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Over the past few months or so, I have been brushing up on my Lie group theory, as part of my project to fully understand the theory surrounding Hilbert’s fifth problem. Every so often, I encounter a basic fact in Lie theory which requires a slightly non-trivial “trick” to prove; I am recording two of them here, so that I can find these tricks again when I need to.

The first fact concerns the exponential map ${\exp: {\mathfrak g} \rightarrow G}$ from a Lie algebra ${{\mathfrak g}}$ of a Lie group ${G}$ to that group. (For this discussion we will only consider finite-dimensional Lie groups and Lie algebras over the reals ${{\bf R}}$.) A basic fact in the subject is that the exponential map is locally a homeomorphism: there is a neighbourhood of the origin in ${{\mathfrak g}}$ that is mapped homeomorphically by the exponential map to a neighbourhood of the identity in ${G}$. This local homeomorphism property is the foundation of an important dictionary between Lie groups and Lie algebras.

It is natural to ask whether the exponential map is globally a homeomorphism, and not just locally: in particular, whether the exponential map remains both injective and surjective. For instance, this is the case for connected, simply connected, nilpotent Lie groups (as can be seen from the Baker-Campbell-Hausdorff formula.)

The circle group ${S^1}$, which has ${{\bf R}}$ as its Lie algebra, already shows that global injectivity fails for any group that contains a circle subgroup, which is a huge class of examples (including, for instance, the positive dimensional compact Lie groups, or non-simply-connected Lie groups). Surjectivity also obviously fails for disconnected groups, since the Lie algebra is necessarily connected, and so the image under the exponential map must be connected also. However, even for connected Lie groups, surjectivity can fail. To see this, first observe that if the exponential map was surjective, then every group element ${g \in G}$ has a square root (i.e. an element ${h \in G}$ with ${h^2 = g}$), since ${\exp(x)}$ has ${\exp(x/2)}$ as a square root for any ${x \in {\mathfrak g}}$. However, there exist elements in connected Lie groups without square roots. A simple example is provided by the matrix

$\displaystyle g = \begin{pmatrix} -4 & 0 \\ 0 & -1/4 \end{pmatrix}$

in the connected Lie group ${SL_2({\bf R})}$. This matrix has eigenvalues ${-4}$, ${-1/4}$. Thus, if ${h \in SL_2({\bf R})}$ is a square root of ${g}$, we see (from the Jordan normal form) that it must have at least one eigenvalue in ${\{-2i,+2i\}}$, and at least one eigenvalue in ${\{-i/2,i/2\}}$. On the other hand, as ${h}$ has real coefficients, the complex eigenvalues must come in conjugate pairs ${\{ a+bi, a-bi\}}$. Since ${h}$ can only have at most ${2}$ eigenvalues, we obtain a contradiction.

However, there is an important case where surjectivity is recovered:

Proposition 1 If ${G}$ is a compact connected Lie group, then the exponential map is surjective.

Proof: The idea here is to relate the exponential map in Lie theory to the exponential map in Riemannian geometry. We first observe that every compact Lie group ${G}$ can be given the structure of a Riemannian manifold with a bi-invariant metric. This can be seen in one of two ways. Firstly, one can put an arbitrary positive definite inner product on ${{\mathfrak g}}$ and average it against the adjoint action of ${G}$ using Haar probability measure (which is available since ${G}$ is compact); this gives an ad-invariant positive-definite inner product on ${{\mathfrak g}}$ that one can then translate by either left or right translation to give a bi-invariant Riemannian structure on ${G}$. Alternatively, one can use the Peter-Weyl theorem to embed ${G}$ in a unitary group ${U(n)}$, at which point one can induce a bi-invariant metric on ${G}$ from the one on the space ${M_n({\bf C}) \equiv {\bf C}^{n^2}}$ of ${n \times n}$ complex matrices.

As ${G}$ is connected and compact and thus complete, we can apply the Hopf-Rinow theorem and conclude that any two points are connected by at least one geodesic, so that the Riemannian exponential map from ${{\mathfrak g}}$ to ${G}$ formed by following geodesics from the origin is surjective. But one can check that the Lie exponential map and Riemannian exponential map agree; for instance, this can be seen by noting that the group structure naturally defines a connection on the tangent bundle which is both torsion-free and preserves the bi-invariant metric, and must therefore agree with the Levi-Civita metric. (Alternatively, one can embed into a unitary group ${U(n)}$ and observe that ${G}$ is totally geodesic inside ${U(n)}$, because the geodesics in ${U(n)}$ can be described explicitly in terms of one-parameter subgroups.) The claim follows. $\Box$

Remark 1 While it is quite nice to see Riemannian geometry come in to prove this proposition, I am curious to know if there is any other proof of surjectivity for compact connected Lie groups that does not require explicit introduction of Riemannian geometry concepts.

The other basic fact I learned recently concerns the algebraic nature of Lie groups and Lie algebras. An important family of examples of Lie groups are the algebraic groups – algebraic varieties with a group law given by algebraic maps. Given that one can always automatically upgrade the smooth structure on a Lie group to analytic structure (by using the Baker-Campbell-Hausdorff formula), it is natural to ask whether one can upgrade the structure further to an algebraic structure. Unfortunately, this is not always the case. A prototypical example of this is given by the one-parameter subgroup

$\displaystyle G := \{ \begin{pmatrix} t & 0 \\ 0 & t^\alpha \end{pmatrix}: t \in {\bf R}^+ \} \ \ \ \ \ (1)$

of ${GL_2({\bf R})}$. This is a Lie group for any exponent ${\alpha \in {\bf R}}$, but if ${\alpha}$ is irrational, then the curve that ${G}$ traces out is not an algebraic subset of ${GL_2({\bf R})}$ (as one can see by playing around with Puiseux series).

This is not a true counterexample to the claim that every Lie group can be given the structure of an algebraic group, because one can give ${G}$ a different algebraic structure than one inherited from the ambient group ${GL_2({\bf R})}$. Indeed, ${G}$ is clearly isomorphic to the additive group ${{\bf R}}$, which is of course an algebraic group. However, a modification of the above construction works:

Proposition 2 There exists a Lie group ${G}$ that cannot be given the structure of an algebraic group.

Proof: We use an example from the text of Tauvel and Yu (that I found via this MathOverflow posting). We consider the subgroup

$\displaystyle G := \{ \begin{pmatrix} 1 & 0 & 0 \\ x & t & 0 \\ y & 0 & t^\alpha \end{pmatrix}: x, y \in {\bf R}; t \in {\bf R}^+ \}$

of ${GL_3({\bf R})}$, with ${\alpha}$ an irrational number. This is a three-dimensional (metabelian) Lie group, whose Lie algebra ${{\mathfrak g} \subset {\mathfrak gl}_3({\bf R})}$ is spanned by the elements

$\displaystyle X := \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \alpha \end{pmatrix}$

$\displaystyle Y := \begin{pmatrix} 0 & 0 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

$\displaystyle Z := \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ -\alpha & 0 & 0 \end{pmatrix}$

with the Lie bracket given by

$\displaystyle [Y,X] = -Y; [Z,X] = -\alpha Z; [Y,Z] = 0.$

As such, we see that if we use the basis ${X, Y, Z}$ to identify ${{\mathfrak g}}$ to ${{\bf R}^3}$, then adjoint representation of ${G}$ is the identity map.

If ${G}$ is an algebraic group, it is easy to see that the adjoint representation ${\hbox{Ad}: G \rightarrow GL({\mathfrak g})}$ is also algebraic, and so ${\hbox{Ad}(G) = G}$ is algebraic in ${GL({\mathfrak g})}$. Specialising to our specific example, in which adjoint representation is the identity, we conclude that if ${G}$ has any algebraic structure, then it must also be an algebraic subgroup of ${GL_3({\bf R})}$; but ${G}$ projects to the group (1) which is not algebraic, a contradiction. $\Box$

A slight modification of the same argument also shows that not every Lie algebra is algebraic, in the sense that it is isomorphic to a Lie algebra of an algebraic group. (However, there are important classes of Lie algebras that are automatically algebraic, such as nilpotent or semisimple Lie algebras.)

In the last few months, I have been working my way through the theory behind the solution to Hilbert’s fifth problem, as I (together with Emmanuel Breuillard, Ben Green, and Tom Sanders) have found this theory to be useful in obtaining noncommutative inverse sumset theorems in arbitrary groups; I hope to be able to report on this connection at some later point on this blog. Among other things, this theory achieves the remarkable feat of creating a smooth Lie group structure out of what is ostensibly a much weaker structure, namely the structure of a locally compact group. The ability of algebraic structure (in this case, group structure) to upgrade weak regularity (in this case, continuous structure) to strong regularity (in this case, smooth and even analytic structure) seems to be a recurring theme in mathematics, and an important part of what I like to call the “dichotomy between structure and randomness”.

The theory of Hilbert’s fifth problem sprawls across many subfields of mathematics: Lie theory, representation theory, group theory, nonabelian Fourier analysis, point-set topology, and even a little bit of group cohomology. The latter aspect of this theory is what I want to focus on today. The general question that comes into play here is the extension problem: given two (topological or Lie) groups ${H}$ and ${K}$, what is the structure of the possible groups ${G}$ that are formed by extending ${H}$ by ${K}$. In other words, given a short exact sequence

$\displaystyle 0 \rightarrow K \rightarrow G \rightarrow H \rightarrow 0,$

to what extent is the structure of ${G}$ determined by that of ${H}$ and ${K}$?

As an example of why understanding the extension problem would help in structural theory, let us consider the task of classifying the structure of a Lie group ${G}$. Firstly, we factor out the connected component ${G^\circ}$ of the identity as

$\displaystyle 0 \rightarrow G^\circ \rightarrow G \rightarrow G/G^\circ \rightarrow 0;$

as Lie groups are locally connected, ${G/G^\circ}$ is discrete. Thus, to understand general Lie groups, it suffices to understand the extensions of discrete groups by connected Lie groups.

Next, to study a connected Lie group ${G}$, we can consider the conjugation action ${g: X \mapsto gXg^{-1}}$ on the Lie algebra ${{\mathfrak g}}$, which gives the adjoint representation ${\hbox{Ad}: G \rightarrow GL({\mathfrak g})}$. The kernel of this representation consists of all the group elements ${g}$ that commute with all elements of the Lie algebra, and thus (by connectedness) is the center ${Z(G)}$ of ${G}$. The adjoint representation is then faithful on the quotient ${G/Z(G)}$. The short exact sequence

$\displaystyle 0 \rightarrow Z(G) \rightarrow G \rightarrow G/Z(G) \rightarrow 0$

then describes ${G}$ as a central extension (by the abelian Lie group ${Z(G)}$) of ${G/Z(G)}$, which is a connected Lie group with a faithful finite-dimensional linear representation.

This suggests a route to Hilbert’s fifth problem, at least in the case of connected groups ${G}$. Let ${G}$ be a connected locally compact group that we hope to demonstrate is isomorphic to a Lie group. As discussed in a previous post, we first form the space ${L(G)}$ of one-parameter subgroups of ${G}$ (which should, eventually, become the Lie algebra of ${G}$). Hopefully, ${L(G)}$ has the structure of a vector space. The group ${G}$ acts on ${L(G)}$ by conjugation; this action should be both continuous and linear, giving an “adjoint representation” ${\hbox{Ad}: G \rightarrow GL(L(G))}$. The kernel of this representation should then be the center ${Z(G)}$ of ${G}$. The quotient ${G/Z(G)}$ is locally compact and has a faithful linear representation, and is thus a Lie group by von Neumann’s version of Cartan’s theorem (discussed in this previous post). The group ${Z(G)}$ is locally compact abelian, and so it should be a relatively easy task to establish that it is also a Lie group. To finish the job, one needs the following result:

Theorem 1 (Central extensions of Lie are Lie) Let ${G}$ be a locally compact group which is a central extension of a Lie group ${H}$ by an abelian Lie group ${K}$. Then ${G}$ is also isomorphic to a Lie group.

This result can be obtained by combining a result of Kuranishi with a result of Gleason; I am recording this argument below the fold. The point here is that while ${G}$ is initially only a topological group, the smooth structures of ${H}$ and ${K}$ can be combined (after a little bit of cohomology) to create the smooth structure on ${G}$ required to upgrade ${G}$ from a topological group to a Lie group. One of the main ideas here is to improve the behaviour of a cocycle by averaging it; this basic trick is helpful elsewhere in the theory, resolving a number of cohomological issues in topological group theory. The result can be generalised to show in fact that arbitrary (topological) extensions of Lie groups by Lie groups remain Lie; this was shown by Gleason. However, the above special case of this result is already sufficient (in conjunction with the rest of the theory, of course) to resolve Hilbert’s fifth problem.

Remark 1 We have shown in the above discussion that every connected Lie group is a central extension (by an abelian Lie group) of a Lie group with a faithful continuous linear representation. It is natural to ask whether this central extension is necessary. Unfortunately, not every connected Lie group admits a faithful continuous linear representation. An example (due to Birkhoff) is the Heisenberg-Weyl group

$\displaystyle G := \begin{pmatrix} 1 & {\bf R} & {{\bf R}/{\bf Z}} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix} / \begin{pmatrix} 1 & 0 & {\bf Z} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.$

Indeed, if we consider the group elements

$\displaystyle A := \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

and

$\displaystyle B := \begin{pmatrix} 1 & 0 & 1/p \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

for some prime ${p}$, then one easily verifies that ${B}$ has order ${p}$ and is central, and that ${AB}$ is conjugate to ${A}$. If we have a faithful linear representation ${\rho: G \rightarrow GL_n({\bf C})}$ of ${G}$, then ${\rho(B)}$ must have at least one eigenvalue ${\alpha}$ that is a primitive ${p^{th}}$ root of unity. If ${V}$ is the eigenspace associated to ${\alpha}$, then ${\rho(A)}$ must preserve ${V}$, and be conjugate to ${\alpha \rho(A)}$ on this space. This forces ${\rho(A)}$ to have at least ${p}$ distinct eigenvalues on ${V}$, and hence ${V}$ (and thus ${{\bf C}^n}$) must have dimension at least ${p}$. Letting ${p \rightarrow \infty}$ we obtain a contradiction. (On the other hand, ${G}$ is certainly isomorphic to the extension of the linear group ${{\bf R}^2}$ by the abelian group ${{\bf R}/{\bf Z}}$.)

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.

Recall that a (complex) abstract Lie algebra is a complex vector space ${{\mathfrak g}}$ (either finite or infinite dimensional) equipped with a bilinear antisymmetric form ${[]: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}}$ that obeys the Jacobi identity

$\displaystyle [[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] = 0. \ \ \ \ \ (1)$

(One can of course define Lie algebras over other fields than the complex numbers ${{\bf C}}$, but in order to avoid some technical issues we shall work solely with the complex case in this post.)

An important special case of the abstract Lie algebras are the concrete Lie algebras, in which ${{\mathfrak g} \subset \hbox{End}(V)}$ is a vector space of linear transformations ${X: V \rightarrow V}$ on a vector space ${V}$ (which again can be either finite or infinite dimensional), and the bilinear form is given by the usual Lie bracket

$\displaystyle [X,Y] := XY-YX.$

It is easy to verify that every concrete Lie algebra is an abstract Lie algebra. In the converse direction, we have

Theorem 1 Every abstract Lie algebra is isomorphic to a concrete Lie algebra.

To prove this theorem, we introduce the useful algebraic tool of the universal enveloping algebra ${U({\mathfrak g})}$ of the abstract Lie algebra ${{\mathfrak g}}$. This is the free (associative, complex) algebra generated by ${{\mathfrak g}}$ (viewed as a complex vector space), subject to the constraints

$\displaystyle [X,Y] = XY - YX. \ \ \ \ \ (2)$

This algebra is described by the Poincaré-Birkhoff-Witt theorem, which asserts that given an ordered basis ${(X_i)_{i \in I}}$ of ${{\mathfrak g}}$ as a vector space, that a basis of ${U({\mathfrak g})}$ is given by “monomials” of the form

$\displaystyle X_{i_1}^{a_1} \ldots X_{i_m}^{a_m} \ \ \ \ \ (3)$

where ${m}$ is a natural number, the ${i_1 < \ldots < i_m}$ are an increasing sequence of indices in ${I}$, and the ${a_1,\ldots,a_m}$ are positive integers. Indeed, given two such monomials, one can express their product as a finite linear combination of further monomials of the form (3) after repeatedly applying (2) (which we rewrite as ${XY = YX + [X,Y]}$) to reorder the terms in this product modulo lower order terms until one all monomials have their indices in the required increasing order. It is then a routine exercise in basic abstract algebra (using all the axioms of an abstract Lie algebra) to verify that this is multiplication rule on monomials does indeed define a complex associative algebra which has the universal properties required of the universal enveloping algebra.

The abstract Lie algebra ${{\mathfrak g}}$ acts on its universal enveloping algebra ${U({\mathfrak g})}$ by left-multiplication: ${X: M \mapsto XM}$, thus giving a map from ${{\mathfrak g}}$ to ${\hbox{End}(U({\mathfrak g}))}$. It is easy to verify that this map is a Lie algebra homomorphism (so this is indeed an action (or representation) of the Lie algebra), and this action is clearly faithful (i.e. the map from ${{\mathfrak g}}$ to ${\hbox{End}(U{\mathfrak g})}$ is injective), since each element ${X}$ of ${{\mathfrak g}}$ maps the identity element ${1}$ of ${U({\mathfrak g})}$ to a different element of ${U({\mathfrak g})}$, namely ${X}$. Thus ${{\mathfrak g}}$ is isomorphic to its image in ${\hbox{End}(U({\mathfrak g}))}$, proving Theorem 1.

In the converse direction, every representation ${\rho: {\mathfrak g} \rightarrow \hbox{End}(V)}$ of a Lie algebra “factors through” the universal enveloping algebra, in that it extends to an algebra homomorphism from ${U({\mathfrak g})}$ to ${\hbox{End}(V)}$, which by abuse of notation we shall also call ${\rho}$.

One drawback of Theorem 1 is that the space ${U({\mathfrak g})}$ that the concrete Lie algebra acts on will almost always be infinite-dimensional, even when the original Lie algebra ${{\mathfrak g}}$ is finite-dimensional. However, there is a useful theorem of Ado that rectifies this:

Theorem 2 (Ado’s theorem) Every finite-dimensional abstract Lie algebra is isomorphic to a concrete Lie algebra over a finite-dimensional vector space ${V}$.

Among other things, this theorem can be used (in conjunction with the Baker-Campbell-Hausdorff formula) to show that every abstract (finite-dimensional) Lie group (or abstract local Lie group) is locally isomorphic to a linear group. (It is well-known, though, that abstract Lie groups are not necessarily globally isomorphic to a linear group, but we will not discuss these global obstructions here.)

Ado’s theorem is surprisingly tricky to prove in general, but some special cases are easy. For instance, one can try using the adjoint representation ${\hbox{ad}: {\mathfrak g} \rightarrow \hbox{End}({\mathfrak g})}$ of ${{\mathfrak g}}$ on itself, defined by the action ${X: Y \mapsto [X,Y]}$; the Jacobi identity (1) ensures that this indeed a representation of ${{\mathfrak g}}$. The kernel of this representation is the centre ${Z({\mathfrak g}) := \{ X \in {\mathfrak g}: [X,Y]=0 \hbox{ for all } Y \in {\mathfrak g}\}}$. This already gives Ado’s theorem in the case when ${{\mathfrak g}}$ is semisimple, in which case the center is trivial.

The adjoint representation does not suffice, by itself, to prove Ado’s theorem in the non-semisimple case. However, it does provide an important reduction in the proof, namely it reduces matters to showing that every finite-dimensional Lie algebra ${{\mathfrak g}}$ has a finite-dimensional representation ${\rho: {\mathfrak g} \rightarrow \hbox{End}(V)}$ which is faithful on the centre ${Z({\mathfrak g})}$. Indeed, if one has such a representation, one can then take the direct sum of that representation with the adjoint representation to obtain a new finite-dimensional representation which is now faithful on all of ${{\mathfrak g}}$, which then gives Ado’s theorem for ${{\mathfrak g}}$.

It remains to find a finite-dimensional representation of ${{\mathfrak g}}$ which is faithful on the centre ${Z({\mathfrak g})}$. In the case when ${{\mathfrak g}}$ is abelian, so that the centre ${Z({\mathfrak g})}$ is all of ${{\mathfrak g}}$, this is again easy, because ${{\mathfrak g}}$ then acts faithfully on ${{\mathfrak g} \times {\bf C}}$ by the infinitesimal shear maps ${X: (Y,t) \mapsto (tX, 0)}$. In matrix form, this representation identifies each ${X}$ in this abelian Lie algebra with an “upper-triangular” matrix:

$\displaystyle X \equiv \begin{pmatrix} 0 & X \\ 0 & 0 \end{pmatrix}.$

This construction gives a faithful finite-dimensional representation of the centre ${Z({\mathfrak g})}$ of any finite-dimensional Lie algebra. The standard proof of Ado’s theorem (which I believe dates back to work of Harish-Chandra) then proceeds by gradually “extending” this representation of the centre ${Z({\mathfrak g})}$ to larger and larger sub-algebras of ${{\mathfrak g}}$, while preserving the finite-dimensionality of the representation and the faithfulness on ${Z({\mathfrak g})}$, until one obtains a representation on the entire Lie algebra ${{\mathfrak g}}$ with the required properties. (For technical inductive reasons, one also needs to carry along an additional property of the representation, namely that it maps the nilradical to nilpotent elements, but we will discuss this technicality later.)

This procedure is a little tricky to execute in general, but becomes simpler in the nilpotent case, in which the lower central series ${{\mathfrak g}_1 := {\mathfrak g}; {\mathfrak g}_{n+1} := [{\mathfrak g}, {\mathfrak g}_n]}$ becomes trivial for sufficiently large ${n}$:

Theorem 3 (Ado’s theorem for nilpotent Lie algebras) Let ${{\mathfrak n}}$ be a finite-dimensional nilpotent Lie algebra. Then there exists a finite-dimensional faithful representation ${\rho: {\mathfrak n} \rightarrow \hbox{End}(V)}$ of ${{\mathfrak n}}$. Furthermore, there exists a natural number ${k}$ such that ${\rho({\mathfrak n})^k = \{0\}}$, i.e. one has ${\rho(X_1) \ldots \rho(X_k)=0}$ for all ${X_1,\ldots,X_k \in {\mathfrak n}}$.

The second conclusion of Ado’s theorem here is useful for induction purposes. (By Engel’s theorem, this conclusion is also equivalent to the assertion that every element of ${\rho({\mathfrak n})}$ is nilpotent, but we can prove Theorem 3 without explicitly invoking Engel’s theorem.)

Below the fold, I give a proof of Theorem 3, and then extend the argument to cover the full strength of Ado’s theorem. This is not a new argument – indeed, I am basing this particular presentation from the one in Fulton and Harris – but it was an instructive exercise for me to try to extract the proof of Ado’s theorem from the more general structural theory of Lie algebras (e.g. Engel’s theorem, Lie’s theorem, Levi decomposition, etc.) in which the result is usually placed. (However, the proof I know of still needs Engel’s theorem to establish the solvable case, and the Levi decomposition to then establish the general case.)

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.

Let ${G}$ be a compact group. (Throughout this post, all topological groups are assumed to be Hausdorff.) Then ${G}$ has a number of unitary representations, i.e. continuous homomorphisms ${\rho: G \rightarrow U(H)}$ to the group ${U(H)}$ of unitary operators on a Hilbert space ${H}$, equipped with the strong operator topology. In particular, one has the left-regular representation ${\tau: G \rightarrow U(L^2(G))}$, where we equip ${G}$ with its normalised Haar measure ${\mu}$ (and the Borel ${\sigma}$-algebra) to form the Hilbert space ${L^2(G)}$, and ${\tau}$ is the translation operation

$\displaystyle \tau(g) f(x) := f(g^{-1} x).$

We call two unitary representations ${\rho: G \rightarrow U(H)}$ and ${\rho': G \rightarrow U(H')}$ isomorphic if one has ${\rho'(g) = U \rho(g) U^{-1}}$ for some unitary transformation ${U: H \rightarrow H'}$, in which case we write ${\rho \equiv \rho'}$.

Given two unitary representations ${\rho: G \rightarrow U(H)}$ and ${\rho': G \rightarrow U(H')}$, one can form their direct sum ${\rho \oplus \rho': G \rightarrow U(H \oplus H')}$ in the obvious manner: ${\rho \oplus \rho'(g)(v) := (\rho(g) v, \rho'(g) v)}$. Conversely, if a unitary representation ${\rho: G \rightarrow U(H)}$ has a closed invariant subspace ${V \subset H}$ of ${H}$ (thus ${\rho(g) V \subset V}$ for all ${g \in G}$), then the orthogonal complement ${V^\perp}$ is also invariant, leading to a decomposition ${\rho \equiv \rho\downharpoonright_V \oplus \rho\downharpoonright_{V^\perp}}$ of ${\rho}$ into the subrepresentations ${\rho\downharpoonright_V: G \rightarrow U(V)}$, ${\rho\downharpoonright_{V^\perp}: G \rightarrow U(V^\perp)}$. Accordingly, we will call a unitary representation ${\rho: G \rightarrow U(H)}$ irreducible if ${H}$ is nontrivial (i.e. ${H \neq \{0\}}$) and there are no nontrivial invariant subspaces (i.e. no invariant subspaces other than ${\{0\}}$ and ${H}$); the irreducible representations play a role in the subject analogous to those of prime numbers in multiplicative number theory. By the principle of infinite descent, every finite-dimensional unitary representation is then expressible (perhaps non-uniquely) as the direct sum of irreducible representations.

The Peter-Weyl theorem asserts, among other things, that the same claim is true for the regular representation:

Theorem 1 (Peter-Weyl theorem) Let ${G}$ be a compact group. Then the regular representation ${\tau: G \rightarrow U(L^2(G))}$ is isomorphic to the direct sum of irreducible representations. In fact, one has ${\tau \equiv \bigoplus_{\xi \in \hat G} \rho_\xi^{\oplus \hbox{dim}(V_\xi)}}$, where ${(\rho_\xi)_{\xi \in \hat G}}$ is an enumeration of the irreducible finite-dimensional unitary representations ${\rho_\xi: G \rightarrow U(V_\xi)}$ of ${G}$ (up to isomorphism). (It is not difficult to see that such an enumeration exists.)

In the case when ${G}$ is abelian, the Peter-Weyl theorem is a consequence of the Plancherel theorem; in that case, the irreducible representations are all one dimensional, and are thus indexed by the space ${\hat G}$ of characters ${\xi: G \rightarrow {\bf R}/{\bf Z}}$ (i.e. continuous homomorphisms into the unit circle ${{\bf R}/{\bf Z}}$), known as the Pontryagin dual of ${G}$. (See for instance my lecture notes on the Fourier transform.) Conversely, the Peter-Weyl theorem can be used to deduce the Plancherel theorem for compact groups, as well as other basic results in Fourier analysis on these groups, such as the Fourier inversion formula.

Because the regular representation is faithful (i.e. injective), a corollary of the Peter-Weyl theorem (and a classical theorem of Cartan) is that every compact group can be expressed as the inverse limit of Lie groups, leading to a solution to Hilbert’s fifth problem in the compact case. Furthermore, the compact case is then an important building block in the more general theory surrounding Hilbert’s fifth problem, and in particular a result of Yamabe that any locally compact group contains an open subgroup that is the inverse limit of Lie groups.

I’ve recently become interested in the theory around Hilbert’s fifth problem, due to the existence of a correspondence principle between locally compact groups and approximate groups, which play a fundamental role in arithmetic combinatorics. I hope to elaborate upon this correspondence in a subsequent post, but I will mention that versions of this principle play a crucial role in Gromov’s proof of his theorem on groups of polynomial growth (discussed previously on this blog), and in a more recent paper of Hrushovski on approximate groups (also discussed previously). It is also analogous in many ways to the more well-known Furstenberg correspondence principle between ergodic theory and combinatorics (also discussed previously).

Because of the above motivation, I have decided to write some notes on how the Peter-Weyl theorem is proven. This is utterly standard stuff in abstract harmonic analysis; these notes are primarily for my own benefit, but perhaps they may be of interest to some readers also.

As is now widely reported, the Fields medals for 2010 have been awarded to Elon Lindenstrauss, Ngo Bao Chau, Stas Smirnov, and Cedric Villani. Concurrently, the Nevanlinna prize (for outstanding contributions to mathematical aspects of information science) was awarded to Dan Spielman, the Gauss prize (for outstanding mathematical contributions that have found significant applications outside of mathematics) to Yves Meyer, and the Chern medal (for lifelong achievement in mathematics) to Louis Nirenberg. All of the recipients are of course exceptionally qualified and deserving for these awards; congratulations to all of them. (I should mention that I myself was only very tangentially involved in the awards selection process, and like everyone else, had to wait until the ceremony to find out the winners. I imagine that the work of the prize committees must have been extremely difficult.)

Today, I thought I would mention one result of each of the Fields medalists; by chance, three of the four medalists work in areas reasonably close to my own. (Ngo is rather more distant from my areas of expertise, but I will give it a shot anyway.) This will of course only be a tiny sample of each of their work, and I do not claim to be necessarily describing their “best” achievement, as I only know a portion of the research of each of them, and my selection choice may be somewhat idiosyncratic. (I may discuss the work of Spielman, Meyer, and Nirenberg in a later post.)

In these notes we lay out the basic theory of the Fourier transform, which is of course the most fundamental tool in harmonic analysis and also of major importance in related fields (functional analysis, complex analysis, PDE, number theory, additive combinatorics, representation theory, signal processing, etc.). The Fourier transform, in conjunction with the Fourier inversion formula, allows one to take essentially arbitrary (complex-valued) functions on a group ${G}$ (or more generally, a space ${X}$ that ${G}$ acts on, e.g. a homogeneous space ${G/H}$), and decompose them as a (discrete or continuous) superposition of much more symmetric functions on the domain, such as characters ${\chi: G \rightarrow S^1}$; the precise superposition is given by Fourier coefficients ${\hat f(\xi)}$, which take values in some dual object such as the Pontryagin dual ${\hat G}$ of ${G}$. Characters behave in a very simple manner with respect to translation (indeed, they are eigenfunctions of the translation action), and so the Fourier transform tends to simplify any mathematical problem which enjoys a translation invariance symmetry (or an approximation to such a symmetry), and is somehow “linear” (i.e. it interacts nicely with superpositions). In particular, Fourier analytic methods are particularly useful for studying operations such as convolution ${f, g \mapsto f*g}$ and set-theoretic addition ${A, B \mapsto A+B}$, or the closely related problem of counting solutions to additive problems such as ${x = a_1 + a_2 + a_3}$ or ${x = a_1 - a_2}$, where ${a_1, a_2, a_3}$ are constrained to lie in specific sets ${A_1, A_2, A_3}$. The Fourier transform is also a particularly powerful tool for solving constant-coefficient linear ODE and PDE (because of the translation invariance), and can also approximately solve some variable-coefficient (or slightly non-linear) equations if the coefficients vary smoothly enough and the nonlinear terms are sufficiently tame.

The Fourier transform ${\hat f(\xi)}$ also provides an important new way of looking at a function ${f(x)}$, as it highlights the distribution of ${f}$ in frequency space (the domain of the frequency variable ${\xi}$) rather than physical space (the domain of the physical variable ${x}$). A given property of ${f}$ in the physical domain may be transformed to a rather different-looking property of ${\hat f}$ in the frequency domain. For instance:

• Smoothness of ${f}$ in the physical domain corresponds to decay of ${\hat f}$ in the Fourier domain, and conversely. (More generally, fine scale properties of ${f}$ tend to manifest themselves as coarse scale properties of ${\hat f}$, and conversely.)
• Convolution in the physical domain corresponds to pointwise multiplication in the Fourier domain, and conversely.
• Constant coefficient differential operators such as ${d/dx}$ in the physical domain corresponds to multiplication by polynomials such as ${2\pi i \xi}$ in the Fourier domain, and conversely.
• More generally, translation invariant operators in the physical domain correspond to multiplication by symbols in the Fourier domain, and conversely.
• Rescaling in the physical domain by an invertible linear transformation corresponds to an inverse (adjoint) rescaling in the Fourier domain.
• Restriction to a subspace (or subgroup) in the physical domain corresponds to projection to the dual quotient space (or quotient group) in the Fourier domain, and conversely.
• Frequency modulation in the physical domain corresponds to translation in the frequency domain, and conversely.

(We will make these statements more precise below.)

On the other hand, some operations in the physical domain remain essentially unchanged in the Fourier domain. Most importantly, the ${L^2}$ norm (or energy) of a function ${f}$ is the same as that of its Fourier transform, and more generally the inner product ${\langle f, g \rangle}$ of two functions ${f}$ is the same as that of their Fourier transforms. Indeed, the Fourier transform is a unitary operator on ${L^2}$ (a fact which is variously known as the Plancherel theorem or the Parseval identity). This makes it easier to pass back and forth between the physical domain and frequency domain, so that one can combine techniques that are easy to execute in the physical domain with other techniques that are easy to execute in the frequency domain. (In fact, one can combine the physical and frequency domains together into a product domain known as phase space, and there are entire fields of mathematics (e.g. microlocal analysis, geometric quantisation, time-frequency analysis) devoted to performing analysis on these sorts of spaces directly, but this is beyond the scope of this course.)

In these notes, we briefly discuss the general theory of the Fourier transform, but will mainly focus on the two classical domains for Fourier analysis: the torus ${{\Bbb T}^d := ({\bf R}/{\bf Z})^d}$, and the Euclidean space ${{\bf R}^d}$. For these domains one has the advantage of being able to perform very explicit algebraic calculations, involving concrete functions such as plane waves ${x \mapsto e^{2\pi i x \cdot \xi}}$ or Gaussians ${x \mapsto A^{d/2} e^{-\pi A |x|^2}}$.

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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This is my final Milliman lecture, in which I talk about the sum-product phenomenon in arithmetic combinatorics, and some selected recent applications of this phenomenon to uniform distribution of exponentials, expander graphs, randomness extractors, and detecting (sieving) almost primes in group orbits, particularly as developed by Bourgain and his co-authors.
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