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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}}.)

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In mathematics, one frequently starts with some space {X} and wishes to extend it to a larger space {Y}. Generally speaking, there are two ways in which one can extend a space {X}:

  • By embedding {X} into a space {Y} that has {X} (or at least an isomorphic copy of {X}) as a subspace.
  • By covering {X} by a space {Y} that has {X} (or an isomorphic copy thereof) as a quotient.

For many important categories of interest (such as abelian categories), the former type of extension can be represented by the exact sequence,

\displaystyle  0 \rightarrow X \rightarrow Y

and the latter type of extension be represented by the exact sequence

\displaystyle  Y \rightarrow X \rightarrow 0.

In some cases, {X} can be both embedded in, and covered by, {Y}, in a consistent fashion; in such cases we sometimes say that the above exact sequences split.

An analogy would be to that of digital images. When a computer represents an image, it is limited both by the scope of the image (what it is picturing), and by the resolution of an image (how much physical space is represented by a given pixel). To make the image “larger”, one could either embed the image in an image of larger scope but equal resolution (e.g. embedding a picture of a {200 \times 200} pixel image of person’s face into a {800 \times 800} pixel image that covers a region of space that is four times larger in both dimensions, e.g. the person’s upper body) or cover the image with an image of higher resolution but of equal scope (e.g. enhancing a {200 \times 200} pixel picture of a face to a {800 \times 800} pixel of the same face). In the former case, the original image is a sub-image (or cropped image) of the extension, but in the latter case the original image is a quotient (or a pixelation) of the extension. In the former case, each pixel in the original image can be identified with a pixel in the extension, but not every pixel in the extension is covered. In the latter case, every pixel in the original image is covered by several pixels in the extension, but the pixel in the original image is not canonically identified with any particular pixel in the extension that covers it; it “loses its identity” by dispersing into higher resolution pixels.

(Note that “zooming in” the visual representation of an image by making each pixel occupy a larger region of the screen neither increases the scope or the resolution; in this language, a zoomed-in version of an image is merely an isomorphic copy of the original image; it carries the same amount of information as the original image, but has been represented in a new coordinate system which may make it easier to view, especially to the visually impaired.)

In the study of a given category of spaces (e.g. topological spaces, manifolds, groups, fields, etc.), embedding and coverings are both important; this is particularly true in the more topological areas of mathematics, such as manifold theory. But typically, the term extension is reserved for just one of these two operations. For instance, in the category of fields, coverings are quite trivial; if one covers a field {k} by a field {l}, the kernel of the covering map {\pi: l \rightarrow k} is necessarily trivial and so {k, l} are in fact isomorphic. So in field theory, a field extension refers to an embedding of a field, rather than a covering of a field. Similarly, in the theory of metric spaces, there are no non-trivial isometric coverings of a metric space, and so the only useful notion of an extension of a metric space is the one given by embedding the original space in the extension.

On the other hand, in group theory (and in group-like theories, such as the theory of dynamical systems, which studies group actions), the term “extension” is reserved for coverings, rather than for embeddings. I think one of the main reasons for this is that coverings of groups automatically generate a special type of embedding (a normal embedding), whereas most embeddings don’t generate coverings. More precisely, given a group extension {G} of a base group {H},

\displaystyle  G \rightarrow H \rightarrow 0,

one can form the kernel {K = \hbox{ker}(\phi)} of the covering map {\pi: G \rightarrow H}, which is a normal subgroup of {G}, and we thus can extend the above sequence canonically to a short exact sequence

\displaystyle  0 \rightarrow K \rightarrow G \rightarrow H \rightarrow 0.

On the other hand, an embedding of {K} into {G},

\displaystyle  0 \rightarrow K \rightarrow G

does not similarly extend to a short exact sequence unless the the embedding is normal.

Another reason for the notion of extension varying between embeddings and coverings from subject to subject is that there are various natural duality operations (and more generally, contravariant functors) which turn embeddings into coverings and vice versa. For instance, an embedding of one vector space {V} into another {W} induces a covering of the dual space {V^*} by the dual space {W^*}, and conversely; similarly, an embedding of a locally compact abelian group {H} in another {G} induces a covering of the Pontryagin dual {\hat H} by the Pontryagin dual {\hat G}. In the language of images, embedding an image in an image of larger scope is largely equivalent to covering the Fourier transform of that image by a transform of higher resolution, and conversely; this is ultimately a manifestation of the basic fact that frequency is inversely proportional to wavelength.

Similarly, a common duality operation arises in many areas of mathematics by starting with a space {X} and then considering a space {C(X)} of functions on that space (e.g. continuous real-valued functions, if {X} was a topological space, or in more algebraic settings one could consider homomorphisms from {X} to some fixed space). Embedding {X} into {Y} then induces a covering of {C(X)} by {C(Y)}, and conversely, a covering of {X} by {Y} induces an embedding of {C(X)} into {C(Y)}. Returning again to the analogy with images, if one looks at the collection of all images of a fixed scope and resolution, rather than just a single image, then increasing the available resolution causes an embedding of the space of low-resolution images into the space of high-resolution images (since of course every low-resolution image is an example of a high-resolution image), whereas increasing the available scope causes a covering of the space of narrow-scope images by the space of wide-scope images (since every wide-scope image can be cropped into a narrow-scope image). Note in the case of images, that these extensions can be split: not only can a low-resolution image be viewed as a special case of a high-resolution image, but any high-resolution image can be pixelated into a low-resolution one. Similarly, not only can any wide-scope image be cropped into a narrow-scope one, a narrow-scope image can be extended to a wide-scope one simply by filling in all the new areas of scope with black (or by using more advanced image processing tools to create a more visually pleasing extension). (In the category of sets, the statement that every covering can be split is precisely the axiom of choice.)

I’ve recently found myself having to deal quite a bit with group extensions in my research, so I have decided to make some notes on the basic theory of such extensions here. This is utterly elementary material for a group theorist, but I found this task useful for organising my own thoughts on this topic, and also in pinning down some of the jargon in this field.

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