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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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Hilbert’s fifth problem asks to clarify the extent that the assumption on a differentiable or smooth structure is actually needed in the theory of Lie groups and their actions. While this question is not precisely formulated and is thus open to some interpretation, the following result of Gleason and Montgomery-Zippin answers at least one aspect of this question:

Theorem 1 (Hilbert’s fifth problem) Let {G} be a topological group which is locally Euclidean (i.e. it is a topological manifold). Then {G} is isomorphic to a Lie group.

Theorem 1 can be viewed as an application of the more general structural theory of locally compact groups. In particular, Theorem 1 can be deduced from the following structural theorem of Gleason and Yamabe:

Theorem 2 (Gleason-Yamabe theorem) Let {G} be a locally compact group, and let {U} be an open neighbourhood of the identity in {G}. Then there exists an open subgroup {G'} of {G}, and a compact subgroup {N} of {G'} contained in {U}, such that {G'/N} is isomorphic to a Lie group.

The deduction of Theorem 1 from Theorem 2 proceeds using the Brouwer invariance of domain theorem and is discussed in this previous post. In this post, I would like to discuss the proof of Theorem 2. We can split this proof into three parts, by introducing two additional concepts. The first is the property of having no small subgroups:

Definition 3 (NSS) A topological group {G} is said to have no small subgroups, or is NSS for short, if there is an open neighbourhood {U} of the identity in {G} that contains no subgroups of {G} other than the trivial subgroup {\{ \hbox{id}\}}.

An equivalent definition of an NSS group is one which has an open neighbourhood {U} of the identity that every non-identity element {g \in G \backslash \{\hbox{id}\}} escapes in finite time, in the sense that {g^n \not \in U} for some positive integer {n}. It is easy to see that all Lie groups are NSS; we shall shortly see that the converse statement (in the locally compact case) is also true, though significantly harder to prove.

Another useful property is that of having what I will call a Gleason metric:

Definition 4 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}.

For instance, the unitary group {U(n)} with the operator norm metric {d(g,h) := \|g-h\|_{op}} can easily verified to be a Gleason metric, with the commutator estimate (1) coming from the inequality

\displaystyle  \| [g,h] - 1 \|_{op} = \| gh - hg \|_{op}

\displaystyle  = \| (g-1) (h-1) - (h-1) (g-1) \|_{op}

\displaystyle  \leq 2 \|g-1\|_{op} \|g-1\|_{op}.

Similarly, any left-invariant Riemannian metric on a (connected) Lie group can be verified to be a Gleason metric. From the escape property one easily sees that all groups with Gleason metrics are NSS; again, we shall see that there is a partial converse.

Remark 1 The escape and commutator properties are meant to capture “Euclidean-like” structure of the group. Other metrics, such as Carnot-Carathéodory metrics on Carnot Lie groups such as the Heisenberg group, usually fail one or both of these properties.

The proof of Theorem 2 can then be split into three subtheorems:

Theorem 5 (Reduction to the NSS case) Let {G} be a locally compact group, and let {U} be an open neighbourhood of the identity in {G}. Then there exists an open subgroup {G'} of {G}, and a compact subgroup {N} of {G'} contained in {U}, such that {G'/N} is NSS, locally compact, and metrisable.

Theorem 6 (Gleason’s lemma) Let {G} be a locally compact metrisable NSS group. Then {G} has a Gleason metric.

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

Clearly, by combining Theorem 5, Theorem 6, and Theorem 7 one obtains Theorem 2 (and hence Theorem 1).

Theorem 5 and Theorem 6 proceed by some elementary combinatorial analysis, together with the use of Haar measure (to build convolutions, and thence to build “smooth” bump functions with which to create a metric, in a variant of the analysis used to prove the Birkhoff-Kakutani theorem); Theorem 5 also requires Peter-Weyl theorem (to dispose of certain compact subgroups that arise en route to the reduction to the NSS case), which was discussed previously on this blog.

In this post I would like to detail the final component to the proof of Theorem 2, namely Theorem 7. (I plan to discuss the other two steps, Theorem 5 and Theorem 6, in a separate post.) The strategy is similar to that used to prove von Neumann’s theorem, as discussed in this previous post (and von Neumann’s theorem is also used in the proof), but with the Gleason metric serving as a substitute for the faithful linear representation. Namely, one first gives the space {L(G)} of one-parameter subgroups of {G} enough of a structure that it can serve as a proxy for the “Lie algebra” of {G}; specifically, it needs to be a vector space, and the “exponential map” needs to cover an open neighbourhood of the identity. This is enough to set up an “adjoint” representation of {G}, whose image is a Lie group by von Neumann’s theorem; the kernel is essentially the centre of {G}, which is abelian and can also be shown to be a Lie group by a similar analysis. To finish the job one needs to use arguments of Kuranishi and of Gleason, as discussed in this previous post.

The arguments here can be phrased either in the standard analysis setting (using sequences, and passing to subsequences often) or in the nonstandard analysis setting (selecting an ultrafilter, and then working with infinitesimals). In my view, the two approaches have roughly the same level of complexity in this case, and I have elected for the standard analysis approach.

Remark 2 From Theorem 7 we see that a Gleason metric structure is a good enough substitute for smooth structure that it can actually be used to reconstruct the entire smooth structure; roughly speaking, the commutator estimate (1) allows for enough “Taylor expansion” of expressions such as {g^n h^n} that one can simulate the fundamentals of Lie theory (in particular, construction of the Lie algebra and the exponential map, and its basic properties. The advantage of working with a Gleason metric rather than a smoother structure, though, is that it is relatively undemanding with regards to regularity; in particular, the commutator estimate (1) is roughly comparable to the imposition {C^{1,1}} structure on the group {G}, as this is the minimal regularity to get the type of Taylor approximation (with quadratic errors) that would be needed to obtain a bound of the form (1). We will return to this point in a later post.

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