Page 153 - Curvature and Homology

= (ad(X) Xp, aa) A:
= (XP, - e(x) a") AP,
P
= (XI A ... A Xp, a1 A ... A - 9(X) ay A ... A ap).
Y =l
It follows that

that is, O(X) is

Lemma 4.2.2.
Indeed,

Lemma 4.2.3.
It suffices to verify this formula for forms of degree 0 and 1 in A(T,*)
-the Grassman algebra associated with L. The identity is trivial for
forms of degree 0 since they are constant functions. In degree 1 we
need only consider the forms ma. Then, 9(XB)wa = CyBawY. But,

Corollary 4.2.3. O(X)d = de(X).
Lemma 4.2.4. d = &(wa)O(Xa).
It is only necessary to verify this formula for the forms of degrees
0 and 1 in A(T,*). Again, since the forms of degree 0 are the constant
functions on G both sides vanish. For a form of Maurer-Cartan wB