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).

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.)

” 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).

— 1. Local groups —

The connection between Lie groups and Lie algebras will be local in nature – the only portion of the Lie group that will be of importance will be the portion that is close to the group identity . To formalise this locality, it is convenient to introduce the notion of a local group and a local Lie group, which are local versions of the concept of a topological group and a Lie group respectively. We will only set up the barest bones of the theory of local groups here; a more detailed discussion may be found at this previous blog post.

Definition 1 (Local group) A local topological group , or local group for short, is a topological space equipped with an identity element , a partially defined but continuous multiplication operation for some domain , and a partially defined but continuous inversion operation , where , obeying the following axioms: (Local closure) is an open neighbourhood of , and is an open neighbourhood of .

is an open neighbourhood of , and is an open neighbourhood of . (Local associativity) If are such that and are both well-defined in , then they are equal. (Note however that it may be possible for one of these products to be defined but not the other.)

are such that and are both well-defined in , then they are equal. (Note however that it may be possible for one of these products to be defined but not the other.) (Identity) For all , .

, . (Local inverse) If and is well-defined in , then . (In particular this, together with the other axioms, forces .) We will sometimes use additive notation for local groups if the groups are locally abelian (thus if is defined, then is also defined and equal to .) A local group is said to be symmetric if , i.e. if every element in has an inverse that is also in . A local Lie group is a local group that is also a smooth manifold, in such a fashion that the partially defined group operations are smooth on their domain of definition.

Clearly, every topological group is a local group, and every Lie group is a local Lie group. We will sometimes refer to the former concepts as global topological groups and global Lie groups in order to distinguish them from their local counterparts. One could also consider local discrete groups, in which the topological structure is just the discrete topology, but we will not need to study such objects in this course.

A model class of examples of a local (Lie) group comes from restricting a global (Lie) group to an open neighbourhood of the identity. Let us formalise this concept:

Definition 2 (Restriction) If is a local group, and is an open neighbourhood of the identity in , then we define the restriction of to to be the topological space with domains and , and with the group operations being the restriction of the group operations of to , respectively. If is symmetric (in the sense that is well-defined and lies in for all ), then this restriction will also be symmetric. If is a global or local Lie group, then will also be a local Lie group. We will sometimes abuse notation and refer to the local group simply as .

Thus, for instance, one can take the Euclidean space , and restrict it to a ball centred at the origin, to obtain an additive local group . In this group, two elements in have a well-defined sum only when their sum in stays inside . Intuitively, this local group behaves like the global group as long as one is close enough to the identity element , but as one gets closer to the boundary of , the group structure begins to break down.

It is natural to ask the question as to whether every local group arises as the restriction of a global group. The answer to this question is somewhat complicated, and can be summarised as “essentially yes in certain circumstances, but not in general”. See this previous blog post for more discussion.

A key example of a local Lie group for this blog post will come from pushing forward a Lie group via a coordinate chart near the origin:

Example 1 Let be a global or local Lie group of some dimension , and let be a smooth coordinate chart from a neighbourhood of the identity in to a neighbourhood of the origin in , such that maps to . Then we can define a local group which is the set (viewed as a smooth submanifold of ) with the local group identity , the local group multiplication law defined by the formula defined whenever are well-defined and lie in , and the local group inversion law defined by the formula defined whenever are well-defined and lie in . One easily verifies that is a local Lie group. We will sometimes denote this local Lie group as , to distinguish it from the additive local Lie group arising by restriction of to . The precise distinction between the two local Lie groups will in fact be a major focus of this post.

Example 2 Let be the Lie group , and let be the ball . If we then let be the ball and be the map , then is a smooth coordinate chart (after identifying with ), and by the construction in the preceding exercise, becomes a local Lie group with the operations (defined whenever all lie in ) and (defined whenever and both lie in ). Note that this Lie group structure is not equal to the additive structure on , nor is it equal to the multiplicative structure on given by matrix multiplication, which is one of the reasons why we use the symbol instead of or for such structures.

Many (though not all) of the familiar constructions in group theory can be generalised to the local setting, though often with some slight additional subtleties. We will not systematically do so here, but we give a single such generalisation for now:

Definition 3 (Homomorphism) A continuous homomorphism between two local groups is a continuous map from to with the following properties: maps the identity of to the identity of : .

maps the identity of to the identity of : . If is such that is well-defined in , then is well-defined in and is equal to .

is such that is well-defined in , then is well-defined in and is equal to . If are such that is well-defined in , then is well-defined and equal to . A smooth homomorphism between two local Lie groups is a continuous homomorphism that is also smooth.

It is easy to see that the composition of two continuous homomorphisms is again a continuous homomorphism; this gives the class of local groups the structure of a category. Similarly, the class of local Lie groups with their smooth homomorphisms is also a category.

Note that homomorphisms on a local group are defined on the entirety of ; it is also natural to consider (continuous or smooth) local homomorphisms, which are only defined on an open neighbourhood of the identity in , with two local homomorphisms considered equivalent if they agree on a (possibly smaller) open neighbourhood of the identity. We will not need to do so for now, however.

Example 3 With the notation of Example 1, is a smooth homomorphism from the local Lie group to the local Lie group . In fact, it is a smooth isomorphism, since provides the inverse homomorphism.

Let us say that a word in a local group is well-defined in (or well-defined, for short) if every possible way of associating this word using parentheses is well-defined from applying the product operation. For instance, in order for to be well-defined, , , , , and must all be well-defined. For instance, in the additive local group (with the group structure restricted from that of the integers ), is not well-defined because one of the ways of associating this sum, namely , is not well-defined (even though is well-defined).

Exercise 1 (Iterating the associative law) Show that if a word in a local group is well-defined, then all ways of associating this word give the same answer, and so we can uniquely evaluate as an element in .

in a local group is well-defined, then all ways of associating this word give the same answer, and so we can uniquely evaluate as an element in . Give an example of a word in a local group which has two ways of being associated that are both well-defined, but give different answers. (Hint: the local associativity axiom prevents this from happening for , so try . A small discrete local group will already suffice to give a counterexample; verifying the local group axioms are easier if one makes the domain of definition of the group operations as small as one can get away with while still having the counterexample.)

— 2. Some differential geometry —

To define the Lie algebra of a Lie group, we must first quickly recall some basic notions from differential geometry associated to smooth manifolds (which are not necessarily embedded in some larger Euclidean space, but instead exist intrinsically as abstract geometric structures). This requires a certain amount of abstract formalism in order to define things rigorously, though for the purposes of visualisation, it is more intuitive to view these concepts from a more informal geometric perspective.

We begin with the concept of the tangent space and related structures.

Definition 4 (Tangent space) Let be a smooth -dimensional manifold. At every point of this manifold, we can define the tangent space of at . Formally, this tangent space can be defined as the space of all continuously differentiable curves defined on an open interval containing with , modulo the relation that two curves are considered equivalent if they have the same derivative at , in the sense that where is a coordinate chart of defined in a neighbourhood of ; it is easy to see from the chain rule that this equivalence is independent of the actual choice of . Using such a coordinate chart, one can identify the tangent space with the Euclidean space , by identifying with . One easily verifies that this gives the structure of a -dimensional vector space, in a manner which is independent of the choice of coordinate chart . Elements of are called tangent vectors of at . If is a continuously differentiable curve with , the equivalence class of in will be denoted . The space of pairs , where is a point in and is a tangent vector of at , is called the tangent bundle. If is a smooth map between two manifolds, we define the derivative map to be the map defined by setting for all continously differentiable curves with for some . We also write for , so that for each , is a map from to . One can easily verify that this latter map is linear. We observe the chain rule for any smooth maps , . (Indeed, one can view the tangent operator and the derivative operator together as a single covariant functor from the category of smooth manifolds to itself, although we will not need to use this perspective here.) Observe that if is an open subset of , then may be identified with . In particular, every coordinate chart of gives rise to a coordinate chart of , which gives the structure of a smooth -dimensional manifold.

Remark 1 Informally, one can think of a tangent vector as an infinitesimal vector from the point of to a nearby point on , where is infinitesimally small; a smooth map then sends to . One can make this informal perspective rigorous by means of nonstandard analysis, but we will not do so here.

Once one has the notion of a tangent bundle, one can define the notion of a smooth vector field:

Definition 5 (Vector fields) A smooth vector field on is a smooth map which is a right inverse for the projection map , thus (by slight abuse of notation) maps to for some . The space of all smooth vector fields is denoted . It is clearly a real vector space. In fact, it is a -module: given a smooth vector field and a smooth function (i.e. a smooth map ), one can define the product in the obvious manner: , and one easily verifies the module axioms. Given a smooth function and a smooth vector field , we define the directional derivative of along by the formula whenever is a continuously differentiable function with and ; one easily verifies that is well-defined and is an element of .

Remark 2 One can define in a more “co-ordinate free” manner as where is the projection map to the second coordinate of ; one can also view as the Lie derivative of along (although, in most texts, the latter definition would be circular, because the Lie derivative is usually defined using the directional derivative).

Remark 3 If is an open subset of , a smooth vector field on can be identified with a smooth map from to . If is a smooth vector field on and is a coordinate chart of , then the pushforward of by is a smooth vector field of . Thus, in coordinates, one can view vector fields as maps from open subsets of to . This perspective is convenient for quick and dirty calculations; for instance, in coordinates, the directional derivative is the same as the familiar directional derivative from several variable calculus. If however one wishes to perform several changes of variable, then the more intrinsically geometric (and “coordinate-free”) perspective outlined above can be more helpful.

There is a fundamental link between smooth vector fields and derivations of :

Exercise 2 (Correspondence between smooth vector fields and derivations) Let be a smooth manifold. If is a smooth vector field, show that is a derivation on the (real) algebra , i.e. a (real) linear map that obeys the Leibniz rule for all .

is a smooth vector field, show that is a derivation on the (real) algebra , i.e. a (real) linear map that obeys the Leibniz rule for all . Conversely, if is a derivation on , show that there exists a unique smooth vector field such that .

We see from the above exercise that smooth vector fields can be interpreted as a purely algebraic construction associated to the real algebra , namely as the space of derivations on that vector space. This can be useful for analysing the algebraic structure of such vector fields. Indeed, we have the following basic algebraic observation:

Exercise 3 (Commutator of derivations is a derivation) Let be two derivations on an algebra . Show that the commutator is also a derivation on .

From the preceding two exercises, we can define the Lie bracket of two vector fields by the formula

This gives the space of smooth vector fields the structure of an (infinite-dimensional) Lie algebra:

Definition 6 (Lie algebra) A (real) Lie algebra is a real vector space (possibly infinite dimensional), together with a bilinear map which is anti-symmetric (thus for all , or equivalently for all ) and obeys the Jacobi identity for all .

Exercise 4 If is a smooth manifold, show that (equipped with the Lie bracket) is a Lie algebra.

— 3. The Lie algebra of a Lie group —

Let be a (global) Lie group. By definition, is then a smooth manifolds, so we can thus define the tangent bundle and smooth vector fields as in the preceding section. In particular, we can define the tangent space of at the identity element .

If , then the left multiplication operation is, by definition of a Lie group, a smooth map from to . This creates a derivative map from the tangent bundle to itself. We say that a vector field is left-invariant if one has for all , or equivalently if for all .

Exercise 5 Let be a (global) Lie group. Show that for every element of there is a unique left-invariant vector field such that .

of there is a unique left-invariant vector field such that . Show that the commutator of two left-invariant vector fields is again a left-invariant vector field.

From the above exercise, we can identify the tangent space with the left-invariant vector fields on , and the Lie bracket structure on the latter then induces a Lie bracket (which we also call ) on . The vector space together with this Lie bracket is then a (finite-dimensional) Lie algebra, which we call the Lie algebra of the Lie group , and we write as .

Remark 4 Informally, an element of the Lie algebra is associated with an infinitesimal perturbation of the identity in the Lie group . This intuition can be formalised fairly easily in the case of matrix Lie groups such as ; for more abstract Lie groups, one can still formalise things using nonstandard analysis, but we will not do so here.

Exercise 6 Show that the Lie algebra of the general linear group can be identified with the space of complex matrices, with the Lie bracket .

of the general linear group can be identified with the space of complex matrices, with the Lie bracket . Describe the Lie algebra of the unitary group .

of the unitary group . Describe the Lie algebra of the special unitary group .

of the special unitary group . Describe the Lie algebra of the orthogonal .

of the orthogonal . Describe the Lie algebra of the special orthogonal .

of the special orthogonal . Describe the Lie algebra of the Heisenberg group .

Exercise 7 Let be a smooth homomorphism between (global) Lie groups. Show that the derivative map at the identity element is then a Lie algebra homomorphism from the Lie algebra of to the Lie algebra of (thus this map is linear and preserves the Lie bracket). (From this and the chain rule (1), we see that the map creates a covariant functor from the category of Lie groups to the category of Lie algebras.)

We have seen that every global Lie group gives rise to a Lie algebra. One can also associate Lie algebras to local Lie groups as follows:

Exercise 8 Let be a local Lie group. Let be a symmetric neighbourhood of the identity in . (It is not difficult to see that least one such neighbourhood exists.) Call a vector field left-invariant if, for every , one has , where is the left-multiplication map , defined on the open set (where we adopt the convention that is shorthand for “ is well-defined and lies in “). Establish the analogue of Exercise 5 in this setting. Conclude that one can give the structure of a Lie algebra, which is independent of the choice of .

the structure of a Lie algebra, which is independent of the choice of . Establish the analogue of Exercise 7 in this setting.

Remark 5 In the converse direction, it is also true that every finite-dimensional Lie algebra can be associated to either a local or a global Lie group; this is known as Lie’s third theorem. However, this theorem is somewhat tricky to prove (particularly if one wants to associate the Lie algebra with a global Lie group), requiring the non-trivial algebraic tool of Ado’s theorem (discussed in this previous blog post); see Exercise 21 below.

— 4. The exponential map —

The exponential map on the reals (or its extension to the complex numbers ) is of course fundamental to modern analysis. It can be defined in a variety of ways, such as the following:

(i) is the differentiable map obeying the ODE and the initial condition .

is the differentiable map obeying the ODE and the initial condition . (ii) is the differentiable map obeying the homomorphism property and the initial condition .

is the differentiable map obeying the homomorphism property and the initial condition . (iii) is the limit of the functions as .

is the limit of the functions as . (iv) is the limit of the infinite series .

We will need to generalise this map to arbitrary Lie algebras and Lie groups. In the case of matrix Lie groups (and matrix Lie algebras), one can use the matrix exponential, which can be defined efficiently by modifying definition (iv) above, and which was already discussed in the previous set of notes. It is however difficult to use this definition for abstract Lie algebras and Lie groups. The definition based on (ii) will ultimately be the best one to use for the purposes of this course, but for foundational purposes (i) or (iii) is initially easier to work with. In most of the foundational literature on Lie groups and Lie algebras, one uses (i), in which case the existence and basic properties of the exponential map can be provided by the Picard existence theorem from the theory of ordinary differential equations. However, we will use (iii), because it relies less heavily on the smooth structure of the Lie group, and will therefore be more aligned with the spirit of Hilbert’s fifth problem (which seeks to minimise the reliance of smoothness hypotheses whenever possible). Actually, for minor technical reasons it is slightly more convenient to work with the limit of rather than .

We turn to the details. It will be convenient to work in local coordinates, and for applications to Hilbert’s fifth problem it will be useful to “forget” almost all of the smooth structure. We make the following definition:

Definition 7 ( local group) A local group is a local group that is an open neighbourhood of the origin in a Euclidean space , with group identity , and whose group operation obeys the estimate for all sufficiently small , where the implied constant in the notation can depend on but is uniform in .

Example 4 Let be a local Lie group of some dimension , and let be a smooth coordinate chart that maps a neighbourhood of the group identity to a neighbourhood of the origin in , with . Then, as explained in Example 1, is a local Lie group with identity ; in particular, one has From Taylor expansion (using the smoothness of ) we thus have (4) for sufficiently small . Thus we see that every local Lie group generates a local group when viewed in coordinates.

Remark 6 In real analysis, a (locally) function is a function on a domain which is continuously differentiable (i.e. in the regularity class ), and whose first derivatives are (locally) Lipschitz (i.e. in the regularity class ) the regularity class is slightly weaker (i.e. larger) than the class of twice continuously differentiable functions, but much stronger than the class of singly continuously differentiable functions. See this previous blog post for more on these sorts of regularity classes. The reason for the terminology in the above definition is that regularity is essentially the minimal regularity for which one has the Taylor expansion for any in the domain of , and any sufficiently close to ; note that the asymptotic (4) is of this form.

We now estimate various expressions in a local group.

Exercise 9 Let be a local group. Throughout this exercise, the implied constants in the notation can depend on , but not on parameters such as . (i) Show that there exists an such that one has whenever and are such that , and the implied constant is uniform in . Here and in the sequel we adopt the convention that a statement such as (5) is automatically false unless all expressions in that statement are well-defined. (Hint: induct on using (4). It is best to replace the asymptotic notation by explicit constants in order to ensure that such constants remain uniform in .) In particular, one has the crude estimate under the same hypotheses as above.

such that one has whenever and are such that , and the implied constant is uniform in . Here and in the sequel we adopt the convention that a statement such as (5) is automatically false unless all expressions in that statement are well-defined. (Hint: induct on using (4). It is best to replace the asymptotic notation by explicit constants in order to ensure that such constants remain uniform in .) In particular, one has the crude estimate (ii) Show that one has for sufficiently close to the origin.

(iii) Show that for sufficiently close to the origin. (Hint: first show that , then express as the product of and .)

(iv) Show that whenever are sufficiently close to the origin.

(v) Show that whenever are sufficiently close to the origin.

(vi) Show that there exists an such that whenever and are such that .

such that (vii) Show that there exists an such that for all and such that , where is the product of copies of (assuming of course that this product is well-defined) for , and .

such that (viii) Show that there exists an such that for all and such that . (Hint: do the case when is positive first. In that case, express as the product of conjugates of by various powers of .)

We can now define the exponential map on this local group by defining

for any in a sufficiently small neighbourhood of the origin in .

Exercise 10 Let be a local group. (i) Show that if is a sufficiently small neighbourhood of the origin in , then the limit in (6) exists for all . (Hint: use the previous exercise to estimate the distance between and .) Establish the additional estimate

is a sufficiently small neighbourhood of the origin in , then the limit in (6) exists for all . (Hint: use the previous exercise to estimate the distance between and .) Establish the additional estimate (ii) Show that if is a smooth curve with , and is sufficiently small, then

is a smooth curve with , and is sufficiently small, then (iii) Show that for all sufficiently small , one has the bilipschitz property Conclude in particular that for sufficiently small, is a homeomorphism between and an open neighbourhood of the origin. (Hint: To show that contains a neighbourhood of the origin, use (7) and the contraction mapping theorem.)

, one has the bilipschitz property Show that for and with sufficiently small. (Hint: first handle the case when are dyadic numbers.)

for and with sufficiently small. (Hint: first handle the case when are dyadic numbers.) (iv) Show that for any sufficiently small , one has Then conclude the stronger estimate

, one has Then conclude the stronger estimate (v) Show that for any sufficiently small , one has (Hint: use the previous part, as well as (viii) of Exercise 9.)

Let us say that a local group is radially homogeneous if one has

whenever and are such that are sufficiently small. (In particular, this implies that for sufficiently small .) From the above exercise, we see that any local group can be made into a radially homogeneous local group by first restricting to an open neighbourhood of the identity, and then applying the logarithmic homeomorphism . Thus:

Corollary 8 Every local group has a neighbourhood of the identity which is isomorphic (as a topological group) to a radially homogeneous local group.

Now we study the exponential map on global Lie groups. If is a global Lie group, and is its Lie algebra, we define the exponential map on a global Lie group by setting

whenever is a smooth curve with .

Exercise 11 Let be a global Lie group. (i) Show that the exponential map is well-defined. (Hint: First handle the case when is small, using the previous exercise, then bootstrap to larger values of .)

is small, using the previous exercise, then bootstrap to larger values of .) (ii) Show that for all and , one has and (Hint: again, begin with the case when are small.)

and , one has and (Hint: again, begin with the case when are small.) (iii) Show that the exponential map is continuous.

(iv) Show that for each , the function is the unique homomorphism from to that is differentiable at with derivative equal to .

Proposition 9 (Lie’s first theorem) Let be a Lie group. Then the exponential map is smooth. Furthermore, there is an open neighbourhood of the origin in and an open neighbourhood of the identity in such that the exponential map is a diffeomorphism from to .

Proof: We begin with the smoothness. From the homomorphism property we see that

for all and . If and are sufficiently small, and one uses a coordinate chart near the origin, the function then satisfies an ODE of the form

for some smooth function , with initial condition ; thus by the fundamental theorem of calculus we have

Now let . An application of the contraction mapping theorem (in the function space localised to small region of spacetime) then shows that lies in for small enough , and by further iteration of the integral equation we then conclude that is times continuously differentiable for small enough . By (8) we then conclude that is smooth everywhere.

Since

we see that the derivative of the exponential map at the origin is the identity map on . The second claim of the proposition thus follows from the inverse function theorem.

In view of this proposition, we see that given a vector space basis for the Lie algebra , we may obtain a smooth coordinate chart for some neighbourhood of the identity and neighbourhood of the origin in by defining

for sufficiently small . These are known as exponential coordinates of the first kind. Although we will not use them much here, we also note that there are exponential coordinates of the second kind, in which the expression is replaced by the slight variant .

Using exponential coordinates of the first kind, we see that we may identify a local piece of the Lie group with the radially homogeneous local group . In the next section, we will analyse such radially homogeneous groups further. For now, let us record some easy consequences of the existence of exponential coordinates. Define a one-parameter subgroup of a topological group to be a continuous homomorphism from to .

Exercise 12 (Classification of one-parameter subgroups) Let be a Lie group. For any , show that the map is a one-parameter subgroup. Conversely, if is a one-parameter subgroup, there exists a unique such that for all . (Hint: mimic the proof of Proposition 1 of Notes 0.)

Proposition 10 (Weak regularity implies strong regularity) Let be global Lie groups, and let be a continuous homomorphism. Then is smooth.

Proof: Since is a continuous homomorphism, it maps one-parameter subgroups of to one-parameter subgroups of . Thus, for every , there exists a unique element such that

for all . In particular, we see that is homogeneous: for all and . Next, we observe using (9) and the fact that is a continuous homomorphism that for any and , one has

and thus is additive:

We conclude that is a linear transformation from the finite-dimensional vector space to the finite-dimensional vector space . In particular, is smooth. On the other hand, we have

Since and are diffeomorphisms near the origin, we conclude that is smooth in a neighbourhood of the identity. Using the homomorphism property (and the fact that the group operations are smooth for both and ) we conclude that is smooth everywhere, as required.

This fact has a pleasant corollary:

Corollary 11 (Uniqueness of Lie structure) Any (global) topological group can be made into a Lie group in at most one manner. More precisely, given a topological group , there is at most one smooth structure one can place on that makes the group operations smooth.

Proof: Suppose for sake of contradiction that one could find two different smooth structures on that make the group operations smooth, leading to two different Lie groups based on . The identity map from to is a continuous homomorphism, and hence smooth by the preceding proposition; similarly for the inverse map from to . This implies that the smooth structures coincide, and the claim follows.

Note that a general high-dimensional topological manifold may have more than one smooth structure, which may even be non-diffeomorphic to each other (as the example of exotic spheres demonstrates), so this corollary is not entirely vacuous.

Exercise 13 Let be a connected (global) Lie group, let be another (global) Lie group, and let be a continuous homomorphism (which is thus smooth by Proposition 10). Show that is uniquely determined by the derivative map . In other words, if is another continuous homomorphism with , then . (Hint: first prove this in a small neighbourhood of the origin. What group does this neighbourhood generate?) What happens if is not connected?

Exercise 14 (Weak regularity implies strong regularity, local version) Let be local Lie groups, and let be a continuous homomorphism. Show that is smooth in a neighbourhood of the identity in .

Exercise 15 (Local Lie implies Lie) Let be a global topological group. Suppose that there is an open neighbourhood of the identity such that the local group can be given the structure of a local Lie group. Show that can be given the structure of a global Lie group. (Hint: We already have at least one coordinate chart on ; translate it around to create an atlas of such charts. To show compatibility of the charts and global smoothness of the group, one needs to show that the conjugation maps are smooth near the origin for any . To prove this, use Exercise 14.)

— 5. The Baker-Campbell-Hausdorff formula —

We now study radially homogeneous local groups in more detail. We will show

Theorem 12 (Baker-Campbell-Hausdorff formula, qualitative version) Let be a radially homogeneous local group. Then the group operation is real analytic near the origin. In particular, after restricting to a sufficiently small neighbourhood of the origin, one obtains a local Lie group.

We will in fact give a more precise formula for , known as the Baker-Campbell-Haudorff-Dynkin formula, in the course of proving Theorem 12.

Remark 7 In the case where comes from viewing a general linear group in local exponential coordinates, the group operation is given by for sufficiently small . Thus, a corollary of Theorem 12 is that this map is real analytic.

We begin the proof of Theorem 12. Throughout this section, is a fixed radially homogeneous local group. We will need some variants of the basic bound (4).

Exercise 16 (Lipschitz bounds) If are sufficiently small, establish the bounds and (Hint: to prove (15), start with the identity .)

Now we exploit the radial homogeneity to describe the conjugation operation as a linear map:

Lemma 13 (Adjoint representation) For all sufficiently close to the origin, there exists a linear transformation such that for all sufficiently close to the origin.

Remark 8 Using the matrix example from Remark 7, we are asserting here that for some linear transform of , and all sufficiently small . Indeed, using the basic matrix identity for invertible (coming from the fact that the conjugation map is a continuous ring homomorphism) we see that we may take here.

Proof: Fix . The map is continuous near the origin, so it will suffice to establish additivity, in the sense that

for sufficiently close to the origin.

Let be a large natural number. Then from (11) we have

Conjugating this by , we see that

But from (4) we have

and thus (by Exercise 16)

But if we split as the product of and and use (4), we have

Putting all this together we see that

sending we obtain the claim.

From (4) we see that

for sufficiently small. Also from the associativity property we see that

for all sufficiently small. Combining these two properties (and using (15)) we conclude in particular that

for sufficiently small. Thus we see that is a (locally) continuous linear representation. In particular, is a (locally) continuous homomorphism into a linear group, and so (by Proposition 1 of Notes 0) we have the Hadamard lemma

for all sufficiently small , where is the linear transformation

From (21), (20), (4) we see that

for sufficiently small, and so by the product rule we have

Also we clearly have for small. Thus we see that is linear in , and so we have

for some bilinear form .

One can show that this bilinear form in fact defines a Lie bracket (i.e. it is anti-symmetric and obeys the Jacobi identity), but for now, all we need is that it is manifestly real analytic (since all bilinear forms are polynomial and thus analytic). In particular and depend analytically on .

We now give an important approximation to in the case when is small:

Lemma 14 For sufficiently small, we have where

Proof: If we write , then (by (4)) and

We will shortly establish the approximation

inverting

we obtain the claim.

It remains to verify (22). Let be a large natural number. We can expand the left-hand side of (22) as a telescoping series

Using (11), the first summand can be expanded as

From (15) one has , so by (17), (18) we can write the preceding expression as

which by definition of can be rewritten as

From (15) one has

while from (20) one has , hence from (4) we can rewrite (24) as

Inserting this back into (23), we can thus write the left-hand side of (22) as

Writing , and then letting , we conclude (from the convergence of the Riemann sum to the Riemann integral) that

and the claim follows.

Remark 9 In the matrix case, the key computation is to show that To see this, we can use the fundamental theorem of calculus to write the left-hand side as Since and , we can rewrite this as Since , this becomes since , we obtain the desired claim.

We can integrate the above formula to obtain an exact formula for :

Corollary 15 (Baker-Campbell-Hausdorff-Dynkin formula) For sufficiently small, one has

The right-hand side is clearly real analytic in and , and Theorem 12 follows.

Proof: Let be a large natural number. We can express as the telescoping sum

From (11) followed by Lemma 14 and (21), one has

We conclude that

Sending , so that the Riemann sum converges to a Riemann integral, we obtain the claim.

Exercise 17 Use the Taylor-type expansion to obtain the explicit expansion where , and show that the series is absolutely convergent for small enough. Invert this to obtain the alternate expansion

Exercise 18 Let be a radially homogeneous local group. By Theorem 12, an open neighbourhood of the origin in has the structure of a local Lie group, and thus by Exercise 8 is associated to a Lie algebra. Show that this Lie algebra is isomorphic to and the Lie bracket is given by (21). Note that this establishes a posteriori the fact that the bracket occurring in (21) is anti-symmetric and obeys the Jacobi identity.

We now record some consequences of the Baker-Campbell-Hausdorff formula.

Exercise 19 (Lie groups are analytic) Let be a global Lie group. Show that is a real analytic manifold (i.e. one can find an atlas of smooth coordinate charts whose transition maps are all real analytic), and that the group operations are also real analytic (i.e. they are real analytic when viewed in the above-mentioned coordinate charts). Furthermore, show that any continuous homomorphism between Lie groups is also real analytic.

Exercise 20 (Lie’s second theorem) Let be global Lie groups, and let be a Lie algebra homomorphism. Show that there exists an open neighbourhood of the identity in and a homomorphism from the local Lie group to such that . If is connected and simply connected, show that one can take to be all of .

Exercise 21 (Lie’s third theorem) Ado’s theorem asserts that every finite-dimensional Lie algebra is isomorphic to a subalgebra of for some . This (somewhat difficult) theorem and its proof is discussed in this previous blog post. Assuming Ado’s theorem as a “black box”, conclude the following claims: (i) (Lie’s third theorem, local version) Every finite-dimensional Lie algebra is isomorphic to the Lie algebra of some local Lie group.

(ii) Every local or global Lie group has a neighbourhood of the identity that is isomorphic to a local linear Lie group (i.e. a local Lie group contained in or for some ).

or for some ). (iii) (Lie’s third theorem, global version) Every finite-dimensional Lie algebra is isomorphic to the Lie algebra of some global Lie group. (Hint: from (i) and (ii), one may identify with the Lie algebra of a local linear Lie group. Now consider the space of all smooth curves in the ambient linear group that are everywhere “tangent” to this local linear Lie group modulo “homotopy”, and use this to build the global Lie group.)

is isomorphic to the Lie algebra of some global Lie group. (Hint: from (i) and (ii), one may identify with the Lie algebra of a local linear Lie group. Now consider the space of all smooth curves in the ambient linear group that are everywhere “tangent” to this local linear Lie group modulo “homotopy”, and use this to build the global Lie group.) (iv) (Lie’s third theorem, simply connected version) Every finite-dimensional Lie algebra is isomorphic to the Lie algebra of some global connected, simply connected Lie group. Furthermore, this Lie group is unique up to isomorphism.

is isomorphic to the Lie algebra of some global connected, simply connected Lie group. Furthermore, this Lie group is unique up to isomorphism. (v) Show that every local Lie group has a neighbourhood of the identity that is isomorphic to a neighbourhood of the identity of a global connected, simply connected Lie group. Furthermore, this Lie group is unique up to isomorphism.

Remark 10 One does not need the full strength of Ado’s theorem to establish conclusion (i) of the above exercise. Indeed, it suffices to show that the operation defined in Exercise 17 is associative near the origin. To do this, it suffices to verify associativity in the sense of formal power series; and then by abstract nonsense one can lift up to the free Lie algebra on generators, and then down to the free nilpotent Lie algebra on generators and of some arbitrary finite step , which one can verify to be a finite dimensional Lie algebra. Applying Ado’s theorem for the special case of nilpotent Lie algebras (which is easier to establish than the general case of Ado’s theorem, as discussed in this previous blog post), one can identify this nilpotent Lie algebra with a subalgebra of for some , and then one can argue as in the above exercise to conclude. However, I do not know how to establish conclusions (ii), (iii) or (iv) without using Ado’s theorem in full generality (and (ii) is in fact equivalent to this theorem, at least in characteristic ).

Remark 11 Lie’s three theorems can be interpreted as establishing an equivalence between three different categories: the category of finite-dimensional Lie algebras; the category of local Lie groups (or more precisely, the category of local Lie group germs, formed by identifying local Lie groups that are identical near the origin); and the category of global connected, simply connected Lie groups. See this blog post for further discussion.

The fact that we were able to establish the Baker-Campbell-Hausdorff formula at the regularity level will be useful for the purposes of proving results related to Hilbert’s fifth problem. In particular, we have the following criterion for a group to be Lie (very much in accordance with the “weak regularity implies strong regularity for group-like objects” principle):

Lemma 16 (Criterion for Lie structure) Let be a topological group. Show that is Lie if and only if there is a neighbourhood of the identity in which is isomorphic (as a topological group) to a local group.

Proof: The “only if” direction is trivial. For the “if” direction, combine Corollary 8 with Theorem 12 and Exercise 15.

Remark 12 Informally, Lemma 16 asserts that regularity can automatically be upgraded to smooth ( ) or even real analytic ( ) regularity for topological groups. In contrast, note that a locally Euclidean group has neighbourhoods of the identity that are isomorphic to a “ local group” (which is the same concept as a local group, but without the asymptotic (4)). Thus we have reduced Hilbert’s fifth problem to the task of boosting regularity to regularity, rather than that of boosting regularity to regularity.

Exercise 22 Let be a Lie group with Lie algebra . For any , show that