Page 152 - Curvature and Homology

134 IV. COMPACT LIE GROUPS
form on the Lie algebra L of G. We may therefore identify the left
invariant forms with the homogeneous elements of the Grassman algebra
associated with L. The number of linearly independent left invariant
p-forms is therefore equal to (%).
Lemma 4.1 .l. The underlying manifold of the Lie group G is orientable.
Indeed, the n-form o1 A ... A un on G is continuous and different
from zero everywhere. G may then be oriented by the requirement
that this form is positive everywhere (cf. $ 1.6).
The Lie group G is thus a compact, connected, orientable analytic
manifold.

4.2. Invariant differential forms
For any X EL, let ad(- be the map Y -+ [X, YJ of L into itself.
It is clear that X -t ad(X) is a linear map, and so, since

we conclude that X -+ ad(X) is a representation. It is called the adjoint
representation of L (cf. 5 3.6).
Let 8(X) be the (unique) derivation of A(Te) which coincides with
ad(X) on T, = A l(T,) defined by

O(X) (XI A ... A X,) = f: Xl A ... A [X,X,J A ... A X,.
a=l
Define the endomorphism B(X) (X E L) of A (T:) by

where or1, ..a, aP are any elements of A'(T,*) (cf. II.A.4).
Lemma 4.2.1. B(X) is a derivation.
If A; denotes the minor obtained by deleting the row a and column
of the matrix ((X,, aa));

= (8(~) (XI A ... A X,), or1 A ... A orp)