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Locally compact groups with faithful finite-dimensional representations

27 May, 2011 in expository, math.GN, math.GR, math.RT | Tags: , , , , | by | 10 comments

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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