Editing Exterior algebra (section)

=== Interior product ===
{{See also|Interior product}}
Suppose that <math>V</math> is finite-dimensional.  If <math>V^*</math> denotes the [[dual space]] to the vector space {{tmath|V}}, then for each {{tmath|\alpha \in V^*}}, it is possible to define an [[derivation (abstract algebra)|antiderivation]] on the algebra {{tmath|{\textstyle\bigwedge}(V)}},
: <math>
\iota_\alpha : {\textstyle\bigwedge}^{\!k}(V)
\rightarrow {\textstyle\bigwedge}^{\!k-1}(V) .
</math>

This derivation is called the '''interior product''' with {{tmath|\alpha}}, or sometimes the '''insertion operator''', or '''contraction''' by {{tmath|\alpha}}.

Suppose that {{tmath|w \in {\textstyle\bigwedge}^{\!k}(V)}}.  Then <math>w</math> is a multilinear mapping of <math>V^*</math> to {{tmath|K}}, so it is defined by its values on the {{math|''k''}}-fold [[Cartesian product]] {{tmath|V^* \times V^* \times \dots \times V^*}}.  If ''u''<sub>1</sub>, ''u''<sub>2</sub>, ..., ''u''<sub>''k''−1</sub> are <math>k - 1</math> elements of {{tmath|V^*}}, then define
: <math>
(\iota_\alpha w)(u_1,u_2,\ldots,u_{k-1})
= w(\alpha,u_1,u_2,\ldots, u_{k-1}).
</math>

Additionally, let <math>\iota_\alpha f = 0</math> whenever <math>f</math> is a pure scalar (i.e., belonging to {{tmath|{\textstyle\bigwedge}^{\!0}(V)}}).

==== Axiomatic characterization and properties ====
The interior product satisfies the following properties:
# For each {{tmath|k}} and each {{tmath|\alpha \in V^*}} (where by convention <math>\Lambda^{-1}(V)=\{0\}</math>),
#: <math>\iota_\alpha : {\textstyle\bigwedge}^{\!k}(V) \rightarrow {\textstyle\bigwedge}^{\!k-1}(V) .</math>
# If <math>v</math> is an element of <math>V</math> ({{tmath|1= = {\textstyle\bigwedge}^{\!1}(V)}}), then {{tmath|1=\iota_\alpha v = \alpha(v)}} is the dual pairing between elements of <math>V</math> and elements of {{tmath|V^*}}.
# For each {{tmath|\alpha \in V^*}}, <math>\iota_\alpha</math> is a [[graded derivation]] of degree −1:
#: <math>\iota_\alpha (a \wedge b)
= (\iota_\alpha a) \wedge b + (-1)^{\deg a}a \wedge (\iota_\alpha b).
</math>

These three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.

Further properties of the interior product include:
* <math> \iota_\alpha\circ \iota_\alpha = 0. </math>
* <math> \iota_\alpha\circ \iota_\beta = -\iota_\beta\circ \iota_\alpha. </math>