability, and that is the setting of topological groups:

Theorem 1 (Birkhoff-Kakutani theorem) Let {G} be a topological group (i.e. a topological space that is also a group, such that the group operations {\cdot: G \times G \rightarrow G} and {()^{-1}: G \rightarrow G} are continuous). Then {G} is metrisable if and only if it is both Hausdorff and first countable.

Remark 1 It is not hard to show that a topological group is Hausdorff if and only if the singleton set {\{\hbox{id}\}} is closed. More generally, in an arbitrary topological group, it is a good exercise to show that the closure of {\{\hbox{id}\}} is always a closed normal subgroup {H} of {G}, whose quotient {G/H} is then a Hausdorff topological group. Because of this, the study of topological groups can usually be reduced immediately to the study of Hausdorff topological groups. (Indeed, in many texts, topological groups are automatically understood to be an abbreviation for “Hausdorff topological group”.)

The standard proof of the Birkhoff-Kakutani theorem (which we have taken from this book of Montgomery and Zippin) relies on the following Urysohn-type lemma:

Lemma 2 (Urysohn-type lemma) Let {G} be a Hausdorff first countable group. Then there exists a bounded continuous function {f: G \rightarrow [0,1]} with the following properties:

  • (Unique maximum) {f(\hbox{id}) = 1}, and {f(x) < 1} for all {x \neq \hbox{id}}.
  • (Neighbourhood base) The sets {\{ x \in G: f(x) > 1-1/n \}} for {n=1,2,\ldots} form a neighbourhood base at the identity.
  • (Uniform continuity) For every {\varepsilon > 0}, there exists an open neighbourhood {U} of the identity such that {|f(gx)-f(x)| \leq \epsilon} for all {g \in U} and {x \in G}.

Note that if {G} had a left-invariant metric, then the function {f(x) := \max( 1 - \hbox{dist}(x,\hbox{id}), 0)} would suffice for this lemma, which already gives some indication as to why this lemma is relevant to the Birkhoff-Kakutani theorem.

Let us assume Lemma 2 for now and finish the proof of the Birkhoff-Kakutani theorem. We only prove the difficult direction, namely that a Hausdorff first countable topological group {G} is metrisable. We let {f} be the function from Lemma 2, and define the function {d_f := G \times G \rightarrow [0,+\infty)} by the formula

\displaystyle  d_f( g, h ) := \| \tau_g f - \tau_h f \|_{BC(G)} = \sup_{x \in G} |f(g^{-1} x) - f(h^{-1} x)| \ \ \ \ \ (1)

where {BC(G)} is the space of bounded continuous functions on {G} (with the supremum norm) and {\tau_g} is the left-translation operator {\tau_g f(x) := f(g^{-1} x)}.

Clearly {d_f} obeys the the identity {d_f(g,g) = 0} and symmetry {d_f(g,h) = d_f(h,g)} axioms, and the triangle inequality {d_f(g,k) \leq d_f(g,h) + d_f(h,k)} is also immediate. This already makes {d_f} a pseudometric. In order for {d_f} to be a genuine metric, what is needed is that {f} have no non-trivial translation invariances, i.e. one has {\tau_g f \neq f} for all {g \neq \hbox{id}}. But this follows since {f} attains its maximum at exactly one point, namely the group identity {\hbox{id}}.

To put it another way: because {f} has no non-trivial translation invariances, the left translation action {\tau} gives an embedding {g \mapsto \tau_g f}, and {G} then inherits a metric {d_f} from the metric structure on {BC(G)}.

Now we have to check whether the metric {d_f} actually generates the topology. This amounts to verifying two things. Firstly, that every ball {B(x,r)} in this metric is open; and secondly, that every open neighbourhood of a point {x \in G} contains a ball {B(x,r)}.

To verify the former claim, it suffices to show that the map {g \mapsto \tau_g f} from {G} to {BC(G)} is continuous, follows from the uniform continuity hypothesis. The second claim follows easily from the neighbourhood base hypothesis, since if {d_f(g,h) < 1/n} then {f(g^{-1} h) > 1-1/n}.

Remark 2 The above argument in fact shows that if a group {G} is metrisable, then it admits a left-invariant metric. The idea of using a suitable continuous function {f} to generate a useful metric structure on a topological group is a powerful one, for instance underlying the Gleason lemmas which are fundamental to the solution of Hilbert’s fifth problem. I hope to return to this topic in a future post.

Now we prove Lemma 2. By first countability, we can find a countable neighbourhood base

\displaystyle  V_1 \supset V_2 \supset \ldots \supset \{\hbox{id}\}

of the identity. As {G} is Hausdorff, we must have

\displaystyle  \bigcap_{n=1}^\infty V_n = \{\hbox{id}\}.

Using the continuity of the group axioms, we can recursively find a sequence of nested open neighbourhoods of the identity

\displaystyle  U_1 \supset U_{1/2} \supset U_{1/4} \supset \ldots \supset \{\hbox{id}\} \ \ \ \ \ (2)

such that each {U_{1/2^n}} is symmetric (i.e. {g \in U_{1/2^n}} if and only if {g^{-1} \in U_{1/2^n}}), is contained in {V_n}, and is such that {U_{1/2^{n+1}} \cdot U_{1/2^{n+1}} \subset U_{1/2^n}} for each {n \geq 0}. In particular the {U_{1/2^n}} are also a neighbourhood base of the identity with

\displaystyle  \bigcap_{n=1}^\infty U_{1/2^n} = \{\hbox{id}\}. \ \ \ \ \ (3)

For every dyadic rational {a/2^n} in {(0,1)}, we can now define the open sets {U_{a/2^n}} by setting

\displaystyle  U_{a/2^n} := U_{1/2^{n_k}} \cdot \ldots \cdot U_{1/2^{n_1}}

where {a/2^n = 2^{-n_1} + \ldots + 2^{-n_k}} is the binary expansion of {a/2^n} with {1 \leq n_1 < \ldots < n_k}. By repeated use of the hypothesis {U_{1/2^{n+1}} \cdot U_{1/2^{n+1}} \subset U_{1/2^n}} we see that the {U_{a/2^n}} are increasing in {a/2^n}; indeed, we have the inclusion

\displaystyle  U_{1/2^n} \cdot U_{a/2^n} \subset U_{(a+1)/2^n} \ \ \ \ \ (4)

for all {n \geq 1} and {1 \leq a < 2^n}.

We now set

\displaystyle  f(x) := \sup \{ 1 - \frac{a}{2^n}: n \geq 1; 1 \leq a < 2^n; x \in U_{a/2^n} \}

with the understanding that {f(x)=0} if the supremum is over the empty set. One easily verifies using (4) that {f} is continuous, and furthermore obeys the uniform continuity property. The neighbourhood base property follows since the {U_{1/2^n}} are a neighbourhood base of the identity, and the unique maximum property follows from (3). This proves Lemma 2.

Remark 3 A very similar argument to the one above also establishes that every topological group {G} is completely regular.

Notice that the function {f} constructed in the above argument was localised to the set {V_1}. As such, it is not difficult to localise the Birkhoff-Kakutani theorem to local groups. A local group is a topological space {G} equipped with an identity {\hbox{id}}, a partially defined inversion operation {()^{-1}: \Lambda \rightarrow G}, and a partially defined product operation {\cdot: \Omega \rightarrow G}, where {\Lambda}, {\Omega} are open subsets of {G} and {G \times G}, obeying the following restricted versions of the group axioms:

  1. (Continuity) {\cdot} and {()^{-1}} are continuous on their domains of definition.
  2. (Identity) For any {g \in G}, {\hbox{id} \cdot g} and {g \cdot \hbox{id}} are well-defined and equal to {g}.
  3. (Inverse) For any {g \in \Lambda}, {g \cdot g^{-1}} and {g^{-1} \cdot g} are well-defined and equal to {\hbox{id}}. {\hbox{id}^{-1}} is well-defined and equal to {\hbox{id}}.
  4. (Local associativity) If {g, h, k \in G} are such that {g \cdot h}, {(g \cdot h) \cdot k}, {h \cdot k}, and {g \cdot (h \cdot k)} are all well-defined, then {(g \cdot h) \cdot k = g \cdot (h \cdot k)}.

Informally, one can view a local group as a topological group in which the closure axiom has been almost completely dropped, but with all the other axioms retained. A basic way to generate a local group is to start with an ordinary topological group {G} and restrict it to an open neighbourhood {U} of the identity, with {\Lambda := \{ g \in U: g^{-1} \in U \}} and {\Omega := \{ (g,h) \in U \times U: gh \in U \}}. However, this is not quite the only way to generate local groups (ultimately because the local associativity axiom does not necessarily imply a (stronger) global associativity axiom in which one considers two different ways to multiply more than three group elements together).

Remark 4 Another important example of a local group is that of a group chunk, in which the sets {\Lambda} and {\Omega} are somehow “generic”; for instance, {G} could be an algebraic variety, {\Lambda, \Omega} Zariski-open, and the group operations birational on their domains of definition. This is somewhat analogous to the notion of a “{99\%} group” in additive combinatorics. There are a number of group chunk theorems, starting with a theorem of Weil in the algebraic setting, which roughly speaking assert that a generic portion of a group chunk can be identified with the generic portion of a genuine group.

We then have

Theorem 3 (Birkhoff-Kakutani theorem for local groups) Let {G} be a local group which is Hausdorff and first countable. Then there exists an open neighbourhood {V_0} of the identity which is metrisable.

Proof: (Sketch) It is not difficult to see that in a local group {G}, one can find a symmetric neighbourhood {V_0} of the identity such that the product of any {100} (say) elements of {V_0} (multiplied together in any order) are well-defined, which effectively allows us to treat elements of {V_0} as if they belonged to a group for the purposes of simple algebraic manipulation, such as applying the cancellation laws {gh=gk \implies h=k} for {g,h,k \in V_0}. Inside this {V_0}, one can then repeat the previous arguments and eventually end up with a continuous function {f \in BC(G)} supported in {V_0} obeying the conclusions of Lemma 2 (but in the uniform continuity conclusion, one has to restrict {x} to, say, {V_0^{10}}, to avoid issues of ill-definedness). The definition (1) then gives a metric on {V_0} with the required properties, where we make the convention that {\tau_g f(x)} vanishes for {x \not \in V_0^{10}} (say) and {g \in V_0}. \Box

My motivation for studying local groups is that it turns out that there is a correspondence (first observed by Hrushovski) between the concept of an approximate group in additive combinatorics, and a locally compact local group in topological group theory; I hope to discuss this correspondence further in a subsequent post.

Recall that a (complex) abstract Lie algebra is a complex vector space {{\mathfrak g}} (either finite or infinite dimensional) equipped with a bilinear antisymmetric form {[]: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}} that obeys the Jacobi identity

\displaystyle  [[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] = 0. \ \ \ \ \ (1)

(One can of course define Lie algebras over other fields than the complex numbers {{\bf C}}, but in order to avoid some technical issues we shall work solely with the complex case in this post.)

An important special case of the abstract Lie algebras are the concrete Lie algebras, in which {{\mathfrak g} \subset \hbox{End}(V)} is a vector space of linear transformations {X: V \rightarrow V} on a vector space {V} (which again can be either finite or infinite dimensional), and the bilinear form is given by the usual Lie bracket

\displaystyle  [X,Y] := XY-YX.

It is easy to verify that every concrete Lie algebra is an abstract Lie algebra. In the converse direction, we have

Theorem 1 Every abstract Lie algebra is isomorphic to a concrete Lie algebra.

To prove this theorem, we introduce the useful algebraic tool of the universal enveloping algebra {U({\mathfrak g})} of the abstract Lie algebra {{\mathfrak g}}. This is the free (associative, complex) algebra generated by {{\mathfrak g}} (viewed as a complex vector space), subject to the constraints

\displaystyle  [X,Y] = XY - YX. \ \ \ \ \ (2)

This algebra is described by the Poincaré-Birkhoff-Witt theorem, which asserts that given an ordered basis {(X_i)_{i \in I}} of {{\mathfrak g}} as a vector space, that a basis of {U({\mathfrak g})} is given by “monomials” of the form

\displaystyle  X_{i_1}^{a_1} \ldots X_{i_m}^{a_m} \ \ \ \ \ (3)

where {m} is a natural number, the {i_1 < \ldots < i_m} are an increasing sequence of indices in {I}, and the {a_1,\ldots,a_m} are positive integers. Indeed, given two such monomials, one can express their product as a finite linear combination of further monomials of the form (3) after repeatedly applying (2) (which we rewrite as {XY = YX + [X,Y]}) to reorder the terms in this product modulo lower order terms until one all monomials have their indices in the required increasing order. It is then a routine exercise in basic abstract algebra (using all the axioms of an abstract Lie algebra) to verify that this is multiplication rule on monomials does indeed define a complex associative algebra which has the universal properties required of the universal enveloping algebra.

The abstract Lie algebra {{\mathfrak g}} acts on its universal enveloping algebra {U({\mathfrak g})} by left-multiplication: {X: M \mapsto XM}, thus giving a map from {{\mathfrak g}} to {\hbox{End}(U({\mathfrak g}))}. It is easy to verify that this map is a Lie algebra homomorphism (so this is indeed an action (or representation) of the Lie algebra), and this action is clearly faithful (i.e. the map from {{\mathfrak g}} to {\hbox{End}(U{\mathfrak g})} is injective), since each element {X} of {{\mathfrak g}} maps the identity element {1} of {U({\mathfrak g})} to a different element of {U({\mathfrak g})}, namely {X}. Thus {{\mathfrak g}} is isomorphic to its image in {\hbox{End}(U({\mathfrak g}))}, proving Theorem 1.

In the converse direction, every representation {\rho: {\mathfrak g} \rightarrow \hbox{End}(V)} of a Lie algebra “factors through” the universal enveloping algebra, in that it extends to an algebra homomorphism from {U({\mathfrak g})} to {\hbox{End}(V)}, which by abuse of notation we shall also call {\rho}.

One drawback of Theorem 1 is that the space {U({\mathfrak g})} that the concrete Lie algebra acts on will almost always be infinite-dimensional, even when the original Lie algebra {{\mathfrak g}} is finite-dimensional. However, there is a useful theorem of Ado that rectifies this:

Theorem 2 (Ado’s theorem) Every finite-dimensional abstract Lie algebra is isomorphic to a concrete Lie algebra over a finite-dimensional vector space {V}.

Among other things, this theorem can be used (in conjunction with the Baker-Campbell-Hausdorff formula) to show that every abstract (finite-dimensional) Lie group (or abstract local Lie group) is locally isomorphic to a linear group. (It is well-known, though, that abstract Lie groups are not necessarily globally isomorphic to a linear group, but we will not discuss these global obstructions here.)

Ado’s theorem is surprisingly tricky to prove in general, but some special cases are easy. For instance, one can try using the adjoint representation {\hbox{ad}: {\mathfrak g} \rightarrow \hbox{End}({\mathfrak g})} of {{\mathfrak g}} on itself, defined by the action {X: Y \mapsto [X,Y]}; the Jacobi identity (1) ensures that this indeed a representation of {{\mathfrak g}}. The kernel of this representation is the centre {Z({\mathfrak g}) := \{ X \in {\mathfrak g}: [X,Y]=0 \hbox{ for all } Y \in {\mathfrak g}\}}. This already gives Ado’s theorem in the case when {{\mathfrak g}} is semisimple, in which case the center is trivial.

The adjoint representation does not suffice, by itself, to prove Ado’s theorem in the non-semisimple case. However, it does provide an important reduction in the proof, namely it reduces matters to showing that every finite-dimensional Lie algebra {{\mathfrak g}} has a finite-dimensional representation {\rho: {\mathfrak g} \rightarrow \hbox{End}(V)} which is faithful on the centre {Z({\mathfrak g})}. Indeed, if one has such a representation, one can then take the direct sum of that representation with the adjoint representation to obtain a new finite-dimensional representation which is now faithful on all of {{\mathfrak g}}, which then gives Ado’s theorem for {{\mathfrak g}}.

It remins to find a finite-dimensional representation of {{\mathfrak g}} which is faithful on the centre {Z({\mathfrak g})}. In the case when {{\mathfrak g}} is abelian, so that the centre {Z({\mathfrak g})} is all of {{\mathfrak g}}, this is again easy, because {{\mathfrak g}} then acts faithfully on {{\mathfrak g} \times {\bf C}} by the infinitesimal shear maps {X: (Y,t) \mapsto (tX, 0)}. In matrix form, this representation identifies each {X} in this abelian Lie algebra with an “upper-triangular” matrix:

\displaystyle  X \equiv \begin{pmatrix} 0 & X \\ 0 & 0 \end{pmatrix}.

This construction gives a faithful finite-dimensional representation of the centre {Z({\mathfrak g})} of any finite-dimensional Lie algebra. The standard proof of Ado’s theorem (which I believe dates back to work of Harish-Chandra) then proceeds by gradually “extending” this representation of the centre {Z({\mathfrak g})} to larger and larger sub-algebras of {{\mathfrak g}}, while preserving the finite-dimensionality of the representation and the faithfulness on {Z({\mathfrak g})}, until one obtains a representation on the entire Lie algebra {{\mathfrak g}} with the required properties. (For technical inductive reasons, one also needs to carry along an additional property of the representation, namely that it maps the nilradical to nilpotent elements, but we will discuss this technicality later.)

This procedure is a little tricky to execute in general, but becomes simpler in the nilpotent case, in which the lower central series {{\mathfrak g}_1 := {\mathfrak g}; {\mathfrak g}_{n+1} := [{\mathfrak g}, {\mathfrak g}_n]} becomes trivial for sufficiently large {n}:

Theorem 3 (Ado’s theorem for nilpotent Lie algebras) Let {{\mathfrak n}} be a finite-dimensional nilpotent Lie algebra. Then there exists a finite-dimensional faithful representation {\rho: {\mathfrak n} \rightarrow \hbox{End}(V)} of {{\mathfrak n}}. Furthermore, there exists a natural number {k} such that {\rho({\mathfrak n})^k = \{0\}}, i.e. one has {\rho(X_1) \ldots \rho(X_k)=0} for all {X_1,\ldots,X_k \in {\mathfrak n}}.

The second conclusion of Ado’s theorem here is useful for induction purposes. (By Engel’s theorem, this conclusion is also equivalent to the assertion that every element of {\rho({\mathfrak n})} is nilpotent, but we can prove Theorem 3 without explicitly invoking Engel’s theorem.)

Below the fold, I give a proof of Theorem 3, and then extend the argument to cover the full strength of Ado’s theorem. This is not a new argument – indeed, I am basing this particular presentation from the one in Fulton and Harris – but it was an instructive exercise for me to try to extract the proof of Ado’s theorem from the more general structural theory of Lie algebras (e.g. Engel’s theorem, Lie’s theorem, Levi decomposition, etc.) in which the result is usually placed. (However, the proof I know of still needs Engel’s theorem to establish the solvable case, and the Levi decomposition to then establish the general case.)

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Let {G} be a compact group. (Throughout this post, all topological groups are assumed to be Hausdorff.) Then {G} has a number of unitary representations, i.e. continuous homomorphisms {\rho: G \rightarrow U(H)} to the group {U(H)} of unitary operators on a Hilbert space {H}, equipped with the strong operator topology. In particular, one has the left-regular representation {\tau: G \rightarrow U(L^2(G))}, where we equip {G} with its normalised Haar measure {\mu} (and the Borel {\sigma}-algebra) to form the Hilbert space {L^2(G)}, and {\tau} is the translation operation

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

We call two unitary representations {\rho: G \rightarrow U(H)} and {\rho': G \rightarrow U(H')} isomorphic if one has {\rho'(g) = U \rho(g) U^{-1}} for some unitary transformation {U: H \rightarrow H'}, in which case we write {\rho \equiv \rho'}.

Given two unitary representations {\rho: G \rightarrow U(H)} and {\rho': G \rightarrow U(H')}, one can form their direct sum {\rho \oplus \rho': G \rightarrow U(H \oplus H')} in the obvious manner: {\rho \oplus \rho'(g)(v) := (\rho(g) v, \rho'(g) v)}. Conversely, if a unitary representation {\rho: G \rightarrow U(H)} has a closed invariant subspace {V \subset H} of {H} (thus {\rho(g) V \subset V} for all {g \in G}), then the orthogonal complement {V^\perp} is also invariant, leading to a decomposition {\rho \equiv \rho\downharpoonright_V \oplus \rho\downharpoonright_{V^\perp}} of {\rho} into the subrepresentations {\rho\downharpoonright_V: G \rightarrow U(V)}, {\rho\downharpoonright_{V^\perp}: G \rightarrow U(V^\perp)}. Accordingly, we will call a unitary representation {\rho: G \rightarrow U(H)} irreducible if {H} is nontrivial (i.e. {H \neq \{0\}}) and there are no nontrivial invariant subspaces (i.e. no invariant subspaces other than {\{0\}} and {H}); the irreducible representations play a role in the subject analogous to those of prime numbers in multiplicative number theory. By the principle of infinite descent, every finite-dimensional unitary representation is then expressible (perhaps non-uniquely) as the direct sum of irreducible representations.

The Peter-Weyl theorem asserts, among other things, that the same claim is true for the regular representation:

Theorem 1 (Peter-Weyl theorem) Let {G} be a compact group. Then the regular representation {\tau: G \rightarrow U(L^2(G))} is isomorphic to the direct sum of irreducible representations. In fact, one has {\tau \equiv \bigoplus_{\xi \in \hat G} \rho_\xi^{\oplus \hbox{dim}(V_\xi)}}, where {(\rho_\xi)_{\xi \in \hat G}} is an enumeration of the irreducible finite-dimensional unitary representations {\rho_\xi: G \rightarrow U(V_\xi)} of {G} (up to isomorphism). (It is not difficult to see that such an enumeration exists.)

In the case when {G} is abelian, the Peter-Weyl theorem is a consequence of the Plancherel theorem; in that case, the irreducible representations are all one dimensional, and are thus indexed by the space {\hat G} of characters {\xi: G \rightarrow {\bf R}/{\bf Z}} (i.e. continuous homomorphisms into the unit circle {{\bf R}/{\bf Z}}), known as the Pontryagin dual of {G}. (See for instance my lecture notes on the Fourier transform.) Conversely, the Peter-Weyl theorem can be used to deduce the Plancherel theorem for compact groups, as well as other basic results in Fourier analysis on these groups, such as the Fourier inversion formula.

Because the regular representation is faithful (i.e. injective), a corollary of the Peter-Weyl theorem (and a classical theorem of Cartan) is that every compact group can be expressed as the inverse limit of Lie groups, leading to a solution to Hilbert’s fifth problem in the compact case. Furthermore, the compact case is then an important building block in the more general theory surrounding Hilbert’s fifth problem, and in particular a result of Yamabe that any locally compact group contains an open subgroup that is the inverse limit of Lie groups.

I’ve recently become interested in the theory around Hilbert’s fifth problem, due to the existence of a correspondence principle between locally compact groups and approximate groups, which play a fundamental role in arithmetic combinatorics. I hope to elaborate upon this correspondence in a subsequent post, but I will mention that versions of this principle play a crucial role in Gromov’s proof of his theorem on groups of polynomial growth (discussed previously on this blog), and in a more recent paper of Hrushovski on approximate groups (also discussed previously). It is also analogous in many ways to the more well-known Furstenberg correspondence principle between ergodic theory and combinatorics (also discussed previously).

Because of the above motivation, I have decided to write some notes on how the Peter-Weyl theorem is proven. This is utterly standard stuff in abstract harmonic analysis; these notes are primarily for my own benefit, but perhaps they may be of interest to some readers also.

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