the tangent space at a point of manifolds likeG, but we do not want to enter
here into the subject of differential geometry in general. One of the standard
definitions of the tangent space is as the space of tangent vectors, with tangent
vectors defined as the possible velocity vectors of parametrized curvesg(t) in
the groupG.
More advanced treatments of Lie group theory develop this point of view (see
for example [98]) which applies to arbitrary Lie groups, whether or not they are
groups of matrices. In our case though, since we are interested in specific groups
that are usually explicitly given as groups of matrices, in such cases we can give
a more concrete definition, using the exponential map on matrices. For a more
detailed exposition of this subject, using the same concrete definition of the Lie
algebra in terms of matrices, see for instance [42] or the abbreviated on-line
version [40].
5.1 Lie algebras
If a Lie groupGis defined as a differentiable manifold with a group law, one
can consider the tangent space at the identity, and that will be the Lie algebra
ofG. We are however interested mainly in cases whereGis a matrix group,
and in such cases the Lie algebra can be defined more concretely:
Definition(Lie algebra).ForGa Lie group ofnbyninvertible matrices, the
Lie algebra ofG(writtenLie(G)org) is the space ofnbynmatricesXsuch
thatetX∈Gfort∈R.
Here the exponential of a matrix is given by usual power series formula for the
exponential
eA= 1 +A+1
2
A^2 +···+
1
n!An+···which can be shown to converge (like the usual exponential), for any matrixA.
While this definition is more concrete than defining a Lie algebra as a tangent
space, it does not make obvious some general properties of a Lie algebra, in
particular that a Lie algebra is a real vector space (see theorem 3.20 of [42]).
Our main interest will be in using it to recognize certain specific Lie algebras
corresponding to specific Lie groups.
Notice that while the groupGdetermines the Lie algebrag, the Lie algebra
does not determine the group. For example,O(n) andSO(n) have the same
tangent space at the identity, and thus the same Lie algebra, but elements in
O(n) not in the component of the identity (i.e., with determinant−1) can’t be
written in the formetX(since then you could make a path of matrices connecting
such an element to the identity by shrinkingtto zero).
Note also that, for a givenX, different values oftmay give the same group
element, and this may happen in different ways for different groups sharing the
same Lie algebra. For example, considerG=U(1) andG=R, which both
have the same Lie algebrag=R. In the first case an infinity of values oftgive
the same group element, in the second, only one does. In chapter 6 we’ll see