next up previous
Next: Lie derivation Up: some linear algebra Previous: pairing of exterior algebras

interior derivation

We employ the same assumption of the previous subsection. For any $ x\in N$ , we define

$\displaystyle i_x: \wedge_A M \to \wedge_A M
$

to be the unique $ A$ -linear map which satisfies the following properties.
  1. $ i_x(f)=0$ for any $ f \in A$ .
  2. $ i_x(m)=\langle m,x\rangle $ for any $ m\in M$ .
  3. $ i_x$ is an odd derivation. That means,

    $\displaystyle i_x(\alpha \wedge \beta)
=(i_x\alpha)\wedge \beta +(-1)^s \alpha ...
...beta)
\quad (\forall \alpha \in \wedge^s_A M, \forall \beta \in \wedge_A M).
$

The proof of the well-definedness of $ i_x$ is similar to the proof of well-definedness of $ d$ . We leave the detail to the reader.

Using the above definition we may also explicitly write down a formula for the interior derivations.

$\displaystyle i_x(m_1\wedge m_2 \wedge \dots \wedge m_s)=
\sum_{j=1}^s (-1)^j
\...
...2 \wedge \dots\wedge \overset{\text{omit}}{\hat{m_j}}\wedge \dots \wedge m_s).
$

This leads to an important adjunction relation

$\displaystyle \langle i_x \alpha, \beta \rangle_\wedge
=\langle \alpha, x\wedge...
...ngle_\wedge \qquad
(\forall \alpha \in \wedge M, \forall \beta \in \wedge N).
$

(We may verify the above equation by using the ``determinant expansions by minors''.)

In particular, we note that for any $ x,y\in N$ , an anti commutation relation

$\displaystyle [i_x, i_y]_+=0
$

holds. Another useful equation is

$\displaystyle i_{x_s} i_{x_{s-1}}\dots i_{x_2} i_{x_1} \alpha
=\langle \alpha,x...
...e}
\qquad (\forall x_1,x_2,\dots,x_s \in N, \forall \alpha \in \wedge^s_A M).
$



2012-02-29