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Hilbert’s fifth problem concerns the minimal hypotheses one needs to place on a topological group {G} to ensure that it is actually a Lie group. In the previous set of notes, we saw that one could reduce the regularity hypothesis imposed on {G} to a “{C^{1,1}}” condition, namely that there was an open neighbourhood of {G} that was isomorphic (as a local group) to an open subset {V} of a Euclidean space {{\bf R}^d} with identity element {0}, and with group operation {\ast} obeying the asymptotic

\displaystyle  x \ast y = x + y + O(|x| |y|)

for sufficiently small {x,y}. We will call such local groups {(V,\ast)} {C^{1,1}} local groups.

We now reduce the regularity hypothesis further, to one in which there is no explicit Euclidean space that is initially attached to {G}. Of course, Lie groups are still locally Euclidean, so if the hypotheses on {G} do not involve any explicit Euclidean spaces, then one must somehow build such spaces from other structures. One way to do so is to exploit an ambient space with Euclidean or Lie structure that {G} is embedded or immersed in. A trivial example of this is provided by the following basic fact from linear algebra:

Lemma 1 If {V} is a finite-dimensional vector space (i.e. it is isomorphic to {{\bf R}^d} for some {d}), and {W} is a linear subspace of {V}, then {W} is also a finite-dimensional vector space.

We will establish a non-linear version of this statement, known as Cartan’s theorem. Recall that a subset {S} of a {d}-dimensional smooth manifold {M} is a {d'}-dimensional smooth (embedded) submanifold of {M} for some {0 \leq d' \leq d} if for every point {x \in S} there is a smooth coordinate chart {\phi: U \rightarrow V} of a neighbourhood {U} of {x} in {M} that maps {x} to {0}, such that {\phi(U \cap S) = V \cap {\bf R}^{d'}}, where we identify {{\bf R}^{d'} \equiv {\bf R}^{d'} \times \{0\}^{d-d'}} with a subspace of {{\bf R}^d}. Informally, {S} locally sits inside {M} the same way that {{\bf R}^{d'}} sits inside {{\bf R}^d}.

Theorem 2 (Cartan’s theorem) If {H} is a (topologically) closed subgroup of a Lie group {G}, then {H} is a smooth submanifold of {G}, and is thus also a Lie group.

Note that the hypothesis that {H} is closed is essential; for instance, the rationals {{\bf Q}} are a subgroup of the (additive) group of reals {{\bf R}}, but the former is not a Lie group even though the latter is.

Exercise 1 Let {H} be a subgroup of a locally compact group {G}. Show that {H} is closed in {G} if and only if it is locally compact.

A variant of the above results is provided by using (faithful) representations instead of embeddings. Again, the linear version is trivial:

Lemma 3 If {V} is a finite-dimensional vector space, and {W} is another vector space with an injective linear transformation {\rho: W \rightarrow V} from {W} to {V}, then {W} is also a finite-dimensional vector space.

Here is the non-linear version:

Theorem 4 (von Neumann’s theorem) If {G} is a Lie group, and {H} is a locally compact group with an injective continuous homomorphism {\rho: H \rightarrow G}, then {H} also has the structure of a Lie group.

Actually, it will suffice for the homomorphism {\rho} to be locally injective rather than injective; related to this, von Neumann’s theorem localises to the case when {H} is a local group rather a group. The requirement that {H} be locally compact is necessary, for much the same reason that the requirement that {H} be closed was necessary in Cartan’s theorem.

Example 1 Let {G = ({\bf R}/{\bf Z})^2} be the two-dimensional torus, let {H = {\bf R}}, and let {\rho: H \rightarrow G} be the map {\rho(x) := (x,\alpha x)}, where {\alpha \in {\bf R}} is a fixed real number. Then {\rho} is a continuous homomorphism which is locally injective, and is even globally injective if {\alpha} is irrational, and so Theorem 4 is consistent with the fact that {H} is a Lie group. On the other hand, note that when {\alpha} is irrational, then {\rho(H)} is not closed; and so Theorem 4 does not follow immediately from Theorem 2 in this case. (We will see, though, that Theorem 4 follows from a local version of Theorem 2.)

As a corollary of Theorem 4, we observe that any locally compact Hausdorff group {H} with a faithful linear representation, i.e. a continuous injective homomorphism from {H} into a linear group such as {GL_n({\bf R})} or {GL_n({\bf C})}, is necessarily a Lie group. This suggests a representation-theoretic approach to Hilbert’s fifth problem. While this approach does not seem to readily solve the entire problem, it can be used to establish a number of important special cases with a well-understood representation theory, such as the compact case or the abelian case (for which the requisite representation theory is given by the Peter-Weyl theorem and Pontryagin duality respectively). We will discuss these cases further in later notes.

In all of these cases, one is not really building up Euclidean or Lie structure completely from scratch, because there is already a Euclidean or Lie structure present in another object in the hypotheses. Now we turn to results that can create such structure assuming only what is ostensibly a weaker amount of structure. In the linear case, one example of this is is the following classical result in the theory of topological vector spaces.

Theorem 5 Let {V} be a locally compact Hausdorff topological vector space. Then {V} is isomorphic (as a topological vector space) to {{\bf R}^d} for some finite {d}.

Remark 1 The Banach-Alaoglu theorem asserts that in a normed vector space {V}, the closed unit ball in the dual space {V^*} is always compact in the weak-* topology. Of course, this dual space {V^*} may be infinite-dimensional. This however does not contradict the above theorem, because the closed unit ball is not a neighbourhood of the origin in the weak-* topology (it is only a neighbourhood with respect to the strong topology).

The full non-linear analogue of this theorem would be the Gleason-Yamabe theorem, which we are not yet ready to prove in this set of notes. However, by using methods similar to that used to prove Cartan’s theorem and von Neumann’s theorem, one can obtain a partial non-linear analogue which requires an additional hypothesis of a special type of metric, which we will call a Gleason metric:

Definition 6 Let {G} be a topological group. A Gleason metric on {G} is a left-invariant metric {d: G \times G \rightarrow {\bf R}^+} which generates the topology on {G} and obeys the following properties for some constant {C>0}, writing {\|g\|} for {d(g,\hbox{id})}:

  • (Escape property) If {g \in G} and {n \geq 1} is such that {n \|g\| \leq \frac{1}{C}}, then {\|g^n\| \geq \frac{1}{C} n \|g\|}.
  • (Commutator estimate) If {g, h \in G} are such that {\|g\|, \|h\| \leq \frac{1}{C}}, then

    \displaystyle  \|[g,h]\| \leq C \|g\| \|h\|, \ \ \ \ \ (1)

    where {[g,h] := g^{-1}h^{-1}gh} is the commutator of {g} and {h}.

Exercise 2 Let {G} be a topological group that contains a neighbourhood of the identity isomorphic to a {C^{1,1}} local group. Show that {G} admits at least one Gleason metric.

Theorem 7 (Building Lie structure from Gleason metrics) Let {G} be a locally compact group that has a Gleason metric. Then {G} is isomorphic to a Lie group.

We will rely on Theorem 7 to solve Hilbert’s fifth problem; this theorem reduces the task of establishing Lie structure on a locally compact group to that of building a metric with suitable properties. Thus, much of the remainder of the solution of Hilbert’s fifth problem will now be focused on the problem of how to construct good metrics on a locally compact group.

In all of the above results, a key idea is to use one-parameter subgroups to convert from the nonlinear setting to the linear setting. Recall from the previous notes that in a Lie group {G}, the one-parameter subgroups are in one-to-one correspondence with the elements of the Lie algebra {{\mathfrak g}}, which is a vector space. In a general topological group {G}, the concept of a one-parameter subgroup (i.e. a continuous homomorphism from {{\bf R}} to {G}) still makes sense; the main difficulties are then to show that the space of such subgroups continues to form a vector space, and that the associated exponential map {\exp: \phi \mapsto \phi(1)} is still a local homeomorphism near the origin.

Exercise 3 The purpose of this exercise is to illustrate the perspective that a topological group can be viewed as a non-linear analogue of a vector space. Let {G, H} be locally compact groups. For technical reasons we assume that {G, H} are both {\sigma}-compact and metrisable.

  • (i) (Open mapping theorem) Show that if {\phi: G \rightarrow H} is a continuous homomorphism which is surjective, then it is open (i.e. the image of open sets is open). (Hint: mimic the proof of the open mapping theorem for Banach spaces, as discussed for instance in these notes. In particular, take advantage of the Baire category theorem.)
  • (ii) (Closed graph theorem) Show that if a homomorphism {\phi: G \rightarrow H} is closed (i.e. its graph {\{ (g, \phi(g)): g \in G \}} is a closed subset of {G \times H}), then it is continuous. (Hint: mimic the derivation of the closed graph theorem from the open mapping theorem in the Banach space case, as again discussed in these notes.)
  • (iii) Let {\phi: G \rightarrow H} be a homomorphism, and let {\rho: H \rightarrow K} be a continuous injective homomorphism into another Hausdorff topological group {K}. Show that {\phi} is continuous if and only if {\rho \circ \phi} is continuous.
  • (iv) Relax the condition of metrisability to that of being Hausdorff. (Hint: Now one cannot use the Baire category theorem for metric spaces; but there is an analogue of this theorem for locally compact Hausdorff spaces.)

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