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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.

For sake of concreteness we will work here over the complex numbers ${{\bf C}}$, although most of this discussion is valid for arbitrary algebraically closed fields (but some care needs to be taken in characteristic ${2}$, as always, particularly when defining the orthogonal and symplectic groups). Then one has the following four infinite families of classical Lie groups for ${n \geq 1}$:

1. (Type ${A_n}$) The special linear group ${SL_{n+1}({\bf C})}$ of volume-preserving linear maps ${T: {\bf C}^{n+1} \rightarrow {\bf C}^{n+1}}$.
2. (Type ${B_n}$) The special orthogonal group ${SO_{2n+1}({\bf C})}$ of (orientation preserving) linear maps ${T: {\bf C}^{2n+1} \rightarrow {\bf C}^{2n+1}}$ preserving a non-degenerate symmetric form ${\langle, \rangle: {\bf C}^{2n+1} \times {\bf C}^{2n+1} \rightarrow {\bf C}}$, such as the standard symmetric form

$\displaystyle \langle (z_1,\ldots,z_{2n+1}), (w_1,\ldots,w_{2n+1}) \rangle := z_1 w_1 + \ldots + z_{2n+1} w_{2n+1}.$

(this is the complexification of the more familiar real special orthogonal group ${SO_{2n+1}({\bf R})}$).

3. (Type ${C_n}$) The symplectic group ${Sp_{2n}({\bf C})}$ of linear maps ${T: {\bf C}^{2n} \rightarrow {\bf C}^{2n}}$ preserving a non-degenerate antisymmetric form ${\omega: {\bf C}^{2n} \times {\bf C}^{2n} \rightarrow {\bf C}}$, such as the standard symplectic form

$\displaystyle \omega((z_1,\ldots,z_{2n}), (w_1,\ldots,w_{2n})) := \sum_{j=1}^n z_j w_{n+j} - z_{n+j} w_j.$

4. (Type ${D_n}$) The special orthogonal group ${SO_{2n}({\bf C})}$ of (orientation preserving) linear maps ${{\bf C}^{2n} \rightarrow {\bf C}^{2n}}$ preserving a non-degenerate symmetric form ${\langle,\rangle: {\bf C}^{2n} \times {\bf C}^{2n} \rightarrow {\bf C}}$ (such as the standard symmetric form).

For this post I will abuse notation somewhat and identify ${A_n}$ with ${SL_{n+1}({\bf C})}$, ${B_n}$ with ${SO_{2n+1}({\bf C})}$, etc., although it is more accurate to say that ${SL_{n+1}({\bf C})}$ is a Lie group of type ${A_n}$, etc., as there are other forms of the Lie algebras associated to ${A_n, B_n, C_n, D_n}$ over various fields. Over a non-algebraically closed field, such as ${{\bf R}}$, the list of Lie groups associated with a given type can in fact get quite complicated; see for instance this list. One can also view the double covers ${Spin_{2n+1}({\bf C})}$ and ${Spin_{2n}({\bf C})}$ of ${SO_{2n+1}({\bf C})}$, ${SO_{2n}({\bf C})}$ (i.e. the spin groups) as being of type ${B_n, D_n}$ respectively; however, I find the spin groups less intuitive to work with than the orthogonal groups and will therefore focus more on the orthogonal model.

The reason for this subscripting is that each of the classical groups ${A_n, B_n, C_n, D_n}$ has rank ${n}$, i.e. the dimension of any maximal connected abelian subgroup of simultaneously diagonalisable elements (also known as a Cartan subgroup) is ${n}$. For instance:

1. (Type ${A_n}$) In ${SL_{n+1}({\bf C})}$, one Cartan subgroup is the diagonal matrices in ${SL_{n+1}({\bf C})}$, which has dimension ${n}$.
2. (Type ${B_n}$) In ${SO_{2n+1}({\bf C})}$, all Cartan subgroups are isomorphic to ${SO_2({\bf C})^n \times SO_1({\bf C})}$, which has dimension ${n}$.
3. (Type ${C_n}$) In ${Sp_{2n}({\bf C})}$, all Cartan subgroups are isomorphic to ${SO_2({\bf C})^n \leq Sp_2({\bf C})^n \leq Sp_{2n}({\bf C})}$, which has dimension ${n}$.
4. (Type ${D_n}$) in ${SO_{2n}({\bf C})}$, all Cartan subgroups are isomorphic to ${SO_2({\bf C})^n}$, which has dimension ${n}$.

(This same convention also underlies the notation for the exceptional simple Lie groups ${G_2, F_4, E_6, E_7, E_8}$, which we will not discuss further here.)

With two exceptions, the classical Lie groups ${A_n,B_n,C_n,D_n}$ are all simple, i.e. their Lie algebras are non-abelian and not expressible as the direct sum of smaller Lie algebras. The two exceptions are ${D_1 = SO_2({\bf C})}$, which is abelian (isomorphic to ${{\bf C}^\times}$, in fact) and thus not considered simple, and ${D_2 = SO_4({\bf C})}$, which turns out to “essentially” split as ${A_1 \times A_1 = SL_2({\bf C}) \times SL_2({\bf C})}$, in the sense that the former group is double covered by the latter (and in particular, there is an isogeny from the latter to the former, and the Lie algebras are isomorphic).

The adjoint action of a Cartan subgroup of a Lie group ${G}$ on the Lie algebra ${{\mathfrak g}}$ splits that algebra into weight spaces; in the case of a simple Lie group, the associated weights are organised by a Dynkin diagram. The Dynkin diagrams for ${A_n, B_n, C_n, D_n}$ are of course well known, and can be found for instance here.

For small ${n}$, some of these Dynkin diagrams are isomorphic; this is a classic instance of the tongue-in-cheek strong law of small numbers, though in this case “strong law of small diagrams” would be more appropriate. These accidental isomorphisms then give rise to the exceptional isomorphisms between Lie algebras (and thence to exceptional isogenies between Lie groups). Excluding those isomorphisms involving the exceptional Lie algebras ${E_n}$ for ${n=3,4,5}$, these isomorphisms are

1. ${A_1 = B_1 = C_1}$;
2. ${B_2 = C_2}$;
3. ${D_2 = A_1 \times A_1}$;
4. ${D_3 = A_3}$.

There is also a pair of exceptional isomorphisms from (the ${Spin_8}$ form of) ${D_4}$ to itself, a phenomenon known as triality.

These isomorphisms are most easily seen via algebraic and combinatorial tools, such as an inspection of the Dynkin diagrams (see e.g. this Wikipedia image). However, the isomorphisms listed above can also be seen by more “geometric” means, using the basic representations of the classical Lie groups on their natural vector spaces (${{\bf C}^{n+1}, {\bf C}^{2n+1}, {\bf C}^{2n}, {\bf C}^{2n}}$ for ${A_n, B_n, C_n, D_n}$ respectively) and combinations thereof (such as exterior powers). (However, I don’t know of a simple way to interpret triality geometrically; the descriptions I have seen tend to involve some algebraic manipulation of the octonions or of a Clifford algebra, in a manner that tended to obscure the geometry somewhat.) These isomorphisms are quite standard (I found them, for instance, in this book of Procesi), but it was instructive for me to work through them (as I have only recently needed to start studying algebraic group theory in earnest), and I am recording them here in case anyone else is interested.

In this final lecture, we establish a Ratner-type theorem for actions of the special linear group $SL_2({\Bbb R})$ on homogeneous spaces. More precisely, we show:

Theorem 1. Let G be a Lie group, let $\Gamma < G$ be a discrete subgroup, and let $H \leq G$ be a subgroup isomorphic to $SL_2({\Bbb R})$. Let $\mu$ be an H-invariant probability measure on $G/\Gamma$ which is ergodic with respect to H (i.e. all H-invariant sets either have full measure or zero measure). Then $\mu$ is homogeneous in the sense that there exists a closed connected subgroup $H \leq L \leq G$ and a closed orbit $Lx \subset G/\Gamma$ such that $\mu$ is L-invariant and supported on Lx.

This result is a special case of a more general theorem of Ratner, which addresses the case when H is generated by elements which act unipotently on the Lie algebra ${\mathfrak g}$ by conjugation, and when $G/\Gamma$ has finite volume. To prove this theorem we shall follow an argument of Einsiedler, which uses many of the same ingredients used in Ratner’s arguments but in a simplified setting (in particular, taking advantage of the fact that H is semisimple with no non-trivial compact factors). These arguments have since been extended and made quantitative by Einsiedler, Margulis, and Venkatesh.
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