Editing Exterior algebra (section)

=== Hodge duality ===
{{main article|Hodge star operator}}
Suppose that <math>V</math> has finite dimension {{tmath|n}}.  Then the interior product induces a canonical isomorphism of vector spaces
: <math>
{\textstyle\bigwedge}^{\!k}(V^*) \otimes {\textstyle\bigwedge}^{\!n}(V)
\to {\textstyle\bigwedge}^{\!n-k}(V)
</math>
by the recursive definition
: <math> \iota_{\alpha \wedge \beta} = \iota_\beta \circ \iota_\alpha. </math>

In the geometrical setting, a non-zero element of the top exterior power <math>{\textstyle\bigwedge}^{\!n}(V)</math> (which is a one-dimensional vector space) is sometimes called a '''[[volume form]]''' (or '''orientation form''', although this term may sometimes lead to ambiguity).  The name orientation form comes from the fact that a choice of preferred top element determines an orientation of the whole exterior algebra, since it is tantamount to fixing an ordered basis of the vector space.  Relative to the preferred volume form {{tmath|\sigma}}, the isomorphism is given explicitly by
: <math>
{\textstyle\bigwedge}^{\!k}(V^*) \to {\textstyle\bigwedge}^{\!n-k}(V)
: \alpha \mapsto \iota_\alpha \sigma .
</math>

If, in addition to a volume form, the vector space ''V'' is equipped with an [[inner product]] identifying <math>V</math> with {{tmath|V^*}}, then the resulting isomorphism is called the '''Hodge star operator''', which maps an element to its '''Hodge dual''':
: <math>\star : {\textstyle\bigwedge}^{\!k}(V) \rightarrow {\textstyle\bigwedge}^{\!n-k}(V) .</math>

The composition of <math> \star </math> with itself maps <math>{\textstyle\bigwedge}^{\!k}(V) \to {\textstyle\bigwedge}^{\!k}(V)</math> and is always a scalar multiple of the identity map.  In most applications, the volume form is compatible with the inner product in the sense that it is an exterior product of an [[orthonormal basis]] of {{tmath|V}}.  In this case,
: <math> \star \circ \star : {\textstyle\bigwedge}^{\!k}(V) \to {\textstyle\bigwedge}^{\!k}(V) = (-1)^{k(n-k) + q}\mathrm{id} </math>
where id is the identity mapping, and the inner product has [[metric signature]] {{nowrap|(''p'', ''q'')}} — ''p'' pluses and ''q'' minuses.