Page 151 - Curvature and Homology

the dual base for the forms of Maurer-Cartan, that is the base such that
wa(X,) = 6; (a, = 1, -a, n). (In the sequel, Greek indices refer to vectors,
tensors, and forms on T, and its dual.) A differential form a is said
to be left invariant if it is invariant by every L,(a E G), that is, if L,*a = a
for every a E G where L: is the induced map in A(T*). The forms of
Maurer-Cartan are left invariant pfaffian forms. For an element X EL
and an element a in the dual space, a(X) is constant on G. Hence, by
lemma 3.5.2

where X, Y are any elements of L and a any element of the dual space.
If we write
[X8, Xy] = cp,Q Xa, (4.1.2)
then, from (4.1.1)

The constants CBya are called the cmtanis of structure of L with respect
to the base {XI, .-., XJ. These constants are not arbitrary since they
must satisfy the relations

a, /3, y = 1, n, that is

and
Capp CYpd + CbYP CWd + Cy/ cDPd = 0. (4.1.7)
The equations (4.1.3) are called the equations of Maurer-Cartan.
Since the induced dual maps L,* (a E G) commute with d, we have

for any Maurer-Cartan form a, that is, if a is a left invariant 1-form,
dor is a left invariant 2-form. This also follows from (4.1.3). More
generally, if Aal.. are any constants, the p-form A, . wal A . .. A was
is a left invariant differential form on G. That any feft rnvariant diffe-
rential form of degree p > 0 may be expressed in this manner is clear.
A left invariant form may be considered as an alternating multilinear