1.1. Hilbert’s fifth problem 9

Remark 1.1.4. In some literature, Lie groups are required to be connected

(and occasionally, are even required to be simply connected), but we will

not adopt this convention here. One can also define infinite-dimensional Lie

groups, but in this text all Lie groups are understood to be finite dimen-

sional.

Example 1.1.5. Every group can be viewed as a Lie group if given the

discrete topology (and the discrete smooth structure). (Note that we are

not requiring Lie groups to be connected.)

Example 1.1.6. Finite-dimensional vector spaces such as

Rd

are (additive)

Lie groups, as are sublattices such as

Zd

or quotients such as

Rd/Zd.

How-

ever, nonclosed subgroups such as

Qd

are not manifolds (at least with the

topology induced from

Rd)

and are thus not Lie groups; similarly, quotients

such as

Rd/Qd

are not Lie groups either (they are not even Hausdorff).

Also, infinite-dimensional topological vector spaces (such as

RN

with the

product topology) will not be Lie groups.

Example 1.1.7. The general linear group GLn(C) of invertible n × n com-

plex matrices is a Lie group. A theorem of Cartan (Theorem 3.0.14) asserts

that any closed subgroup of a Lie group is a smooth submanifold of that

Lie group and is in particular also a Lie group. In particular, closed lin-

ear groups (i.e., closed subgroups of a general linear group) are Lie groups;

examples include the real general linear group GLn(R), the unitary group

Un(C), the special unitary group SUn(C), the orthogonal group On(R), the

special orthogonal group SOn(R), and the Heisenberg group

⎛

⎝0

1 R R

1

R⎠

0 0 1

⎞

of unipotent upper triangular 3 × 3 real matrices. Many Lie groups are

isomorphic to closed linear groups; for instance, the additive group R can

be identified with the closed linear group

1 R

0 1

,

the circle R/Z can be identified with SO2(R) (or U1(C)), and so forth.

However, not all Lie groups are isomorphic to closed linear groups. A some-

what trivial example is that of a discrete group with cardinality larger than

the continuum, which is simply too large to fit inside any linear group. A

less pathological example is provided by the Weil-Heisenberg group

(1.3) G :=

⎛

⎝0

1 R R/Z

1 R

0 0 1

⎞

⎠

:=

⎛

⎝0

1 R R

1

R⎠

0 0 1

⎞

/

⎛

⎝0

1 0 Z

1

0⎠

0 0 1

⎞