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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 -adic field for some prime , 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 be a locally compact Hausdorff topological group that has a faithful finite-dimensional linear representation, i.e. an injective continuous homomorphism into some linear group. Then can be given the structure of a Lie group. Furthermore, after giving this Lie structure, becomes smooth (and even analytic) and non-degenerate (the Jacobian always has full rank).
This result is closely related to a theorem of Cartan:
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 be the real line , which we faithfully represent in the -torus using an irrational embedding for some fixed irrational . The -torus can in turn be embedded in a linear group (e.g. by identifying it with , or ), thus giving a faithful linear representation of . However, the image is not closed (it is a dense subgroup of a -torus), and so Cartan’s theorem does not directly apply ( fails to be a Lie group). Nevertheless, Theorem 1 still applies and guarantees that the original group is a Lie group.
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 is a continuous homomorphism from the real line (with the additive group structure) to .
Remark 1 Technically, is a parameterisation of a subgroup , rather than a subgroup itself, but we will abuse notation and refer to as the subgroup.
In a Lie group , the one-parameter subgroups are in one-to-one correspondence with the Lie algebra , with each element giving rise to a one-parameter subgroup , and conversely each one-parameter subgroup giving rise to an element 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 as a Lie group:
- First, form the space of one-parameter subgroups of .
- Show that has the structure of a (finite-dimensional) Lie algebra.
- Show that “behaves like” the tangent space of at the identity (in particular, the one-parameter subgroups in should cover a neighbourhood of the identity in ).
- Conclude that 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.