Editing Exterior algebra (section)

== Duality ==

=== Alternating operators ===
Given two vector spaces ''V'' and ''X'' and a natural number ''k'', an '''alternating operator''' from ''V''<sup>''k''</sup> to ''X'' is a [[multilinear map]]
: <math> f : V^k \to X </math>
such that whenever ''v''<sub>1</sub>, ..., ''v''<sub>''k''</sub> are [[linearly dependent]] vectors in ''V'', then
: <math> f(v_1,\ldots, v_k) = 0. </math>

The map
: <math> w : V^k \to {\textstyle\bigwedge}^{\!k}(V),</math>
which associates to <math> k </math> vectors from <math> V </math> their exterior product, i.e. their corresponding <math> k </math>-vector, is also alternating.  In fact, this map is the "most general" alternating operator defined on <math> V^k; </math> given any other alternating operator <math> f : V^k \rightarrow X, </math> there exists a unique [[linear map]] <math> \phi : {\textstyle\bigwedge}^{\!k}(V) \rightarrow X </math> with <math> f = \phi \circ w. </math>  This [[universal property]] characterizes the space of alternating operators on <math> V^k </math> and can serve as its definition.

=== Alternating multilinear forms ===
<!-- Alternating form redirects here -->
{{See also|Alternating multilinear map}}

[[File:N-form.svg|thumb|upright=0.6|Geometric interpretation for the '''exterior product''' of {{math|''n''}} [[1-form]]s ({{math|''ε''}}, {{math|''η''}}, {{math|''ω''}}) to obtain an {{math|''n''}}-form ("mesh" of [[coordinate surface]]s, here planes),<ref name=Penrose07/> for {{math|1=''n'' = 1, 2, 3}}.  The "circulations" show [[Orientation (vector space)|orientation]].<ref>''Note'': The orientations shown here are not correct; the diagram simply gives a sense that an orientation is defined for every {{math|''k''}}-form.</ref><ref>{{cite book|title=Gravitation |first1=J.A. |last1=Wheeler |first2=C. |last2=Misner |first3=K.S. |last3=Thorne |publisher=W.H. Freeman & Co|year=1973|pages=58–60, 83, 100–9, 115–9|isbn=0-7167-0344-0}}</ref>]]

The above discussion specializes to the case when {{tmath|1=X = K}}, the base field.  In this case an alternating multilinear function
: <math> f : V^k \to K</math>
is called an '''alternating multilinear form'''.  The set of all [[Alternating map|alternating]] [[multilinear form]]s is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating.  By the universal property of the exterior power, the space of alternating forms of degree <math>k</math> on <math>V</math> is [[natural transformation|naturally]] isomorphic with the [[dual vector space]] {{tmath|\bigl({\textstyle\bigwedge}^{\!k}(V)\bigr)^*}}.  If <math>V</math> is finite-dimensional, then the latter is {{Clarify|date=December 2021|reason=By what isomorphism? See, for example, https://mathoverflow.net/q/68004|text=naturally isomorphic}} to {{tmath|{\textstyle\bigwedge}^{\!k}\left(V^*\right)}}.  In particular, if <math>V</math> is <math>n</math>-dimensional, the dimension of the space of alternating maps from <math>V^k</math> to <math>K</math> is the [[binomial coefficient]] {{tmath|1=\textstyle\binom{n}{k} }}.

Under such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones.  Suppose {{nowrap|''ω'' : ''V''<sup>''k''</sup> → ''K''}} and {{nowrap|''η'' : ''V''<sup>''m''</sup> → ''K''}} are two anti-symmetric maps.  As in the case of [[tensor product]]s of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables.  Depending on the choice of identification of elements of exterior power with multilinear forms, the exterior product is defined as
: <math> \omega \wedge \eta = \operatorname{Alt}(\omega \otimes \eta) </math>
or as
: <math>
\omega \dot{\wedge} \eta
= \frac{(k+m)!}{k!\,m!}\operatorname{Alt}(\omega \otimes \eta),
</math>
where, if the characteristic of the base field <math>K</math> is 0, the alternation Alt of a multilinear map is defined to be the average of the sign-adjusted values over all the [[permutation]]s of its variables:
: <math>
\operatorname{Alt}(\omega)(x_1,\ldots,x_k)
= \frac{1}{k!}\sum_{\sigma \in S_k}\operatorname{sgn}(\sigma)\, \omega(x_{\sigma(1)}, \ldots, x_{\sigma(k)}).
</math>

When the [[field (mathematics)|field]] <math>K</math> has [[characteristic of a field|finite characteristic]], an equivalent version of the second expression without any factorials or any constants is well-defined:
: <math>
{\omega \dot{\wedge} \eta(x_1,\ldots,x_{k+m})}
= \sum_{\sigma \in \mathrm{Sh}_{k,m}} \operatorname{sgn}(\sigma)\, \omega(x_{\sigma(1)}, \ldots, x_{\sigma(k)})\, \eta(x_{\sigma(k+1)}, \ldots, x_{\sigma(k+m)}),
</math>
where here {{nowrap|Sh<sub>''k'',''m''</sub> ⊂ ''S''<sub>''k''+''m''</sub>}} is the subset of [[(p,q) shuffle|{{math|(''k'', ''m'')}} shuffles]]: [[permutation]]s ''σ'' of the set {{nowrap|{{mset|1, 2, ..., ''k'' + ''m''}}}} such that {{nowrap|''σ''(1) < ''σ''(2) < ⋯ < ''σ''(''k'')}}, and {{nowrap|''σ''(''k'' + 1) < ''σ''(''k'' + 2) < ... < ''σ''(''k'' + ''m'')}}. As this might look very specific and fine tuned, an equivalent raw version is to sum in the above formula over permutations in left cosets of {{nowrap|''S''<sub>''k''+''m''</sub> / (''S''<sub>''k''</sub> × ''S''<sub>''m''</sub>)}}.

=== 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>

=== 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.

=== Inner product ===
For {{tmath|V}} a finite-dimensional space, an [[inner product]] (or a [[Pseudo-Euclidean space|pseudo-Euclidean]] inner product) on {{tmath|V}} defines an isomorphism of <math>V</math> with {{tmath|V^*}}, and so also an isomorphism of <math>{\textstyle\bigwedge}^{\!k}(V)</math> with {{tmath|\bigl({\textstyle\bigwedge}^{\!k} V\bigr)^*}}.  The pairing between these two spaces also takes the form of an inner product.  On decomposable {{tmath|k}}-vectors,
: <math>
\left\langle v_1 \wedge \cdots \wedge v_k, w_1 \wedge \cdots \wedge w_k\right\rangle 
= \det\bigl(\langle v_i,w_j\rangle\bigr),
</math>

the determinant of the matrix of inner products.  In the special case {{nowrap|1=''v''<sub>''i''</sub> = ''w''<sub>''i''</sub>}}, the inner product is the square norm of the ''k''-vector, given by the determinant of the [[Gramian matrix]] {{nowrap|({{langle}}''v''<sub>''i''</sub>, ''v''<sub>''j''</sub>{{rangle}})}}.  This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on <math>{\textstyle\bigwedge}^{\!k}(V). </math>  If ''e''<sub>''i''</sub>, {{nowrap|1=''i'' = 1, 2, ..., ''n''}}, form an [[orthonormal basis]] of {{tmath|V}}, then the vectors of the form
: <math> e_{i_1} \wedge \cdots \wedge e_{i_k},\quad i_1 < \cdots < i_k, </math>
constitute an orthonormal basis for {{tmath|{\textstyle\bigwedge}^{\!k}(V)}}, a statement equivalent to the [[Cauchy–Binet formula]].

With respect to the inner product, exterior multiplication and the interior product are mutually adjoint.  Specifically, for {{tmath|\mathbf{v}\in {\textstyle\bigwedge}^{\!k-1}(V)}}, {{tmath|\mathbf{w} \in {\textstyle\bigwedge}^{\!k}(V)}}, and {{tmath|x \in V}},
: <math>
\langle x \wedge \mathbf{v}, \mathbf{w}\rangle
= \langle \mathbf{v}, \iota_{x^\flat}\mathbf{w}\rangle
</math>
where {{nowrap|''x''<sup>♭</sup> ∈ ''V''<sup>∗</sup>}} is the [[musical isomorphism]], the linear functional defined by
: <math> x^\flat(y) = \langle x, y\rangle </math>
for all {{tmath|y \in V}}.  This property completely characterizes the inner product on the exterior algebra.

Indeed, more generally for {{tmath|\mathbf{v}\in {\textstyle\bigwedge}^{\!k-l}(V)}}, {{tmath|\mathbf{w}\in {\textstyle\bigwedge}^{\!k}(V)}}, and {{tmath|\mathbf{x}\in {\textstyle\bigwedge}^{\!l}(V)}}, iteration of the above adjoint properties gives
: <math>
\langle \mathbf{x} \wedge \mathbf{v}, \mathbf{w}\rangle
= \langle \mathbf{v}, \iota_{\mathbf{x}^\flat}\mathbf{w}\rangle
</math>
where now <math>\mathbf{x}^\flat \in {\textstyle\bigwedge}^{\!l}\left(V^*\right) \simeq \bigl({\textstyle\bigwedge}^{\!l}(V)\bigr)^* </math> is the dual {{tmath|l}}-vector defined by
: <math>
\mathbf{x}^\flat(\mathbf{y}) = \langle \mathbf{x}, \mathbf{y}\rangle
</math>
for all {{tmath|\mathbf{y} \in {\textstyle\bigwedge}^{\!l}(V)}}.

=== Bialgebra structure ===
There is a correspondence between the graded dual of the graded algebra <math>{\textstyle\bigwedge}(V)</math> and alternating multilinear forms on {{tmath|V}}.  The exterior algebra (as well as the [[symmetric algebra]]) inherits a bialgebra structure, and, indeed, a [[Hopf algebra]] structure, from the [[tensor algebra]].  See the article on [[tensor algebra]]s for a detailed treatment of the topic.

The exterior product of multilinear forms defined above is dual to a [[coproduct]] defined on {{tmath|{\textstyle\bigwedge}(V)}}, giving the structure of a [[coalgebra]].  The '''coproduct''' is a linear function {{tmath|\Delta : {\textstyle\bigwedge}(V) \to {\textstyle\bigwedge}(V) \otimes {\textstyle\bigwedge}(V)}}, which is given by
: <math> \Delta(v) = 1 \otimes v + v \otimes 1 </math>
on elements {{tmath|v \in V}}.  The symbol <math>1</math> stands for the unit element of the field {{tmath|K}}.  Recall that {{tmath|K \simeq {\textstyle\bigwedge}^{\!0}(V) \subseteq {\textstyle\bigwedge}(V)}}, so that the above really does lie in {{tmath|{\textstyle\bigwedge}(V) \otimes {\textstyle\bigwedge}(V)}}. This definition of the coproduct is lifted to the full space <math>{\textstyle\bigwedge}(V)</math> by (linear) homomorphism. 
The correct form of this homomorphism is not what one might naively write, but has to be the one carefully defined in the [[coalgebra]] article.  In this case, one obtains
: <math> \Delta(v \wedge w) = 1 \otimes (v \wedge w) + v \otimes w - w \otimes v + (v \wedge w) \otimes 1 .</math>

Expanding this out in detail, one obtains the following expression on decomposable elements:
: <math>
\Delta(x_1 \wedge \cdots \wedge x_k)
= \sum_{p=0}^k \; \sum_{\sigma \in Sh(p,k-p)} \; \operatorname{sgn}(\sigma) (x_{\sigma(1)} \wedge \cdots \wedge x_{\sigma(p)}) \otimes (x_{\sigma(p+1)} \wedge \cdots \wedge x_{\sigma(k)}).
</math>
where the second summation is taken over all [[(p,q) shuffle|{{math|(''p'', ''k''−''p'')}}-shuffles]].  By convention, one takes that Sh(''k,''0) and Sh(0,''k'') equals {id: {1, ..., ''k''} → {1, ..., ''k''}}. It is also convenient to take the pure wedge products <math>v_{\sigma(1)}\wedge\dots\wedge v_{\sigma(p)}</math> and <math>v_{\sigma(p+1)}\wedge\dots\wedge v_{\sigma(k)}</math>
to equal 1 for ''p'' = 0 and ''p'' = ''k'', respectively (the empty product in <math>{\textstyle\bigwedge}(V)</math>).  The shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements <math>x_k</math> is ''preserved'' in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.

Observe that the coproduct preserves the grading of the algebra.  Extending to the full space <math display=inline> {\textstyle\bigwedge}(V), </math> one has
: <math>
\Delta : {\textstyle\bigwedge}^k(V)
\to \bigoplus_{p=0}^k {\textstyle\bigwedge}^p(V) \otimes {\textstyle\bigwedge}^{k-p}(V)
</math>

The tensor symbol ⊗ used in this section should be understood with some caution: it is ''not'' the same tensor symbol as the one being used in the definition of the alternating product.  Intuitively, it is perhaps easiest to think it as just another, but different, tensor product: it is still (bi-)linear, as tensor products should be, but it is the product that is appropriate for the definition of a bialgebra, that is, for creating the object {{tmath|{\textstyle\bigwedge}(V) \otimes {\textstyle\bigwedge}(V)}}. Any lingering doubt can be shaken by pondering the equalities {{nowrap|1=(1 ⊗ ''v'') ∧ (1 ⊗ ''w'') = 1 ⊗ (''v'' ∧ ''w'')}} and {{nowrap|1=(''v'' ⊗ 1) ∧ (1 ⊗ ''w'') = ''v'' ⊗ ''w''}}, which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols.  This distinction is developed in greater detail in the article on [[tensor algebra]]s.  Here, there is much less of a problem, in that the alternating product <math>\wedge</math> clearly corresponds to multiplication in the exterior algebra, leaving the symbol <math>\otimes</math> free for use in the definition of the bialgebra.  In practice, this presents no particular problem, as long as one avoids the fatal trap of replacing alternating sums of <math>\otimes</math> by the wedge symbol, with one exception.  One can construct an alternating product from {{tmath|\otimes}}, with the understanding that it works in a different space.  Immediately below, an example is given: the alternating product for the ''dual space'' can be given in terms of the coproduct.  The construction of the bialgebra here parallels the construction in the [[tensor algebra]] article almost exactly, except for the need to correctly track the alternating signs for the exterior algebra.

In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:
: <math>
(\alpha \wedge \beta)(x_1 \wedge \cdots \wedge x_k)
= (\alpha \otimes \beta)\left(\Delta(x_1 \wedge \cdots \wedge x_k)\right)
</math>

where the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely, {{nowrap|1=''α'' ∧ ''β'' = ''ε'' ∘ (''α'' ⊗ ''β'') ∘ Δ}}, where <math>\varepsilon</math> is the counit, as defined presently).

The '''counit''' is the homomorphism <math>\varepsilon : {\textstyle\bigwedge}(V) \to K</math> that returns the 0-graded component of its argument.  The coproduct and counit, along with the exterior product, define the structure of a [[bialgebra]] on the exterior algebra.

With an '''antipode''' defined on homogeneous elements by {{tmath|1=S(x) = (-1)^{\binom{\text{deg}\, x\, + 1}{2} }x}}, the exterior algebra is furthermore a [[Hopf algebra]].<ref>Indeed, the exterior algebra of {{tmath|V}} is the [[Universal enveloping algebra|enveloping algebra]] of the abelian [[Lie superalgebra]] structure on {{tmath|V}}.</ref>
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It may be worth saying something about this, but has already been mentioned in passing above.

=== The duality isomorphism ===
In general, there are two different kinds of alternating structures defined via duality:
* The structure of alternating multilinear forms on <math display=inline>{\textstyle\bigwedge}(V). </math>  The space of all such forms is the graded dual <mathdisplay=inline>{\textstyle\bigwedge}(V) </math><sup>∗</sup>, and the product of such forms dualizes the coproduct on the exterior algebra.
* The exterior algebra of the dual vector space <math display=inline>{\textstyle\bigwedge}\left(V^*\right). </math>
If ''V'' is finite-dimensional, then these two exterior algebras are naturally isomorphic.
-->