John Stillwell - Naive Lie Theory.pdf - Index of

7.4 The log function into the tangent space 149
so they sum to zero. (I thank Brian Hall for this observation.) Therefore,
T m − 1 Y m
lim = lim
m→∞ |Y m | m→∞ |Y m | = Y.
Since each T m ∈ G, it follows that the sequential tangent
T m − 1
lim = Y
m→∞ |Y m |
is in T 1 (G) by the smoothness of sequential tangents proved in Section 7.3.
But Y ∉ T 1 (G), as observed above. This contradiction shows that our
original assumption was false, so there is a neighborhood N δ (1) mapped
into T 1 (G) by log.
□
Corollary. The log function gives a bijection, continuous in both directions,
between N δ (1) in G and logN δ (1) in T 1 (G).
Proof. The continuity of log, and of its inverse function exp, shows that
there is a 1-to-1 correspondence, continuous in both directions, between
N δ (1) and its image logN δ (1) in T 1 (G).
□
If N δ (1) in G is mapped into T 1 (G) by log, then each A ∈ N δ (1) has the
form A = e X ,whereX = logA ∈ T 1 (G). Thus the paradise of SO(2) and
SU(2)—where each group element is the exponential of a tangent vector—
is partly regained by the theorem above. Any matrix Lie group G has at
least a neighborhood of 1 in which each element is the exponential of a
tangent vector.
The corollary tells us that the set log N δ (1) is a “neighborhood” of 0
in T 1 (G) in a more general sense—the topological sense—that we will
discuss in Chapter 8. The existence of this continuous bijection between
neighborhoods finally establishes that G has a topological dimension equal
to the real vector space dimension of T 1 (G), thanks to the deep theorem
of Brouwer [1911] on the invariance of topological dimension. This gives
a broad justification for the Lie theory convention, already mentioned in
Section 5.5, of defining the dimension of a Lie group to be the dimension
of its Lie algebra. In practice, arguments about dimension are made at the
Lie algebra level, where we can use linear algebra, so we will not actually
need the topological concept of dimension.
Exercises
The continuous bijection between neighborhoods of 1 in G and of 0 in T 1 (G)
enables us to show the existence of nth roots in a matrix Lie group.