10 1. Introduction

which is isomorphic to the image of the Heisenberg group under the Weil

representation, or equivalently the group of isometries of

L2(R)

generated

by translations and modulations. Despite this, though, it is helpful to think

of closed linear groups and Lie groups as being almost the same concept

as a first approximation. For instance, one can show using Ado’s theorem

(Theorem 13.0.4) that every Lie group is locally isomorphic to a linear local

group (a concept we will discuss in Section 2.1).

An important subclass of the closed linear groups are the linear algebraic

groups, in which the group is also a real or complex algebraic variety (or at

least an algebraically constructible set). All of the examples of closed linear

groups given above are linear algebraic groups, although there exist closed

linear groups that are not isomorphic to any algebraic group; see Proposition

21.0.4.

Exercise 1.1.1 (Weil-Heisenberg group is not linear). Show that there is

no injective homomorphism ρ : G → GLn(C) from the Weil-Heisenberg

group (1.3) to a general linear group GLn(C) for any finite n. (Hint: The

centre [G, G] maps via ρ to a circle subgroup of GLn(C); diagonalise this

subgroup and reduce to the case when the image of the centre consists of

multiples of the identity. Now, use the fact that commutators in GLn(C)

have determinant one.) This fact was first observed by Birkhoff.

Hilbert’s fifth problem, like many of Hilbert’s problems, does not have a

unique interpretation, but one of the most commonly accepted interpreta-

tions of the question posed by Hilbert is to determine if the requirement of

smoothness in the definition of a Lie group is redundant. (There is also an

analogue of Hilbert’s fifth problem for group actions, known as the Hilbert-

Smith conjecture; see Chapter 17.) To answer this question, we need to relax

the notion of a Lie group to that of a topological group.

Definition 1.1.8 (Topological group). A topological group is a group G =

(G, ·) that is also a topological space, in such a way that the group operations

· : G×G → G and

()−1

: G → G are continuous. (As before, we also consider

additive topological groups G = (G, +) provided that they are abelian.)

Clearly, every Lie group is a topological group if one simply forgets the

smooth structure, retaining only the topological and group structures. Fur-

thermore, such topological groups remain locally Euclidean. It was estab-

lished by Montgomery-Zippin [MoZi1952] and Gleason [Gl1952] that the

converse statement holds, thus solving at least one formulation of Hilbert’s

fifth problem:

Theorem 1.1.9 (Hilbert’s fifth problem). Let G be an object with the fol-

lowing properties: