Page 115 - Curvature and Homology

3.3. DERIVATIONS IN A GRADED ALGEBRA
where *-I denotes the inverse of the star operator:

for p-forms. We define the operator i(X) on p-forms as follows:

That i(X) is an endomorphism of A(T*) is clear. Since

we conclude that i(X) is the dual of the exterior product by 6 operator.
Evidently, i(X) lowers the degree by one. The operator i(X) is called
the interior product by X. From (3.3.5) we obtain
~(5) = (- 1~9+~+l *i(x)*

on forms of degree p.

Lemma 3.3.1. For every 1-form or and infinitesimal transformation X
i(X) a = (X, a).
From (3.3.5)

Lemma 3.3.2. i(X), X E T is an anti-derivation of the algebra A(T*).
For, let (XI, .-:, X,) and (wl, --., wn) be dual bases. Then, by (1.5.1)
and (II.A.l)

where or1, ..., arp are any covectors in T*. Moreover, from (1.5.1)
(XI A ... A Xi A ... A X,, a1 A ... A 04 A ... A aP) = det((Xi, d)).

Hence, for any decomposable element X, A ... A X, E A(T), if we
apply (3.3.6) and then develop the determinant by the row i = 1

(X, A ... A Xi A ... A X,, i(Xl) (alA ... A ari A ... A a,))
= (w" ... A wi r\ ... A wp, ;(xl) (a1 A ... A O11 A ... A a,))
= (e(w1)wS A ... A wi A ... A wp, a1 A ... A aj A ... A aP)
= (XI A ... A Xi A ... A X,, a' A ... A d A ... A aP)