EXTERIOR ALGEBRA

(Redirected from Exterior power)
In mathematics, the 'exterior product' or 'wedge product' of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of vectors is used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. In linear algebra, the exterior product provides an abstract algebraic basis-independent manner for describing the determinant and the minors of a linear transformation, and is fundamentally related to ideas of rank and linear independence. The 'exterior algebra' (also known as the 'Grassmann algebra', after Hermann Grassmann^[1]) of a given vector space ''V'' is the algebra generated by the exterior product. It is widely used in contemporary geometry, especially differential geometry and algebraic geometry through the algebra of differential forms, as well as in multilinear algebra and related fields.
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 $wedge$. The wedge product is an associative and bilinear operation
:$wedge:Lambda(V)\; imesLambda(V)\; oLambda(V)$.
:::.
Its essential feature is that it is ''alternating'' on ''V'':
:(1) $vwedge\; v\; =\; 0\; mbox\{\; for\; all\; \}vin\; V,$
which implies in particular
:(2) $uwedge\; v\; =\; -\; vwedge\; u$ for all $u,vin\; V$, and
:(3) $v\_1wedge\; v\_2wedgecdots\; wedge\; v\_k\; =\; 0$ whenever $v\_1,\; ldots,\; v\_k\; in\; V$ are linearly dependent.^[2]
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 forms 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''.

Contents

Motivating examples

Areas in the plane

Cross and triple products

Formal definitions and algebraic properties

Anticommutativity of the wedge product

The exterior power

Basis and dimension

Graded structure

Universal property

Generalizations

Duality

Alternating operators

Alternating multilinear forms

Bialgebra structure

The interior product

Axiomatic characterization and properties

Hodge duality

Functoriality

Exactness

The alternating tensor algebra

Index notation

Applications

Linear geometry

Differential geometry

Representation theory

Physics

History

Notes

References

Mathematical references

Historical references

Other references and further reading

See also

Motivating examples

Areas in the plane

The Cartesian plane 'R'² is a vector space equipped with a basis consisting of a pair of unit vectors
:$\{mathbf\; e\}\_1\; =\; (1,0),quad\; \{mathbf\; e\}\_2\; =\; (0,1).$
Suppose that
:$\{mathbf\; v\}\; =\; v\_1\{mathbf\; e\}\_1\; +\; v\_2\{mathbf\; e\}\_2,\; quad\; \{mathbf\; w\}\; =\; w\_1\{mathbf\; e\}\_1\; +\; w\_2\{mathbf\; e\}\_2$
are a pair of given vectors in 'R'², 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:
:
ight| = |v_1w_2 - v_2w_1|.
Consider now the exterior product of 'v' and 'w':
:$\{mathbf\; v\}wedge\; \{mathbf\; w\}\; =\; (v\_1\{mathbf\; e\}\_1\; +\; v\_2\{mathbf\; e\}\_2)wedge\; (w\_1\{mathbf\; e\}\_1\; +\; w\_2\{mathbf\; e\}\_2)=v\_1w\_1\{mathbf\; e\}\_1wedge\{mathbf\; e\}\_1+\; v\_1w\_2\{mathbf\; e\}\_1wedge\; \{mathbf\; e\}\_2+v\_2w\_1\{mathbf\; e\}\_2wedge\; \{mathbf\; e\}\_1+v\_2w\_2\{mathbf\; e\}\_2wedge\; \{mathbf\; e\}\_2$
:$=(v\_1w\_2-v\_2w\_1)\{mathbf\; e\}\_1wedge\{mathbf\; e\}\_2$
where the first step uses the distributive law for the wedge product, and the last uses the fact that the wedge product is alternating. Note that the coefficient in this last expression is precisely the determinant of the matrix ['v' 'w']. The fact that this may be positive or negative has the intuitive meaning that 'v' and 'w' may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the 'signed area' of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation.
The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if A('v','w') denotes the signed area of the parallelogram determined by the pair of vectors 'v' and 'w', then A must satisfy the following properties:
# A(''a'''v',''b'''w') = ''a b'' A('v','w') for any real numbers ''a'' and ''b'', since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram).
# A('v','v') = 0, since the area of the degenerate parallelogram determined by 'v' (i.e., a line segment) is zero.
# A('w','v') = -A('v','w'), since interchanging the roles of 'v' and 'w' reverses the orientation of the parallelogram.
# A('v' + ''a'''w','w') = A('v','w'), since adding a multiple of 'w' to 'v' affects neither the base nor the height of the parallelogram and consequently preserves its area.
# A('e'₁, 'e'₂) = 1, since the area of the unit square is one.
With the exception of the last property, the wedge product satisfies the same formal properties as the area. In a certain sense, the wedge product generalizes the final property by allowing the area of a parallelogram to be compared to that of any "standard" chosen parallelogram. In other words, the exterior product in two-dimensions is a ''basis-independent'' formulation of area.^[3]

Cross and triple products

For vectors in 'R'³, the exterior algebra is closely related to the cross product and triple product. Using the standard basis {'e'₁, 'e'₂, 'e'₃}, the wedge product of a pair of vectors
:$mathbf\{u\}\; =\; u\_1\; mathbf\{e\}\_1\; +\; u\_2\; mathbf\{e\}\_2\; +\; u\_3\; mathbf\{e\}\_3$
and
:$mathbf\{v\}\; =\; v\_1\; mathbf\{e\}\_1\; +\; v\_2\; mathbf\{e\}\_2\; +\; v\_3\; mathbf\{e\}\_3$
is
:$mathbf\{u\}\; wedge\; mathbf\{v\}\; =\; (u\_1\; v\_2\; -\; u\_2\; v\_1)\; (mathbf\{e\}\_1\; wedge\; mathbf\{e\}\_2)\; +\; (u\_1\; v\_3\; -\; u\_3\; v\_1)\; (mathbf\{e\}\_1\; wedge\; mathbf\{e\}\_3)\; +\; (u\_2\; v\_3\; -\; u\_3\; v\_2)\; (mathbf\{e\}\_2\; wedge\; mathbf\{e\}\_3)$
where {'e'₁ Λ 'e'₂, 'e'₁ Λ 'e'₃, 'e'₂ Λ 'e'₃} is the basis for the three-dimensional space Λ²('R'³). This imitates the usual definition of the cross product of vectors in three dimensions.
Bringing in a third vector
:$mathbf\{w\}\; =\; w\_1\; mathbf\{e\}\_1\; +\; w\_2\; mathbf\{e\}\_2\; +\; w\_3\; mathbf\{e\}\_3$,
the wedge product of three vectors is
:$mathbf\{u\}\; wedge\; mathbf\{v\}\; wedge\; mathbf\{w\}\; =\; (u\_1\; v\_2\; w\_3\; +\; u\_2\; v\_3\; w\_1\; +\; u\_3\; v\_1\; w\_2\; -\; u\_1\; v\_3\; w\_2\; -\; u\_2\; v\_1\; w\_3\; -\; u\_3\; v\_2\; w\_1)\; (mathbf\{e\}\_1\; wedge\; mathbf\{e\}\_2\; wedge\; mathbf\{e\}\_3)$
where 'e'₁ Λ 'e'₂ Λ 'e'₃ is the basis vector for the one-dimensional space Λ³('R'³). This imitates the usual definition of the triple product.
The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product 'u'×'v' can be interpreted as a vector which is perpendicular to both 'u' and 'v' and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the minors of the matrix with columns 'u' and 'v'. The triple product of 'u', 'v', and 'w' is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns 'u', 'v', and 'w'. The exterior product in three-dimensions allows for similar interpretations. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions.

Formal definitions and algebraic properties

The exterior algebra Λ(''V'') over a vector space ''V'' is defined as the quotient algebra of the tensor algebra by the two-sided ideal ''I'' generated by all elements of the form $x\; otimes\; x$ such that ''x'' ∈ ''V''. Symbolically,
:$Lambda(V)\; :=\; T(V)/I.,$
The wedge product ∧ of two elements of Λ(''V'') is defined by
:

Anticommutativity of the wedge product

This product is anticommutative on elements of ''V'', for supposing that ''x'', ''y'' ∈ ''V'',
:$0\; equiv\; (x+y)wedge\; (x+y)\; =\; xwedge\; x\; +\; xwedge\; y\; +\; ywedge\; x\; +\; ywedge\; y\; equiv\; xwedge\; y\; +\; ywedge\; x\; pmod\; I$
whence
:$xwedge\; y\; =\; -\; ywedge\; x.$
More generally, if ''x''₁, ''x''₂, ..., ''x''_k are elements of ''V'', and σ is a permutation of the integers [1,...,''k''], then
:$x\_\{sigma(1)\}wedge\; x\_\{sigma(2)\}wedgedotswedge\; x\_\{sigma(k)\}\; =\; \{$
m sgn}(sigma)x_1wedge x_2wedgedots x_k,
where sgn(σ) is the signature of the permutation σ.

The exterior power

The ''k''-th 'exterior power' of ''V'', denoted Λ^''k''(''V''), is the vector subspace of Λ(''V'') spanned by elements of the form
:$x\_1wedge\; x\_2wedgedotswedge\; x\_k,quad\; x\_iin\; V,\; i=1,2,dots,\; k.$
If α ∈ Λ^''k''(''V''), then α is said to be a ''k''-'multivector'. If, furthermore, α can be expressed as a wedge product of ''k'' elements of ''V'', then α is said to be 'decomposable'. Although decomposable multivectors span Λ^''k''(''V''), not every element of Λ^''k''(''V'') is decomposable. For example, in 'R'⁴, the following 2-multivector is not decomposable:
:
(This is in fact a symplectic form. To see this, one need only check that α ∧ α ≠ 0.)

Basis and dimension

If the dimension of ''V'' is ''n'' and {''e''₁,...,''e''_''n''} is a basis of ''V'', then the set
:
is a basis for Λ^''k''(''V''). The reason is the following: given any wedge product of the form
:$v\_1wedgecdotswedge\; v\_k$
then every vector ''v''_''j'' can be written as a linear combination of the basis vectors ''e''_''i''; using the bilinearity of the wedge product, this can be expanded to a linear combination of wedge products of those basis vectors. Any wedge product in which the same basis vector appears more than once is zero; any wedge product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis ''k''-vectors can be computed as the minors of the matrix that describes the vectors ''v''_''j'' in terms of the basis ''e''_''i''.
Counting the basis elements, we see that the dimension of Λ^''k''(''V'') is the binomial coefficient ''n'' choose ''k''. In particular, Λ^''k''(''V'') = {0} for ''k'' > ''n''.
Any element of the exterior algebra can be written as a sum of multivectors. Hence, as a vector space the exterior algebra is a direct sum
:$Lambda(V)\; =\; Lambda^0(V)oplus\; Lambda^1(V)\; oplus\; Lambda^2(V)\; oplus\; cdots\; oplus\; Lambda^n(V)$
(where we set Λ⁰(''V'') = ''K'' and Λ¹(''V'') = ''V''), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2^''n''.

Graded structure

The wedge product of a ''k''-multivector with a ''p''-multivector is a (''k''+''p'')-multivector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section
:$Lambda(V)\; =\; Lambda^0(V)oplus\; Lambda^1(V)\; oplus\; Lambda^2(V)\; oplus\; cdots\; oplus\; Lambda^n(V)$
gives the exterior algebra the additional structure of a graded algebra. Symbolically,
:$left(Lambda^k(V)$
ight)wedgeleft(Lambda^p(V)
ight)sub Lambda^{k+p}(V).
Moreover, the wedge product is graded anticommutative, meaning that if α ∈ Λ^k(''V'') and β ∈ Λ^p(''V''), then
:

Universal property

Let ''V'' be a vector space over the field ''K''. Informally, multiplication in Λ(''V'') is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identities ''v'' ∧ ''v'' = 0 for ''v'' ∈ ''V'' and ''v'' ∧ ''w'' = -''w'' ∧ ''v'' for ''v'', ''w'' ∈ ''V''. Formally, Λ(''V'') is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative ''K''-algebra containing ''V'' with alternating multiplication on ''V'' must contain a homomorphic image of Λ(''V''). In other words, the exterior algebra has the following universal property:^[4]

Given any unital associative ''K''-algebra ''A'' and any ''K''-linear map ''j'' : ''V'' → ''A'' such that ''j''(''v'')''j''(''v'') = 0 for every ''v'' in ''V'', then there exists ''precisely one'' unital algebra homomorphism ''f'' : Λ(''V'') → ''A'' such that ''f''(''v'') = ''j''(''v'') for all ''v'' in ''V''.


To construct the most general algebra that contains ''V'' and whose multiplication is alternating on ''V'', it is natural to start with the most general algebra that contains ''V'', the tensor algebra ''T''(''V''), and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided ideal ''I'' in ''T''(''V'') generated by all elements of the form ''v''⊗''v'' for ''v'' in ''V'', and define Λ(''V'') as the quotient
:Λ(''V'') = T(''V'')/''I''
(and use Λ as the symbol for multiplication in Λ(''V'')). It is then straightforward to show that Λ(''V'') contains ''V'' and satisfies the above universal property.
As a consequence of this construction, the operation of assigning to a vector space ''V'' its exterior algebra Λ(''V'') is a functor from the category of vector spaces to the category of algebras.
Rather than defining Λ(''V'') first and then identifying the exterior powers Λ^''k''(''V'') as certain subspaces, one may alternatively define the spaces Λ^''k''(''V'') first and then combine them to form the algebra Λ(''V''). This approach is often used in differential geometry and is described in the next section.

Generalizations

Given a commutative ring ''R'' and an ''R''-module ''M'', we can define the exterior algebra Λ(''M'') just as above, as a suitable quotient of the tensor algebra 'T'(''M''). It will satisfy the analogous universal property. Many of the properties of Λ(''M'') also require that ''M'' be a projective module. Where finite-dimensionality is used, the properties further require that ''M'' be finitely generated and projective.

Duality

Alternating operators

Given two vector spaces ''V'' and ''X'', an 'alternating operator' (or ''anti-symmetric operator'') from ''V''^''k'' to ''X'' is a multilinear map
:''f'': ''V''^''k'' → ''X ''
such that whenever ''v''₁,...,''v''_''k'' are linearly dependent vectors in ''V'', then
:''f''(''v''₁,...,''v''_''k'') = 0.
The most famous example is the determinant, an alternating operator from (''K''^''n'')^''n'' to ''K''.
The map
:''w'': ''V''^''k'' → Λ^''k''(''V'')
which associates to ''k'' vectors from ''V'' their wedge product, i.e. their corresponding ''k''-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on ''V''^''k'': given any other alternating operator ''f'' : ''V''^''k'' → ''X'', there exists a unique linear map φ: Λ^''k''(''V'') → ''X'' with ''f'' = φ o ''w''. This universal property characterizes the space Λ^''k''(''V'') and can serve as its definition.

Alternating multilinear forms

The above discussion specializes to the case when ''X'' = ''K'', the base field. In this case an alternating multilinear function
:''f'' : ''V''^k → ''K''
is called an 'alternating multilinear form'. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the multiplication of such a map with a scalar, is again alternating. If ''V'' has finite dimension ''n'', then this space can be identified with Λ^''k''(''V''^∗), where ''V''^∗ denotes the dual space of ''V''. In particular, the dimension of the space of anti-symmetric maps from ''V''^''k'' to ''K'' is the binomial coefficient ''n'' choose ''k''.
Under this identification, and if the base field is 'R' or 'C', the wedge product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : ''V''^''k'' → ''K'' and η : ''V''^''m'' → ''K'' are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their wedge product is the sum of the numbers of their variables. It is defined as follows:
:
m Alt}(omegaotimeseta)
where the alternation Alt of a multilinear map is defined to be the signed average of the values over all the permutations of its variables:
:$\{$
m Alt}(omega)(x_1,ldots,x_k)=rac{1}{k!}sum_{sigmain S_k}{
m sgn}(sigma),omega(x_{sigma(1)},ldots,x_{sigma(k)})
This definition of the wedge product is well-defined even if the fields ''K'' has finite characteristic, if
one considers an equivalent version of the above that does not use factorials or any constants:
:$omega\; wedge\; eta(x\_1,ldots,x\_\{k+m\})\; =\; sum\_\{sigma\; in\; Sh\_\{k,m\}\}\; \{$
m sgn}(sigma),omega(x_{sigma(1)}, ldots, x_{sigma(k)}) eta(x_{sigma(k+1)}, ldots, x_{sigma(k+m)}),
where here $Sh\_\{k,m\}\; subset\; S\_\{k+m\}$ is the subset of ''k,m shuffles'': permutations $sigma$
sending $1,2,ldots,k$ to numbers $sigma(1)$
< sigma(2) < cdots < sigma(k), and $k+1,k+2,ldots,k+m$ to numbers
('Note.' Some conventions, particularly in physics, define the wedge product as
:$omegawedgeeta=\{$
m Alt}(omegaotimeseta).
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
:$Delta(x\_1wedgedotswedge\; x\_k)\; =\; sum\_\{p=0\}^k\; sum\_\{sigmain\; Sh\_\{p,k-p\}\}\; \{$
m sgn}(sigma) (x_{sigma(1)}wedgedotswedge x_{sigma(p)})otimes (x_{sigma(p+1)}wedgedotswedge x_{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:
:
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

Suppose that ''V'' is finite-dimensional. If ''V
★ '' 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),
:
ightarrowLambda^{k-1}V.
This derivation is called the 'interior product' with α, or sometimes the 'insertion operator'.
Suppose that 'w' ∈ Λ^k''V''. Then 'w' is a multilinear mapping of ''V''
^★ 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
:
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:
# For each ''k'' and each α ∈ V
^★ ,
#::
ightarrow Lambda^{k-1}V.
#:(By convention, Λ^-1 = 0.)
# If ''v'' is an element of ''V'' ( = Λ¹''V''), then ''i''_α''v'' = α(''v'') is the dual pairing between elements of ''V'' and elements of ''V''
^★ .
# For each α ∈ ''V''
^★ , ''i''α is a graded derivation of degree -1:
#::.
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:
:
★
:
★

Hodge duality

Main articles: Hodge dual

Suppose that ''V'' has finite dimension ''n''. Then the interior product induces a canonical isomorphism of vector spaces
:$Lambda^k(V^$
★ ) 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_lphasigma in Lambda^{n-k}(V).
If, in addition to a volume form, the vector space ''V'' is equipped with an inner product identifying ''V'' with ''V''
^★ , then the resulting isomorphism is called the 'Hodge dual' (or more commonly the 'Hodge star operator')
:$$
★ : Lambda^k(V)
ightarrow Lambda^{n-k}(V).
The composite of
★ with itself maps Λ^k(''V'') → Λ^k(''V'') and is always a scalar multiple of the identity map. In most applications, the volume form is compatible with the inner product in the sense that it is a wedge product of an orthonormal basis of ''V''. In this case,
:$$
★ 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
:$Lambda(f)\; :\; Lambda(V)$
ightarrow Lambda(W)
such that
:$Lambda(f)|\_\{Lambda^1(V)\}\; =\; f\; :\; V=Lambda^1(V)$
ightarrow W=Lambda^1(W).
In particular, Λ(''f'') preserves homogeneous degree. The ''k''-graded components of Λ(''f'') are given on decomposable elements by
:$Lambda(f)(x\_1wedge\; dots\; wedge\; x\_k)\; =\; f(x\_1)wedgedotswedge\; f(x\_k).$
Let
:$Lambda^k(f)\; =\; Lambda(f)\_\{Lambda^k(V)\}\; :\; Lambda^k(V)$
ightarrow Lambda^k(W).
The components of the transformation Λ(''k'') relative to a basis of ''V'' and ''W'' is the matrix of ''k'' × ''k'' minors of ''f''. In particular, if ''V'' = ''W'' and ''V'' is of finite dimension ''n'', then Λⁿ(''f'') is a mapping of a one-dimensional vector space Λⁿ to itself, and is therefore given by a scalar: the determinant of ''f''.

Exactness

The functor Λ is exact, meaning that if
:$0$
ightarrow U
ightarrow V
ightarrow W
ightarrow 0
is a short exact sequence of vector spaces, then
:$0$
ightarrow Lambda(U)
ightarrow Lambda(V)
ightarrow Lambda(W)
ightarrow 0
is also exact.^[5]
One consequence of the exactness is that the exterior powers of a direct sum of two vector spaces decompose into tensor products:
:
In particular, if
:$0\; o\; U\; o\; V\; o\; W\; o\; 0$
is a short exact sequence of vector spaces, with dim(''U'') = ''a'', dim(''V'') = ''a''+''b'', and dim(''W'') = ''b'', then
:$Lambda^\{a+b\}(V)=Lambda^a(U)otimesLambda^b(W).$

The alternating tensor algebra

If ''K'' is a field of characteristic 0,^[6] then the exterior algebra of a vector space ''V'' can be canonically identified with the vector subspace of T(''V'') consisting of antisymmetric tensors. Recall that the exterior algebra is the quotient of T(''V'') by the ideal ''I'' generated by ''x'' ⊗ ''x''.
Let T^r(''V'') be the space of homogeneous tensors of rank ''r''. This is spanned by decomposable tensors
:$v\_1otimesdotsotimes\; v\_r,quad\; v\_iin\; V.$
The 'antisymmetrization' (or sometimes the 'skew-symmetrization') of a decomposable tensor is defined by
:
m sgn}(sigma) v_{sigma(1)}otimesdotsotimes v_{sigma(r)}
where the sum is taken over the symmetric group of permutations on the symbols {1,...,r}. This extends by linearity and homogeneity to an operation, also denoted by ''Alt'', on the full tensor algebra T(''V''). The image ''Alt''(T(''V'')) is the 'alternating tensor algebra', denoted A(''V''). This is a vector subspace of T(''V''), and it inherits the structure of a graded vector space from that on T(''V''). It carries an associative graded product $widehat\{otimes\}$ defined by
:$t\; widehat\{otimes\}\; s\; =\; ext\{Alt\}(totimes\; s).$
Although this product differs from the tensor product, the kernel of ''Alt'' is precisely the ideal ''I'' (again, assuming that ''K'' has characteristic 0), and there is a canonical isomorphism
:$A(V)cong\; Lambda(V).$

Index notation

Suppose that ''V'' has finite dimension ''n'', and that a basis 'e'₁, ..., 'e'_n of ''V'' is given. then any alternating tensor ''t'' ∈ A^r(''V'') ⊂ T^r(''V'') can be written in index notation as
:$t\; =\; t^\{i\_1i\_2dots\; i\_r\},\; \{mathbf\; e\}\_\{i\_1\}otimes\; \{mathbf\; e\}\_\{i\_2\}otimesdotsotimes\; \{mathbf\; e\}\_\{i\_r\}$
where ''t''^i₁...i_r is completely antisymmetric in its indices.
The wedge product of two alternating tensors ''t'' and ''s'' of ranks ''r'' and ''p'' is given by
:
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:
:$(twidehat\{otimes\}\; s)^\{i\_1dots\; i\_\{r+p\}\}\; =\; t^\{[i\_1dots\; i\_r\}s^\{i\_\{r+1\}dots\; i\_\{r+p\}]\}.$
The interior product may also be described in index notation as follows. Let $t\; =\; t^\{i\_0i\_1dots\; i\_\{r-1\}\}$ be an antisymmetric tensor of rank ''r''. Then, for α ∈ ''V''
^★ , ''i''_α't' is an alternating tensor of rank ''r''-1, given by
:.
where ''n'' is the dimension of ''V''.

Applications

Linear geometry

The decomposable ''k''-vectors have geometric interpretations: the bivector $uwedge\; v$ 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 $uwedge\; vwedge\; w$ 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 forms. 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 functors on the category of vector spaces, the other being the symmetric algebra. Together, these constructions are used to generate the irreducible representations 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 fermions 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''.^[7] 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 multivectors. It was thus a ''calculus'', much like the propositional calculus, except focused exclusively on the task of formal reasoning in geometrical terms.^[8] 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,^[9] 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 forms.
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

1. Grassmann (1844) introduced these as ''extended'' algebras (cf. Clifford, 1878). He used the word ''äußere'' (literally translated as ''outer'', or ''exterior'') only to indicate the ''produkt'' he defined, which is nowadays conventionally called ''exterior product'', probably to distinguish it from the ''outer product'' as defined in modern linear algebra.
2. 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.
3. This axiomatization of areas is due to Leopold Kronecker and Karl Weierstrass (see Bourbaki (1989), Historical Note). For a modern treatment, see MacLane and Birkhoff (1999) Theorem IX.2.2. For an elementary treatment, see Strang (1993), Chapter 5.
4. See Bourbaki (1989) III.7.1, and MacLane and Birkhoff (1999) Theorem XVI.6.8. More detail on universal properties in general can be found in MacLane and Birkhoff (1999) Chapter VI, and throughout the works of Bourbaki.
5. This also holds in greater generality if ''U'', ''V'', and ''W'' are projective modules over a commutative ring. In the non-projective case, Λ is only right-exact.
6. See Bourbaki (1989) III.7.5 for generalizations.
7. Kannenberg (2000) published a translation of Grassmann's work in English; he translated ''Ausdehnungslehre'' as ''Extension Theory''.
8. 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.
9. Bourbaki, ''Algebra'' (1989) p. 661.

References

Mathematical references

★ Tensor analysis on manifolds, Bishop, R. and Goldberg, S.I., , , Dover, 1980,
:: 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.

★ Elements of mathematics, Algebra I, Bourbaki, Nicolas, , , Springer-Verlag, 1989,
:: 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.

★ Algebra, MacLane, S. and Birkhoff, G., , , AMS Chelsea, 1999,
:: Chapter XVI sections 6-10 give a more elementary account of the exterior algebra, including duality, determinants and minors, and alternating forms.

★ Lectures on Differential Geometry, Sternberg, Shlomo, , , Prentice Hall, 1964,
:: Contains a classical treatment of the exterior algebra as alternating tensors, and applications to differential geometry.

Historical references

★ Elements of mathematics, Algebra I, , Nicolas, Bourbaki, Springer-Verlag, 1989,

★ Applications of Grassmann's Extensive Algebra, Clifford, W., , , American Journal of Mathematics, 1878

★ The Calculus of Extension, , H. G., Forder, Cambridge University Press, 1941,

★ Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik, , Hermann, Grassmann, , 1844, (The Linear Extension Theory - A new Branch of Mathematics)

★ Extension Theory (translation by L.C. Kannenberg), , Hermann, Grassmann, American Mathematical Society, 2000,

★ Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva, , Giuseppe, Peano, , 1888, [Geometric Calculus according to Grassmann's Ausdehnungslehre, preceded by the Operations of Deductive Logic]

★ A Treatise on Universal Algebra, with Applications, , Alfred North, Whitehead, Cambridge, 1898,

Other references and further reading

★ Grassmann algebra - Exploring applications of Extended Vector Algebra with Mathematica, , J.M., Browne, Published on line, 2007,
:: An introduction to the exterior algebra, and geometric algebra, with a focus on applications. Also includes a history section and bibliography.

★ Introduction to linear algebra, Strang, G., , , Wellesley-Cambridge Press, 1993,
:: 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