John Stillwell - Naive Lie Theory.pdf - Index of

8.8 Discussion 183
existence of a tangent space at the identity (and hence at every point) for
each matrix Lie group G. As we saw in Chapter 7, every matrix Lie group
G has a tangent space T 1 (G) at the identity, and T 1 (G) equals some R k .
Even finite groups, such as G = {1}, have a tangent space at the identity;
not surprisingly it is the space R 0 .
Topology gives a way to describe all the matrix Lie groups with zero
tangent space: they are the discrete groups, where a group H is called
discrete if there is a neighborhood of 1 not containing any elements of G
except 1 itself. Every finite group is obviously discrete, but there are also
infinite discrete groups; for example, Z is a discrete subgroup of R. The
groups Z and R can be viewed as matrix groups by associating each x ∈ R
with the matrix ( 1 x
01)
(because multiplying two such matrices results in
addition of their x entries).
It follows immediately from the definition of discreteness that T 1 (H)=
{0} for a discrete group H. It also follows that if H is a discrete subgroup
of a matrix Lie group G then G/H is “locally isomorphic” to G in some
neighborhood of 1. This is because every element of G in some neighborhood
of 1 belongs to a different coset. From this we conclude that G/H
and G have the same tangent space at 1, and hence the same Lie algebra.
This result shows, once again, why Lie algebras are simpler than Lie
groups—they do not “see” discrete subgroups.
Apart from the existence of a tangent space, there is an algebraic reason
for including the discrete matrix groups among the matrix Lie groups: they
occur as kernels of “Lie homomorphisms.” Since everything in Lie theory
is supposed to be smooth, the only homomorphisms between Lie groups
that belong to Lie theory are the smooth ones. We will not attempt a general
definition of smooth homomorphism here, but merely give an example: the
map Φ : R → S 1 defined by
Φ(θ)=e iθ .
This is surely a smooth map because Φ is a differentiable function of θ.
The kernel of this Φ is the discrete subgroup of R (isomorphic to Z) consisting
of the integer multiples of 2π. We would like any natural aspect of a
Lie “thing” to be another Lie “thing,” so the kernel of a smooth homomorphism
ought to be a Lie group. This is an algebraic reason for considering
the discrete group Z to be a Lie group.
The concepts of compactness, path-connectedness, simple connectedness,
and coverings play a fundamental role in topology, as a glance at any