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:
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