Editing Exterior algebra (section)

== Applications ==
=== Oriented volume in affine space ===
The natural setting for (oriented) <math>k</math>-dimensional volume and exterior algebra is [[affine space]]. This is also the intimate connection between exterior algebra and [[differential forms]], as to integrate we need a 'differential' object to measure infinitesimal volume. If <math> \mathbb{A}</math> is an affine space over the vector space {{tmath|V}}, and a ([[simplex]]) collection of ordered <math>k+1</math> points <math> A_0, A_1, ... , A_k</math>, we can define its oriented <math>k</math>-dimensional volume as the exterior product of vectors <math> A_0A_1\wedge A_0A_2\wedge \cdots\wedge A_0A_k ={}</math> <math>(-1)^jA_jA_0\wedge A_jA_1\wedge A_jA_2\wedge \cdots\wedge A_jA_k</math> (using concatenation <math>PQ</math> to mean the [[displacement vector]] from point <math>P</math> to <math>Q</math>); if the order of the points is changed, the oriented volume changes by a sign, according to the parity of the permutation. In {{tmath|n}}-dimensional space, the volume of any <math>n</math>-dimensional simplex is a scalar multiple of any other.

The sum of the <math>(k-1)</math>-dimensional oriented areas of the boundary simplexes of a {{tmath|k}}-dimensional simplex is zero, as for the sum of vectors around a triangle or the oriented triangles bounding the tetrahedron in the previous section.

The vector space structure on <math>{\textstyle\bigwedge}(V)</math> generalises addition of vectors in {{tmath|V}}: we have <math>(u_1 + u_2) \wedge v = u_1 \wedge v + u_2 \wedge v</math> and similarly a {{math|''k''}}-blade <math>v_1 \wedge \dots \wedge v_k</math> is linear in each factor.

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=== Linear algebra ===
In applications to [[linear algebra]], the exterior product provides an abstract algebraic manner for describing the [[determinant]] and the [[minor (matrix)|minors]] of a [[matrix (mathematics)|matrix]].  For instance, it is well known that the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix (with a sign to track orientation).  This suggests that the determinant can be ''defined'' in terms of the exterior product of the column vectors.  Likewise, the {{math|''k'' × ''k''}} minors of a matrix can be defined by looking at the exterior products of column vectors chosen {{math|''k''}} at a time.  These ideas can be extended not just to matrices but to [[linear transformation]]s as well: the determinant of a linear transformation is the factor by which it scales the oriented volume of any given reference parallelotope.  So the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power.  The action of a transformation on the lesser exterior powers gives a [[basis of a vector space|basis]]-independent way to talk about the minors of the transformation.

=== Physics ===
{{main|Electromagnetic tensor}}
In physics, many quantities are naturally represented by alternating operators.  For example, if the motion of a charged particle is described by velocity and acceleration vectors in four-dimensional spacetime, then normalization of the velocity vector requires that the electromagnetic force must be an alternating operator on the velocity.  Its six degrees of freedom are identified with the electric and magnetic fields.

=== Electromagnetic field ===
In [[Theory of relativity|Einstein's theories of relativity]], the [[electromagnetic field]] is generally given as a [[differential form|differential 2-form]] <math> F = dA </math> in [[4-space]] or as the equivalent [[Antisymmetric tensor|alternating tensor field]] <math> F_{ij} = A_{[i,j]} = A_{[i;j]}, </math> the [[electromagnetic tensor]].  Then <math> dF = ddA = 0 </math> or the equivalent Bianchi identity <math> F_{[ij,k]} = F_{[ij;k]} = 0. </math>
None of this requires a metric.

Adding the [[Lorentz metric]] and an [[Orientability#Orientability of differentiable manifolds|orientation]] provides the [[Hodge star operator]] <math> \star </math> and thus makes it possible to define <math> J = {\star}d{\star}F </math> or the equivalent tensor [[divergence]] <math> J^i = F^{ij}_{,j} = F^{ij}_{;j} </math> where <math> F^{ij} = g^{ik}g^{jl}F_{kl}. </math>

=== Linear geometry ===
The decomposable {{math|''k''}}-vectors have geometric interpretations: the bivector <math>u \wedge v</math> represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented [[parallelogram]] with sides <math>u</math> and {{tmath|v}}.  Analogously, the 3-vector <math>u \wedge v \wedge w</math> represents the spanned 3-space weighted by the volume of the oriented [[parallelepiped]] with edges {{tmath|u}}, {{tmath|v}}, and {{tmath|w}}.

=== Projective geometry ===
Decomposable {{math|''k''}}-vectors in <math>{\textstyle\bigwedge}^{\!k}(V)</math> correspond to weighted {{Math|''k''}}-dimensional [[linear subspace]]s of {{tmath|V}}.  In particular, the [[Grassmannian]] of {{math|''k''}}-dimensional subspaces of {{tmath|V}}, denoted {{tmath|\operatorname{Gr}_k(V)}}, can be naturally identified with an [[algebraic variety|algebraic subvariety]] of the [[projective space]] {{nowrap|<math display=inline>\mathbf{P}\bigl({\textstyle\bigwedge}^{\!k}(V)\bigr)</math>}}.  This is called the [[Plücker embedding]], and the image of the embedding can be characterized by the [[Plücker relations]].

=== Differential geometry ===
The exterior algebra has notable applications in [[differential geometry]], where it is used to define [[differential form]]s.<ref>{{cite book |first=A.T. |last=James |chapter=On the Wedge Product |title=Studies in Econometrics, Time Series, and Multivariate Statistics |editor-first=Samuel |editor-last=Karlin |editor2-first=Takeshi |editor2-last=Amemiya |editor3-first=Leo A. |editor3-last=Goodman |publisher=Academic Press |year=1983 |isbn=0-12-398750-4 |pages=455–464 |chapter-url=https://books.google.com/books?id=-hDjBQAAQBAJ&pg=PA455 }}</ref> Differential forms are mathematical objects that evaluate the length of vectors, areas of parallelograms, and volumes of [[Parallelepiped#Parallelotope|higher-dimensional bodies]], so they can be [[integral|integrated]] over curves, surfaces and higher dimensional [[manifold]]s in a way that generalizes the [[line integral]]s and [[surface integral]]s from calculus.  A [[differential form]] at a point of a [[differentiable manifold]] is an alternating multilinear form on the [[tangent space]] at the point.  Equivalently, a differential form of degree {{math|''k''}} is a [[linear functional]] on the {{math|''k''}}th exterior power of the tangent space.  As a consequence, the exterior product of multilinear forms defines a natural exterior product for differential forms.  Differential forms play a major role in diverse areas of differential geometry.

An [[Differential (mathematics)#Differentials as germs of functions|alternate approach]] defines differential forms in terms of [[Germ (mathematics)|germs of functions]].

In particular, the [[exterior derivative]] gives the exterior algebra of differential forms on a manifold the structure of a [[differential graded algebra]].  The exterior derivative commutes with [[pullback (differential geometry)|pullback]] along smooth mappings between manifolds, and it is therefore a [[natural transformation|natural]] [[differential operator]].  The exterior algebra of differential forms, equipped with the exterior derivative, is a [[cochain complex]] whose cohomology is called the [[de Rham cohomology]] of the underlying manifold and plays a vital role in the [[algebraic topology]] of differentiable manifolds.

=== 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]] (see ''[[Fundamental representation]]'').

=== Superspace ===
The exterior algebra over the complex numbers is the archetypal example of a [[superalgebra]], which plays a fundamental role in physical theories pertaining to [[fermion]]s and [[supersymmetry]].  A single element of the exterior algebra is called a '''supernumber'''<ref>{{cite book |author-link=Bryce DeWitt |first=Bryce |last=DeWitt  |title=Supermanifolds |year=1984 |publisher=Cambridge University Press  |isbn=0-521-42377-5 |chapter=Chapter 1 |page=1}}</ref> or [[Grassmann number]].  The exterior algebra itself is then just a one-dimensional [[superspace]]: it is just the set of all of the points in the exterior algebra.  The topology on this space is essentially the [[weak topology]], the [[open sets]] being the [[cylinder set]]s.  An {{math|''n''}}-dimensional superspace is just the {{tmath|n}}-fold product of exterior algebras.

=== Lie algebra homology ===
Let <math>L</math> be a Lie algebra over a field {{tmath|K}}, then it is possible to define the structure of a [[chain complex]] on the exterior algebra of {{tmath|L}}.  This is a {{tmath|K}}-linear mapping
: <math>
\partial : {\textstyle\bigwedge}^{\!p+1}(L) \to {\textstyle\bigwedge}^{\!p}(L)
</math>
defined on decomposable elements by
: <math>
\partial (x_1 \wedge \cdots \wedge x_{p+1})
= \frac{1}{p+1}\sum_{j<\ell}(-1)^{j+\ell+1}[x_j,x_\ell] \wedge x_1 \wedge \cdots \wedge \hat{x}_j \wedge \cdots \wedge \hat{x}_\ell \wedge \cdots \wedge x_{p+1}.
</math>

The [[Jacobi identity]] holds if and only if {{tmath|\partial\partial = 0}}, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra <math>L</math> to be a Lie algebra.  Moreover, in that case <math display=inline>{\textstyle\bigwedge}(L) </math> is a [[chain complex]] with boundary operator {{tmath|\partial}}.  The [[homology theory|homology]] associated to this complex is the [[Lie algebra homology]].

=== Homological algebra ===
The exterior algebra is the main ingredient in the construction of the [[Koszul complex]], a fundamental object in [[homological algebra]].