Page 122 - Curvature and Homology

T, and L are isomorphic. Moreover, the conditions X EL, Y EL
imply [X, EL. In fact,

It follows that the left invariant infinitesimal transformations of the
Lie group G form a Lie algebra L called the Lie algebra of G. That the
right invariant infinitesimal transformations also form a Lie algebra is
clear. However, this Lie algebra is isomorphic with L (cf. Chapter IV).
To an element A of L we associate the local 1-parameter group of
transformations 9, generated by A in a neighborhood of e E G. We show
that vt is a global 1-parameter group of transformations on G and that
it defines a 1-parameter subgroup of G. Since A is invariant by L,,
(x E G), it follows from lemma 3.4.1 that y, commutes with L, for
every x E G. Hence, A generates a global 1-parameter group of trans-
formations ?, on G. The subgroup a, of G defined by a, = rpde) satisfies
at+. = at as; moreover, v,(x) = Rap(= x a3 for every x E G. We call
a, the I-parameter subgroup of G generated by A.
More generally, we define a Lie subgroup G' of G to be a subgroup
of G which is simultaneously a submanifold of G. G' is itself a Lie
group with respect to the differentiable structure induced by G. Evidently,
the subspace L' of left invariant infinitesimal transformations cor-
responding to the tangent vectors at e E G' is a subalgebra of L, namely,
the Lie algebra of G'.
Let f be an element of the group of automorphisms of a Lie group G.
Then, f, is an automorphism of L: Since f(e) = e, if we identify the
vector space L with T, we see that f, induces an endomorphism of T,.
Since f-I f = identity automorphism of G, it follows that f, is an
automorphism. In particular, if f is an inner automorphism: x -+ axa-l
defined by a E G, the induced automorphism of L is called the adjoint
representation of G and is denoted by ad(a). For an element B EL,
ad(a)B = Ra-i*B, since axa-I = Ra-I Lax. If a, is the 1-parameter
subgroup of G generated by A E L we conclude from lemma 3.4.3 that

for every B E L.
Consider a differentiable manifold M on which a connected Lie group
G acts differentiably. G is said to be a Lie transformation group on M
if the following conditions hold:
(i) To each a E G there corresponds a homeomorphism R, of M
onto itself such that RaRb = R,;