John Stillwell - Naive Lie Theory.pdf - Index of

5.4 Algebraic properties of the tangent space 105
Definition. A matrix Lie algebra is a vector space of matrices that is closed
under the Lie bracket [X,Y ]=XY −YX.
All the Lie algebras we have seen so far have been matrix Lie algebras,
and in fact there is a theorem (Ado’s theorem) saying that every Lie algebra
is isomorphic to a matrix Lie algebra. Thus it is not wrong to say simply
“Lie algebra” rather than “matrix Lie algebra,” and we will usually do so.
Perhaps the most important idea in Lie theory is to study Lie groups
by looking at their Lie algebras. This idea succeeds because vector spaces
are generally easier to work with than curved objects—which Lie groups
usually are—and the Lie bracket captures most of the group structure.
However, it should be emphasized at the outset that g does not always
capture G entirely, because different Lie groups can have the same Lie
algebra. We have already seen one class of examples. For all n, O(n) is
different from SO(n), but they have the same tangent space at 1 and hence
the same Lie algebra. There is a simple geometric reason for this: SO(n)
is the subgroup of O(n) whose members are connected by paths to 1. The
tangent space to O(n) at 1 is therefore the tangent space to SO(n) at 1.
Exercises
If, instead of considering the path C s (t) =A(s)B(t)A(s) −1 in G we consider the
path
D s (t)=A(s)B(t)A(s) −1 B(t) −1 for some fixed value of s,
then we can relate the Lie bracket [X,Y] of X,Y ∈ T 1 (G) to the so-called commutator
A(s)B(t)A(s) −1 B(t) −1 of smooth paths A(s) and B(t) through 1 in G.
5.4.1 Find D ′ s(t), and hence show that D ′ s(0)=A(s)YA(s) −1 −Y.
5.4.2 D ′ s (0) ∈ T 1(G) (why?) and hence, as s varies, we have a smooth path E(s)=
D ′ s(0) in T 1 (G) (why?).
5.4.3 Show that the velocity E ′ (0) equals XY −YX, and explain why E ′ (0) is in
T 1 (G).
The tangent space at 1 is the most natural one to consider, but in fact all
elements of G have the “same” tangent space.
5.4.4 Show that the smooth paths through any g ∈ G are of the form gA(t),where
A(t) is a smooth path through 1.
5.4.5 Deduce from Exercise 5.4.4 that the space of tangents to G at g is isomorphic
to the space of tangents to G at 1.