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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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