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In this set of notes, we describe the basic analytic structure theory of Lie groups, by relating them to the simpler concept of a Lie algebra. Roughly speaking, the Lie algebra encodes the “infinitesimal” structure of a Lie group, but is a simpler object, being a vector space rather than a nonlinear manifold. Nevertheless, thanks to the fundamental theorems of Lie, the Lie algebra can be used to reconstruct the Lie group (at a local level, at least), by means of the exponential map and the Baker-Campbell-Hausdorff formula. As such, the local theory of Lie groups is completely described (in principle, at least) by the theory of Lie algebras, which leads to a number of useful consequences, such as the following:

- (Local Lie implies Lie) A topological group is Lie (i.e. it is isomorphic to a Lie group) if and only if it is locally Lie (i.e. the group operations are smooth near the origin).
- (Uniqueness of Lie structure) A topological group has at most one smooth structure on it that makes it Lie.
- (Weak regularity implies strong regularity, I) Lie groups are automatically real analytic. (In fact one only needs a “local ” regularity on the group structure to obtain real analyticity.)
- (Weak regularity implies strong regularity, II) A continuous homomorphism from one Lie group to another is automatically smooth (and real analytic).

The connection between Lie groups and Lie algebras also highlights the role of *one-parameter subgroups* of a topological group, which will play a central role in the solution of Hilbert’s fifth problem.

We note that there is also a very important *algebraic* structure theory of Lie groups and Lie algebras, in which the Lie algebra is split into solvable and semisimple components, with the latter being decomposed further into simple components, which can then be completely classified using Dynkin diagrams. This classification is of fundamental importance in many areas of mathematics (e.g. representation theory, arithmetic geometry, and group theory), and many of the deeper facts about Lie groups and Lie algebras are proven via this classification (although in such cases it can be of interest to also find alternate proofs that avoid the classification). However, it turns out that we will not need this theory in this course, and so we will not discuss it further here (though it can of course be found in any graduate text on Lie groups and Lie algebras).

This fall (starting Monday, September 26), I will be teaching a graduate topics course which I have entitled “Hilbert’s fifth problem and related topics.” The course is going to focus on three related topics:

- Hilbert’s fifth problem on the topological description of Lie groups, as well as the closely related (local) classification of locally compact groups (the Gleason-Yamabe theorem).
- Approximate groups in nonabelian groups, and their classification via the Gleason-Yamabe theorem (this is very recent work of Emmanuel Breuillard, Ben Green, Tom Sanders, and myself, building upon earlier work of Hrushovski);
- Gromov’s theorem on groups of polynomial growth, as proven via the classification of approximate groups (as well as some consequences to fundamental groups of Riemannian manifolds).

I have already blogged about these topics repeatedly in the past (particularly with regard to Hilbert’s fifth problem), and I intend to recycle some of that material in the lecture notes for this course.

The above three families of results exemplify two broad principles (part of what I like to call “the dichotomy between structure and randomness“):

- (Rigidity) If a group-like object exhibits a weak amount of regularity, then it (or a large portion thereof) often automatically exhibits a strong amount of regularity as well;
- (Structure) This strong regularity manifests itself either as Lie type structure (in continuous settings) or nilpotent type structure (in discrete settings). (In some cases, “nilpotent” should be replaced by sister properties such as “abelian“, “solvable“, or “polycyclic“.)

Let me illustrate what I mean by these two principles with two simple examples, one in the continuous setting and one in the discrete setting. We begin with a continuous example. Given an complex matrix , define the matrix exponential of by the formula

which can easily be verified to be an absolutely convergent series.

Exercise 1Show that the map is a real analytic (and even complex analytic) map from to , and obeys the restricted homomorphism property

Proposition 1 (Rigidity and structure of matrix homomorphisms)Let be a natural number. Let be the group of invertible complex matrices. Let be a map obeying two properties:

- (Group-like object) is a homomorphism, thus for all .
- (Weak regularity) The map is continuous.
Then:

- (Strong regularity) The map is smooth (i.e. infinitely differentiable). In fact it is even real analytic.
- (Lie-type structure) There exists a (unique) complex matrix such that for all .

*Proof:* Let be as above. Let be a small number (depending only on ). By the homomorphism property, (where we use here to denote the identity element of ), and so by continuity we may find a small such that for all (we use some arbitrary norm here on the space of matrices, and allow implied constants in the notation to depend on ).

The map is real analytic and (by the inverse function theorem) is a diffeomorphism near . Thus, by the inverse function theorem, we can (if is small enough) find a matrix of size such that . By the homomorphism property and (1), we thus have

On the other hand, by another application of the inverse function theorem we see that the squaring map is a diffeomorphism near in , and thus (if is small enough)

We may iterate this argument (for a fixed, but small, value of ) and conclude that

for all . By the homomorphism property and (1) we thus have

whenever is a dyadic rational, i.e. a rational of the form for some integer and natural number . By continuity we thus have

for all real . Setting we conclude that

for all real , which gives existence of the representation and also real analyticity and smoothness. Finally, uniqueness of the representation follows from the identity

Exercise 2Generalise Proposition 1 by replacing the hypothesis that is continuous with the hypothesis that is Lebesgue measurable (Hint:use the Steinhaus theorem.). Show that the proposition fails (assuming the axiom of choice) if this hypothesis is omitted entirely.

Note how one needs both the group-like structure and the weak regularity in combination in order to ensure the strong regularity; neither is sufficient on its own. We will see variants of the above basic argument throughout the course. Here, the task of obtaining smooth (or real analytic structure) was relatively easy, because we could borrow the smooth (or real analytic) structure of the domain and range ; but, somewhat remarkably, we shall see that one can still build such smooth or analytic structures even when none of the original objects have any such structure to begin with.

Now we turn to a second illustration of the above principles, namely Jordan’s theorem, which uses a discreteness hypothesis to upgrade Lie type structure to nilpotent (and in this case, abelian) structure. We shall formulate Jordan’s theorem in a slightly stilted fashion in order to emphasise the adherence to the above-mentioned principles.

Theorem 2 (Jordan’s theorem)Let be an object with the following properties:

- (Group-like object) is a group.
- (Discreteness) is finite.
- (Lie-type structure) is contained in (the group of unitary matrices) for some .
Then there is a subgroup of such that

- ( is close to ) The index of in is (i.e. bounded by for some quantity depending only on ).
- (Nilpotent-type structure) is abelian.

A key observation in the proof of Jordan’s theorem is that if two unitary elements are close to the identity, then their commutator is even closer to the identity (in, say, the operator norm ). Indeed, since multiplication on the left or right by unitary elements does not affect the operator norm, we have

and so by the triangle inequality

Now we can prove Jordan’s theorem.

*Proof:* We induct on , the case being trivial. Suppose first that contains a central element which is not a multiple of the identity. Then, by definition, is contained in the centraliser of , which by the spectral theorem is isomorphic to a product of smaller unitary groups. Projecting to each of these factor groups and applying the induction hypothesis, we obtain the claim.

Thus we may assume that contains no central elements other than multiples of the identity. Now pick a small (one could take in fact) and consider the subgroup of generated by those elements of that are within of the identity (in the operator norm). By considering a maximal -net of we see that has index at most in . By arguing as before, we may assume that has no central elements other than multiples of the identity.

If consists only of multiples of the identity, then we are done. If not, take an element of that is not a multiple of the identity, and which is as close as possible to the identity (here is where we crucially use that is finite). By (2), we see that if is sufficiently small depending on , and if is one of the generators of , then lies in and is closer to the identity than , and is thus a multiple of the identity. On the other hand, has determinant . Given that it is so close to the identity, it must therefore be the identity (if is small enough). In other words, is central in , and is thus a multiple of the identity. But this contradicts the hypothesis that there are no central elements other than multiples of the identity, and we are done.

Commutator estimates such as (2) will play a fundamental role in many of the arguments we will see in this course; as we saw above, such estimates combine very well with a discreteness hypothesis, but will also be very useful in the continuous setting.

Exercise 3Generalise Jordan’s theorem to the case when is a finite subgroup of rather than of . (Hint:The elements of are not necessarily unitary, and thus do not necessarily preserve the standard Hilbert inner product of . However, if one averages that inner product by the finite group , one obtains a new inner product on that is preserved by , which allows one to conjugate to a subgroup of . This averaging trick is (a small) part of Weyl’s unitary trick in representation theory.)

Exercise 4 (Inability to discretise nonabelian Lie groups)Show that if , then the orthogonal group cannot contain arbitrarily dense finite subgroups, in the sense that there exists an depending only on such that for every finite subgroup of , there exists a ball of radius in (with, say, the operator norm metric) that is disjoint from . What happens in the case?

Remark 1More precise classifications of the finite subgroups of are known, particularly in low dimensions. For instance, one can show that the only finite subgroups of (which is a double cover of) are isomorphic to either a cyclic group, a dihedral group, or the symmetry group of one of the Platonic solids.

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