Wedge Product

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Formally, the exterior algebra is a certain Unital Associative Algebra over a Field ''K'' that contains ''V'' as a Subspace . It is denoted by Λ(''V'') or Λ•(''V'') and its multiplication is also known as the ''wedge product'' or the ''exterior product'' and is written as $\backslash wedge$. The wedge product is an Associative and Bilinear operation :$\backslash wedge:\backslash Lambda(V)\; imes\backslash Lambda(V)\; o\backslash Lambda(V)$. :::$(\backslash alpha,\backslash beta)\backslash mapsto\; \backslash alpha\backslash wedge\backslash beta$. Its essential feature is that it is ''alternating'' on ''V'': :(1) $v\backslash wedge\; v\; =\; 0\; \backslash mbox\{\; for\; all\; \}v\backslash in\; V,$ which implies in particular :(2) $u\backslash wedge\; v\; =\; -\; v\backslash wedge\; u$ for all $u,v\backslash in\; V$, and :(3) $v\_1\backslash wedge\; v\_2\backslash wedge\backslash cdots\; \backslash wedge\; v\_k\; =\; 0$ whenever $v\_1,\; \backslash ldots,\; v\_k\; \backslash in\; V$ are linearly dependent.Note that these three properties are only valid for the vectors in ''V'', ''not'' for all elements of the algebra Λ(''V''). The defining property (1) and property (3) are equivalent; properties (1) and (2) are equivalent unless the Characteristic of ''K'' is two. In terms of Category Theory , the exterior algebra is a type of Functor on vector spaces, given by a Universal Construction . It is one example of a Bialgebra , meaning that its Dual Space also possesses a product, and this dual product is compatible with the wedge product. This dual algebra is precisely the algebra of Alternating Multilinear Form s on ''V'', and the pairing between the exterior algebra and its dual is given by the Interior Product . With the additional structure of a Volume Form , the exterior algebra becomes a Hopf Algebra whose antipode is the Hodge Dual . In many cases, the exterior algebra is naturally realized as a certain subspace of the Tensor Algebra of ''V''. MOTIVATING EXAMPLES Areas in the plane The Cartesian Plane R2 is a vector space equipped with a Basis consisting of a pair of Unit Vector s :$\{\backslash mathbf\; e\}\_1\; =\; (1,0),\backslash quad\; \{\backslash mathbf\; e\}\_2\; =\; (0,1).$ Suppose that :$\{\backslash mathbf\; v\}\; =\; v\_1\{\backslash mathbf\; e\}\_1\; +\; v\_2\{\backslash mathbf\; e\}\_2,\; \backslash quad\; \{\backslash mathbf\; w\}\; =\; w\_1\{\backslash mathbf\; e\}\_1\; +\; w\_2\{\backslash mathbf\; e\}\_2$ are a pair of given vectors in R2, written in components. There is a unique parallelogram having '''v''' and '''w''' as two of its sides. The ''area'' of this parallelogram is given by the standard Determinant formula:

where here $Sh\_\{k,m\}\; \backslash subset\; S\_\{k+m\}$ is the subset of ''k,m Shuffles'' : permutations $\backslash sigma$
sending $1,2,\backslash ldots,k$ to numbers $\backslash sigma(1)$
< \sigma(2) < \cdots < \sigma(k), and $k+1,k+2,\backslash ldots,k+m$ to numbers .

(Note. Some conventions, particularly in physics, define the wedge product as
:$\backslash omega\backslash wedge\backslash eta=\{\; m\; Alt\}(\backslash omega\backslash otimes\backslash eta).$
This convention is not adopted here, but see the Alternating tensor algebra section below for further details.)


Bialgebra structure

In formal terms, there is a correspondence between the graded dual of the graded algebra Λ(''V'') and alternating multilinear forms on ''V''. The wedge product of multilinear forms defined above is dual to a Coproduct defined on Λ(''V''), giving the structure of a Coalgebra .

The coproduct is a linear function Δ : Λ(''V'') → Λ(''V'') ⊗ Λ(''V'') given on decomposable elements by
:$\backslash Delta(x\_1\backslash wedge\backslash dots\backslash wedge\; x\_k)\; =\; \backslash sum\_\{p=0\}^k\; \backslash sum\_\{\backslash sigma\backslash in\; Sh\_\{p,k-p\}\}\; \{\; m\; sgn\}(\backslash sigma)\; (x\_\{\backslash sigma(1)\}\backslash wedge\backslash dots\backslash wedge\; x\_\{\backslash sigma(p)\})\backslash otimes\; (x\_\{\backslash sigma(p+1)\}\backslash wedge\backslash dots\backslash wedge\; x\_\{\backslash sigma(k)\}).$
This extends by linearity to an operation defined on the whole exterior algebra. In terms of the coproduct, the wedge product on the dual space is just the graded dual of the coproduct:
:$(\backslash alpha\backslash wedge\backslash beta)(x\_1\backslash wedge\backslash dots\backslash wedge\; x\_k)\; =\; (\backslash alpha\backslash otimes\backslash beta)\backslash left(\backslash Delta(x\_1\backslash wedge\backslash dots\backslash wedge\; x\_k)\; ight)$
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, α∧β = ε o (α⊗β) o Δ, where ε is the counit, as defined presently).

The counit is the homomorphism ε : Λ(''V'') → ''K'' which returns the 0-graded component of its argument. The coproduct and counit, along with the wedge product, define the structure of a Bialgebra on the exterior algebra.


The interior product

See Also: interior product

• '' denotes the Dual Space to the vector space ''V'', then for each α ∈ ''V''^---, it is possible to define an Antiderivation on the algebra Λ(V),

:$i\_\backslash alpha:\backslash Lambda^k\; V\; ightarrow\backslash Lambda^\{k-1\}V.$

This derivation is called the interior product with α, or sometimes the '''insertion operator'''.

• to '''R''', so it is defined by its values on the ''k''-fold Cartesian Product ''V''^---× ''V''^---× ... × ''V''^---. If ''u''₁, ''u''₂, ..., ''u''_k-1 are ''k-1'' elements of ''V''^---, then define

:$(i\_\backslash alpha\; \{\backslash bold\; w\})(u\_1,u\_2\backslash dots,u\_\{k-1\})=\{\backslash bold\; w\}(\backslash alpha,u\_1,u\_2,\backslash dots,\; u\_\{k-1\})$

Additionally, let ''i''_α''f'' = 0 whenever ''f'' is a pure scalar (i.e., belonging to Λ⁰''V'').


Axiomatic characterization and properties

The interior product satisfies the following properties:

• ,

#::$i\_\backslash alpha:\backslash Lambda^kV\; ightarrow\; \backslash Lambda^\{k-1\}V.$

#:(By convention, Λ^-1 = 0.)

• .


• , ''i''_{α is a Graded Derivation of degree -1:}

#::$i\_\backslash alpha\; (a\backslash wedge\; b)\; =\; (i\_\backslash alpha\; a)\backslash wedge\; b\; +\; (-1)^\{\backslash deg\; a\}a\backslash wedge\; (i\_\backslash alpha\; b)$.

In fact, 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:

• $i\_\backslash alpha\backslash circ\; i\_\backslash alpha\; =\; 0.$


• $i\_\backslash alpha\backslash circ\; i\_\backslash beta\; =\; -i\_\backslash beta\backslash circ\; i\_\backslash alpha.$

Hodge duality

See Also: Hodge dual


Suppose that ''V'' has finite dimension ''n''. Then the interior product induces a canonical isomorphism of vector spaces

• ) \otimes \Lambda^n(V) o \Lambda^{n-k}(V).

In the geometrical setting, a non-zero element of the top exterior power Λⁿ(V) (which is a one-dimensional vector space) is sometimes called a Volume Form (or '''orientation form''', although this term may sometimes lead to ambiguity.) Relative to a given volume form σ, the isomorphism is given explicitly by

• ) \mapsto i_\alpha\sigma \in \Lambda^{n-k}(V).


• , then the resulting isomorphism is called the Hodge dual (or more commonly the '''Hodge star operator''')


• : \Lambda^k(V) ightarrow \Lambda^{n-k}(V).

• with itself maps Λ^k(''V'') → Λ^k(''V'') 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 a wedge product of an Orthonormal Basis of ''V''. In this case,


• \circ --- : \Lambda^k(V) o \Lambda^k(V) = (-1)^{k(n-k) + q}I

where ''I'' is the identity, and the inner product has Metric Signature (''p'',''q'') — ''p'' plusses and ''q'' minuses.

Along with the bialgebra structure, the Hodge star operator on Λ(''V'') defines the antipode map for a Hopf Algebra on the exterior algebra.


FUNCTORIALITY

Suppose that ''V'' and ''W'' are a pair of vector spaces and ''f'' : ''V'' → ''W'' is a Linear Transformation . Then, by the universal construction, there exists a unique homomorphism of graded algebras
:$\backslash Lambda(f)\; :\; \backslash Lambda(V)\; ightarrow\; \backslash Lambda(W)$
such that



The components of this tensor are precisely the skew part of the components of the tensor product ''s'' ⊗ ''t'', denoted by square brackets on the indices:

:$(t\backslash widehat\{\backslash otimes\}\; s)^\{i\_1\backslash dots\; i\_\{r+p\}\}\; =\; t^\{i\_r\}s^\{i\_\{r+1\}\backslash dots\; i\_\{r+p\}\}.$

• , ''i''_αt is an alternating tensor of rank ''r''-1, given by

:$(i\_\backslash alpha\; t)^\{i\_1\backslash dots\; i\_\{r-1\}\}=r\backslash sum\_\{j=0\}^n\backslash alpha\_j\; t^\{ji\_1\backslash dots\; i\_\{r-1\}\}$.

where ''n'' is the dimension of ''V''.


APPLICATIONS


Linear geometry

The decomposable ''k''-vectors have geometric interpretations: the bivector $u\backslash wedge\; v$ represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented Parallelogram with sides ''u'' and ''v''. Analogously, the 3-vector $u\backslash wedge\; v\backslash wedge\; w$ represents the spanned 3-space weighted by the volume of the oriented Parallelepiped with edges ''u'', ''v'', and ''w''.


Differential geometry

The exterior algebra has notable applications in Differential Geometry , where it is used to define Differential Form s. A Differential Form can intuitively be interpreted as a function on weighted subspaces of the Tangent Space of a Differentiable Manifold . As a consequence, there is a natural wedge product for differential forms. Differential forms play a major role in diverse areas of differential geometry.


Representation theory

In Representation Theory , the exterior algebra is one of the two fundamental Schur Functor s on the category of vector spaces, the other being the Symmetric Algebra . Together, these constructions are used to generate the Irreducible Representation s of the General Linear Group .


Physics

The exterior algebra is an archetypal example of a Superalgebra , which plays a fundamental role in physical theories pertaining to Fermion s and Supersymmetry . For a physical discussion, see Grassmann Number . For various other applications of related ideas to physics, see Superspace and Supergroup (physics) .


HISTORY

The exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of ''Ausdehnungslehre'', or ''Theory of Extension''.Kannenberg (2000) published a translation of Grassmann's work in English; he translated ''Ausdehnungslehre'' as ''Extension Theory''. This referred more generally to an algebraic (or axiomatic) theory of extended quantities and was one of the early precursors to the modern notion of a Vector Space .

The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of Multivector s. It was thus a ''calculus'', much like the Propositional Calculus , except focused exclusively on the task of formal reasoning in geometrical terms.Authors have in the past referred to this calculus variously as the ''calculus of extension'' ( Whitehead , 1898; Forder, 1941), or ''extensive algebra'' ( Clifford , 1878), and recently as ''extended vector algebra'' (Browne, 2007), not to be confused with the modern notion of Algebra Over A Field . In particular, this new development allowed for an ''axiomatic'' characterization of dimension, a property that had previously only been examined from the coordinate point of view.

The import of this new theory of vectors and multivectors was lost to mid 19th century mathematicians,Bourbaki, ''Algebra'' (1989) p. 661. until being thoroughly vetted by Giuseppe Peano in 1888. Peano's work also remained somewhat obscure until the turn of the century, when the subject was unified by members of the French geometry school (notably Henri Poincaré , Elie Cartan , and Gaston Darboux ) who applied Grassmann's ideas to the calculus of Differential Form s.

A short while later, Alfred North Whitehead , borrowing from the ideas of Peano and Grassmann, introduced his Universal Algebra . This then paved the way for the 20th century developments of Abstract Algebra by placing the axiomatic notion of an algebraic system on a firm logical footing.


NOTES




REFERENCES


Mathematical references

• ¹

:: Includes a treatment of alternating tensors and alternating forms, as well as a detailed discussion of Hodge duality from the perspective adopted in this article.

• ²

:: This is the ''main mathematical reference'' for the article. It introduces the exterior algebra of a module over a commutative ring (although this article specializes primarily to the case when the ring is a field), including a discussion of the universal property, functoriality, duality, and the bialgebra structure. See chapters III.7 and III.11.

• ³

:: Chapter XVI sections 6-10 give a more elementary account of the exterior algebra, including duality, determinants and minors, and alternating forms.

• ⁴

:: Contains a classical treatment of the exterior algebra as alternating tensors, and applications to differential geometry.

Historical references

• ⁵


• ⁶


• ⁷


• ⁸ (The Linear Extension Theory - A new Branch of Mathematics)


• ⁹


• ¹⁰ Calculus according to Grassmann's Ausdehnungslehre, preceded by the Operations of Deductive Logic


• ¹¹

Other references and further reading

• ¹²

:: An introduction to the exterior algebra, and Geometric Algebra , with a focus on applications. Also includes a history section and bibliography.

• ¹³

:: Includes an elementary treatment of the axiomatization of determinants as signed areas, volumes, and higher-dimensional volumes.

SEE ALSO

• Multilinear Algebra


• Tensor Algebra


• Symmetric Algebra


• Clifford Algebra


• Geometric Algebra


• Koszul Complex