Editing Exterior algebra (section)

=== 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.
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