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| {{Redirect|Wedge product|the wedge sum, sometimes called wedge product, in [[Topology]]|wedge sum}}
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| [[File:N-vector.svg|thumb|125px|Geometric interpretation for the '''exterior product''' of ''n'' [[vector (geometry)|vector]]s ('''u''', '''v''', '''w''') to obtain an ''n''-vector ([[parallelotope]] elements), where ''n'' = [[graded algebra|grade]],<ref>{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}</ref> for ''n'' = 1, 2, 3. The "circulations" show [[Orientation (vector space)|orientation]].<ref>{{cite book|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|page=83|isbn=0-7167-0344-0}}</ref>]]
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| In [[mathematics]], the '''exterior product''' or '''wedge product''' of vectors is an algebraic construction used in [[Euclidean geometry]] to study [[area]]s, [[volume]]s, and their higher-dimensional analogs. The exterior product of two vectors ''u'' and ''v'', denoted by ''u'' ∧ ''v'', is called a [[bivector]] and lives in a space called the ''exterior square'', a geometrical [[vector space]] that differs from the original space of vectors. The [[magnitude (mathematics)|magnitude]]<ref>Strictly speaking, the magnitude depends on some additional structure, namely that the vectors be in a [[Euclidean space]]. We do not generally assume that this structure is available, except where it is helpful to develop intuition on the subject.</ref> of ''u'' ∧ ''v'' can be interpreted as the area of the parallelogram with sides ''u'' and ''v'', which in three dimensions can also be computed using the [[cross product]] of the two vectors. Also like the cross product, the exterior product is [[anticommutativity|anticommutative]], meaning that {{nowrap|1=''u'' ∧ ''v'' = −(''v'' ∧ ''u'')}} for all vectors ''u'' and ''v''. One way to visualize a bivector is as a family of [[parallelograms]] all lying in the same plane, having the same area, and with the same [[orientation (mathematics)|orientation]] of their boundaries—a choice of clockwise or counterclockwise.
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| When regarded in this manner the exterior product of two vectors is called a [[blade (geometry)|2-blade]]. More generally, the exterior product of any number ''k'' of vectors can be defined and is sometimes called a ''k''-blade. It lives in a geometrical space known as the ''k''-th exterior power. The magnitude of the resulting ''k''-blade is the volume of the ''k''-dimensional [[parallelotope]] whose sides are the given vectors, just as the magnitude of the [[scalar triple product]] of vectors in three dimensions gives the volume of the parallelepiped spanned by those vectors.
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| The '''exterior algebra''', or '''Grassmann algebra''' after [[Hermann Grassmann]],<ref>{{harvcoltxt|Grassmann|1844}} introduced these as ''extended'' algebras (cf. {{harvnb|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]].</ref> is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, whereas blades have a concrete geometrical interpretation, objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just ''k''-blades, but sums of ''k''-blades; such a sum is called a [[p-vector|''k''-vector]].<ref>The term ''k-vector'' is not equivalent to and should not be confused with similar terms such as ''[[4-vector]]'', which in a different context could mean a 4-dimensional vector. A minority of authors use the term ''k''-multivector instead of ''k''-vector, which avoids this confusion.</ref> The ''k''-blades, because they are simple products of vectors, are called the simple elements of the algebra. The ''rank'' of any ''k''-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an [[associative algebra]], which means that {{nowrap|1=α ∧ (β ∧ γ) = (α ∧ β) ∧ γ}} for any elements α, β, γ. The ''k''-vectors have degree ''k'', meaning that they are sums of products of ''k'' vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a [[graded algebra]].
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| In a precise sense, given by what is known as a [[universal property|universal construction]], the exterior algebra is the ''largest'' algebra that supports an alternating product on vectors, and can be easily defined in terms of other known objects such as [[tensor]]s. The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as [[vector field]]s or [[function (mathematics)|functions]]. In full generality, the exterior algebra can be defined for [[module (mathematics)|modules]] over a [[commutative ring]], and for other structures of interest in [[abstract algebra]]. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of [[differential forms]] that is fundamental in areas that use [[differential geometry]]. Differential forms are mathematical objects that represent [[infinitesimal]] areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be [[integral|integrated]] over surfaces and higher dimensional [[manifold]]s in a way that generalizes the [[line integral]]s from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of [[functor]] on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a [[bialgebra]], meaning that its [[dual space]] also possesses a product, and this dual product is compatible with the exterior 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]].
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| ==Motivating examples==
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| <!--The purpose of this section is to motivate the skewness of the exterior product on vectors in ''V''-->
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| ===Areas in the plane===
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| [[Image:Area parallellogram as determinant.svg|thumb|right|The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices.]]
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| The [[Cartesian plane]] '''R'''<sup>2</sup> is a vector space equipped with a [[basis of a vector space|basis]] consisting of a pair of [[unit vector]]s
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| :<math>{\mathbf e}_1 = \begin{bmatrix}1\\0\end{bmatrix},\quad {\mathbf e}_2 = \begin{bmatrix}0\\1\end{bmatrix}.</math>
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| Suppose that
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| :<math>{\mathbf v} = \begin{bmatrix}a\\b\end{bmatrix} = a {\mathbf e}_1 + b {\mathbf e}_2, \quad {\mathbf w} = \begin{bmatrix}c\\d\end{bmatrix} = c {\mathbf e}_1 + d {\mathbf e}_2</math>
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| are a pair of given vectors in '''R'''<sup>2</sup>, 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:
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| :<math>\text{Area} = \left|\det\begin{bmatrix}{\mathbf v}& {\mathbf w}\end{bmatrix}\right| = \left|\det\begin{bmatrix} a & c\\b & d \end{bmatrix}\right| = \left| ad - bc \right|.</math>
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| Consider now the exterior product of '''v''' and '''w''':
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| :<math>
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| \begin{align}
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| {\mathbf v}\wedge {\mathbf w} & = (a{\mathbf e}_1 + b{\mathbf e}_2)\wedge (c{\mathbf e}_1 + d{\mathbf e}_2) \\
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| & = ac{\mathbf e}_1\wedge{\mathbf e}_1+ ad{\mathbf e}_1\wedge {\mathbf e}_2+bc{\mathbf e}_2\wedge {\mathbf e}_1+bd{\mathbf e}_2\wedge {\mathbf e}_2 \\
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| & =(ad-bc){\mathbf e}_1\wedge{\mathbf e}_2
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| \end{align}
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| </math>
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| where the first step uses the distributive law for the [[#Formal_definitions_and_algebraic_properties|exterior product]], and the last uses the fact that the exterior product is alternating, and in particular {{nowrap|1='''e'''<sub>2</sub> ∧ '''e'''<sub>1</sub> = −'''e'''<sub>1</sub> ∧ '''e'''<sub>2</sub>}}. Note that the coefficient in this last expression is precisely the determinant of the matrix {{nowrap|1=['''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.
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| 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 {{nowrap|1=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:
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| # A(''j'''''v''', ''k'''''w''') = ''j k'' A('''v''', '''w''') for any real numbers ''j'' and ''k'', 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).
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| # A('''v''','''v''') = 0, since the area of the [[degenerate (mathematics)|degenerate]] parallelogram determined by '''v''' (i.e., a [[line segment]]) is zero.
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| # A('''w''','''v''') = −A('''v''','''w'''), since interchanging the roles of '''v''' and '''w''' reverses the orientation of the parallelogram.
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| # A('''v''' + ''j'''''w''','''w''') = A('''v''','''w'''), for real ''j'', since adding a multiple of '''w''' to '''v''' affects neither the base nor the height of the parallelogram and consequently preserves its area.
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| # A('''e'''<sub>1</sub>, '''e'''<sub>2</sub>) = 1, since the area of the unit square is one.
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| With the exception of the last property, the exterior product satisfies the same formal properties as the area. In a certain sense, the exterior product generalizes the final property by allowing the area of a parallelogram to be compared to that of any "standard" chosen parallelogram (here, the one with sides '''e'''<sub>1</sub> and '''e'''<sub>2</sub>). In other words, the exterior product in two dimensions provides a ''basis-independent'' formulation of area.<ref>This axiomatization of areas is due to [[Leopold Kronecker]] and [[Karl Weierstrass]]; see {{harvtxt|Bourbaki|1989|loc=Historical Note}}. For a modern treatment, see {{harvtxt|Mac Lane|Birkhoff|1999|loc=Theorem IX.2.2}}. For an elementary treatment, see {{harvtxt|Strang|1993|loc=Chapter 5}}.</ref>
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| ===Cross and triple products===
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| [[Image:Exterior calc cross product.svg|250px|thumb|The cross product ('''<span style="color:blue;">blue</span>''' vector) in relation to the exterior product ('''<span style="color:#779ECB;">light blue</span>''' parallelogram). The length of the cross product is to the length of the parallel unit vector ('''<span style="color:#CC0000;">red</span>''') as the size of the exterior product is to the size of the reference parallelogram ('''<span style="color:#CC4E5C;">light red</span>''').]]
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| For vectors in '''R'''<sup>3</sup>, the exterior algebra is closely related to the [[cross product]] and [[triple product]]. Using the standard basis {'''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>}, the exterior product of a pair of vectors
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| :<math> \mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3 </math> | |
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| and
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| :<math> \mathbf{v} = v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2 + v_3 \mathbf{e}_3 </math>
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| is
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| :<math> \mathbf{u} \wedge \mathbf{v} = (u_1 v_2 - u_2 v_1) (\mathbf{e}_1 \wedge \mathbf{e}_2) + (u_3 v_1 - u_1 v_3) (\mathbf{e}_3 \wedge \mathbf{e}_1) + (u_2 v_3 - u_3 v_2) (\mathbf{e}_2 \wedge \mathbf{e}_3) </math>
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| where {'''e'''<sub>1</sub> ∧ '''e'''<sub>2</sub>, '''e'''<sub>3</sub> ∧ '''e'''<sub>1</sub>, '''e'''<sub>2</sub> ∧ '''e'''<sub>3</sub>} is the basis for the three-dimensional space Λ<sup>2</sup>('''R'''<sup>3</sup>). The coefficients above are the same as those in the usual definition of the [[cross product]] of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a 2-vector.
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| Bringing in a third vector
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| :<math> \mathbf{w} = w_1 \mathbf{e}_1 + w_2 \mathbf{e}_2 + w_3 \mathbf{e}_3, </math> | |
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| the exterior product of three vectors is
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| :<math> \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) </math> | |
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| where '''e'''<sub>1</sub> ∧ '''e'''<sub>2</sub> ∧ '''e'''<sub>3</sub> is the basis vector for the one-dimensional space Λ<sup>3</sup>('''R'''<sup>3</sup>). The scalar coefficient is the [[triple product]] of the three vectors.
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| The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product {{nowrap|1='''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 [[minor (mathematics)|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.
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| ==Formal definitions and algebraic properties==
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| The exterior algebra Λ(''V'') over a vector space ''V'' over a [[field (mathematics)|field]] ''K'' is defined as the [[Quotient ring|quotient algebra]] of the [[tensor algebra]] by the two-sided [[Ideal (ring theory)|ideal]] ''I'' generated by all elements of the form {{nowrap|1=''x'' ⊗ ''x''}} such that {{nowrap|1=''x'' ∈ ''V''}}.<ref>This definition is a standard one. See, for instance, {{harvtxt|Mac Lane|Birkhoff|1999}}.</ref> Symbolically,
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| :<math>\Lambda(V) := T(V)/I.\, </math> | |
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| The exterior product ∧ of two elements of Λ(''V'') is defined by
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| :<math>\alpha\wedge\beta = \alpha\otimes\beta \pmod I.</math>
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| ===Anticommutativity of the exterior product===
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| The exterior product is ''alternating'' on elements of ''V'', which means that {{nowrap|1=''x'' ∧ ''x'' = 0}} for all {{nowrap|1=''x'' ∈ ''V''}}. It follows that the product is also [[anticommutative]] on elements of ''V'', for supposing that {{nowrap|1=''x'', ''y'' ∈ ''V''}},
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| :<math>0 = (x+y)\wedge (x+y) = x\wedge x + x\wedge y + y\wedge x + y\wedge y = x\wedge y + y\wedge x</math>
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| hence
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| :<math> x \wedge y = - y \wedge x. </math>
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| Conversely, it follows from the anticommutativity of the product that the product is alternating, unless ''K'' has [[characteristic of a field|characteristic]] two.
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| More generally, if ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''k''</sub> are elements of ''V'', and σ is a [[permutation group|permutation]] of the integers [1,...,''k''], then
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| :<math>x_{\sigma(1)}\wedge x_{\sigma(2)}\wedge\dots\wedge x_{\sigma(k)} = \operatorname{sgn}(\sigma)x_1\wedge x_2\wedge\dots \wedge x_k,</math>
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| where sgn(σ) is the [[signature of a permutation|signature of the permutation]] σ.<ref>A proof of this can be found in more generality in {{harvtxt|Bourbaki|1989}}.</ref>
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| ===The exterior power===
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| The ''k''th '''exterior power''' of ''V'', denoted Λ<sup>''k''</sup>(''V''), is the [[vector subspace]] of Λ(''V'') [[linear span|spanned]] by elements of the form
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| :<math>x_1\wedge x_2\wedge\dots\wedge x_k,\quad x_i\in V, i=1,2,\dots, k.</math>
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| If {{nowrap|1=α ∈ Λ<sup>''k''</sup>(''V'')}}, then α is said to be a '''[[p-vector|''k''-vector]]'''. If, furthermore, α can be expressed as an exterior product of ''k'' elements of ''V'', then α is said to be '''decomposable'''. Although decomposable ''k''-vectors span Λ<sup>''k''</sup>(''V''), not every element of Λ<sup>''k''</sup>(''V'') is decomposable. For example, in '''R'''<sup>4</sup>, the following 2-vector is not decomposable:
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| :<math>\alpha = e_1\wedge e_2 + e_3\wedge e_4.</math>
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| (This is in fact a [[symplectic form]], since α ∧ α ≠ 0.<ref>See {{harvtxt|Sternberg|1964|loc=§III.6}}.</ref>)
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| ====Basis and dimension====
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| If the [[dimension (linear algebra)|dimension]] of ''V'' is ''n'' and {''e''<sub>1</sub>,...,''e''<sub>''n''</sub>} is a [[basis (linear algebra)|basis]] of ''V'', then the set
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| :<math>\{e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_k} \mid 1\le i_1 < i_2 < \cdots < i_k \le n\}</math>
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| is a basis for Λ<sup>''k''</sup>(''V''). The reason is the following: given any exterior product of the form
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| :<math>v_1\wedge\cdots\wedge v_k</math>
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| then every vector ''v''<sub>''j''</sub> can be written as a [[linear combination]] of the basis vectors ''e''<sub>''i''</sub>; using the bilinearity of the exterior product, this can be expanded to a linear combination of exterior products of those basis vectors. Any exterior product in which the same basis vector appears more than once is zero; any exterior 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 [[minor (linear algebra)|minor]]s of the [[matrix (mathematics)|matrix]] that describes the vectors ''v''<sub>''j''</sub> in terms of the basis ''e''<sub>''i''</sub>.
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| By counting the basis elements, the dimension of Λ<sup>''k''</sup>(''V'') is equal to a [[binomial coefficient]]:
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| :<math>\operatorname{dim}(\Lambda^k(V)) = \binom{n}{k}</math>
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| In particular, Λ<sup>''k''</sup>(''V'') = {0} for ''k'' > ''n''.
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| Any element of the exterior algebra can be written as a sum of [[p-vector|''k''-vector]]s. Hence, as a vector space the exterior algebra is a [[Direct sum of modules|direct sum]]
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| :<math>\Lambda(V) = \Lambda^0(V)\oplus \Lambda^1(V) \oplus \Lambda^2(V) \oplus \cdots \oplus \Lambda^n(V)</math>
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| (where by convention Λ<sup>0</sup>(''V'') = ''K'' and Λ<sup>1</sup>(''V'') = ''V''), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2<sup>''n''</sup>.
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| ====Rank of a ''k''-vector====
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| If α ∈ Λ<sup>''k''</sup>(''V''), then it is possible to express α as a linear combination of decomposable [[p-vector|''k''-vector]]s:
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| :<math> \alpha = \alpha^{(1)} + \alpha^{(2)} + \cdots + \alpha^{(s)}</math>
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| where each α<sup>(''i'')</sup> is decomposable, say
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| :<math>\alpha^{(i)} = \alpha^{(i)}_1\wedge\cdots\wedge\alpha^{(i)}_k,\quad i=1,2,\dots, s.</math>
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| The '''rank''' of the ''k''-vector α is the minimal number of decomposable ''k''-vectors in such an expansion of α. This is similar to the notion of [[tensor rank]].
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| Rank is particularly important in the study of 2-vectors {{harv|Sternberg|1974|loc=§III.6}} {{harv|Bryant|Chern|Gardner|Goldschmidt|1991}}. The rank of a 2-vector α can be identified with half the [[rank of a matrix|rank of the matrix]] of coefficients of α in a basis. Thus if ''e''<sub>''i''</sub> is a basis for ''V'', then α can be expressed uniquely as
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| :<math>\alpha = \sum_{i,j}a_{ij}e_i\wedge e_j</math>
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| where ''a''<sub>''ij''</sub> = −''a''<sub>''ji''</sub> (the matrix of coefficients is [[skew-symmetric matrix|skew-symmetric]]). The rank of the matrix ''a''<sub>''ij''</sub> is therefore even, and is twice the rank of the form α.
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| In characteristic 0, the 2-vector α has rank ''p'' if and only if
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| :<math>\underset{p}{\underbrace{\alpha\wedge\cdots\wedge\alpha}}\not= 0</math>
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| and
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| :<math>\underset{p+1}{\underbrace{\alpha\wedge\cdots\wedge\alpha}} = 0.</math>
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| ===Graded structure===
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| The exterior product of a ''k''-vector with a ''p''-vector is a (''k''+''p'')-vector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section
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| :<math>\Lambda(V) = \Lambda^0(V)\oplus \Lambda^1(V) \oplus \Lambda^2(V) \oplus \cdots \oplus \Lambda^n(V)</math>
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| gives the exterior algebra the additional structure of a [[graded algebra]]. Symbolically,
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| :<math>\left(\Lambda^k(V)\right)\wedge\left(\Lambda^p(V)\right)\sub \Lambda^{k+p}(V).</math>
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| Moreover, the exterior product is graded anticommutative, meaning that if α ∈ Λ<sup>k</sup>(''V'') and β ∈ Λ<sup>p</sup>(''V''), then
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| :<math>\alpha\wedge\beta = (-1)^{kp}\beta\wedge\alpha.</math>
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| In addition to studying the graded structure on the exterior algebra, {{harvtxt|Bourbaki|1989}} studies additional graded structures on exterior algebras, such as those on the exterior algebra of a [[graded module]] (a module that already carries its own gradation).
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| ===Universal property===
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| 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 identity ''v'' ∧ ''v'' = 0 for ''v'' ∈ ''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]]:<ref>See {{harvtxt|Bourbaki|1989|loc=III.7.1}}, and {{harvtxt|Mac Lane|Birkhoff|1999|loc=Theorem XVI.6.8}}. More detail on universal properties in general can be found in {{harvtxt|Mac Lane|Birkhoff|1999|loc=Chapter VI}}, and throughout the works of Bourbaki.</ref>
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| <div style="margin-left: 2em; margin-right: 2em">
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| Given any unital associative ''K''-algebra ''A'' and any ''K''-[[linear map]] {{nowrap|1=''j'' : ''V'' → ''A''}} such that {{nowrap|1=''j''(''v'')''j''(''v'') = 0}} for every ''v'' in ''V'', then there exists ''precisely one'' unital [[algebra homomorphism]] {{nowrap|1=''f'' : Λ(''V'') → ''A''}} such that {{nowrap|1=''j''(''v'') = ''f''(''i''(''v''))}} for all ''v'' in ''V''.
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| </div>
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| [[Image:ExteriorAlgebra-01.png|center|Universal property of the exterior algebra]]
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| 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 ring|quotient]]. We thus take the two-sided [[ideal (ring theory)|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
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| :<math>\Lambda(V) = T(V)/I\ </math>
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| (and use ∧ 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.
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| 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 (mathematics)|category]] of vector spaces to the category of algebras.
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| Rather than defining Λ(''V'') first and then identifying the exterior powers Λ<sup>''k''</sup>(''V'') as certain subspaces, one may alternatively define the spaces Λ<sup>''k''</sup>(''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.
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| ===Generalizations===
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| Given a [[commutative ring]] ''R'' and an ''R''-[[module (mathematics)|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 module|finitely generated]] and projective. Generalizations to the most common situations can be found in {{harv|Bourbaki|1989}}.
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| | |
| Exterior algebras of [[vector bundle]]s are frequently considered in geometry and topology. There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the [[Serre–Swan theorem]]. More general exterior algebras can be defined for [[sheaf (mathematics)|sheaves]] of modules.
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| ==Duality==
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| ===Alternating operators===
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| Given two vector spaces ''V'' and ''X'', an '''alternating operator''' from ''V''<sup>''k''</sup> to ''X'' is a [[multilinear]] map
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| :<math> f: V^k \to X </math>
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| such that whenever ''v''<sub>1</sub>,...,''v''<sub>''k''</sub> are [[linearly dependent]] vectors in ''V'', then
| |
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| :<math> f(v_1,\ldots, v_k)=0</math>
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| The map
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| :<math> w: V^k \to \Lambda^k(V) </math>
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| which associates to ''k'' vectors from ''V'' their exterior product, i.e. their corresponding ''k''-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on ''V''<sup>''k''</sup>: given any other alternating operator {{nowrap|1=''f'' : ''V''<sup>''k''</sup> → ''X''}}, there exists a unique [[linear map]] {{nowrap|1=φ : Λ<sup>''k''</sup>(''V'') → ''X''}} with {{nowrap|1=''f'' = φ ∘ ''w''}}. This [[universal property]] characterizes the space Λ<sup>''k''</sup>(''V'') and can serve as its definition.
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| === Alternating multilinear forms ===<!-- Alternating form redirects here -->
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| [[File:N-form.svg|thumb|125px|Geometric interpretation for the '''exterior product''' of ''n'' [[1-form]]s ('''ε''', '''η''', '''ω''') to obtain an ''n''-form ("mesh" of [[coordinate surface]]s, here planes),<ref>{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}</ref> for ''n'' = 1, 2, 3. The "circulations" show [[Orientation (vector space)|orientation]].<ref>{{cite book|title=Gravitation|author=J.A. Wheeler, C. Misner, K.S. Thorne|publisher=W.H. Freeman & Co|year=1973|pages=58–60, 83, 100–109, 115–119|isbn=0-7167-0344-0}}</ref>]]
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| The above discussion specializes to the case when {{nowrap|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 multilinear forms 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 ''k'' on ''V'' is [[natural transformation|naturally]] isomorphic with the [[dual vector space]] (Λ<sup>''k''</sup>''V'')<sup>∗</sup>. If ''V'' is finite-dimensional, then the latter is naturally isomorphic to Λ<sup>''k''</sup>(''V''<sup>∗</sup>). In particular, the dimension of the space of anti-symmetric maps from ''V''<sup>''k''</sup> to ''K'' is the [[binomial coefficient]] ''n'' choose ''k''.
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| Under this identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose {{nowrap|1=ω : ''V''<sup>''k''</sup> → ''K''}} and {{nowrap|1=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. It is defined as follows:<ref>Some conventions, particularly in physics, define the exterior product as
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| :<math>\omega\wedge\eta=\operatorname{Alt}(\omega\otimes\eta).</math>
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| | |
| This convention is not adopted here, but is discussed in connection with [[Exterior algebra#The alternating tensor algebra|alternating tensors]].</ref>
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| | |
| :<math>\omega\wedge\eta=\frac{(k+m)!}{k!\,m!}\operatorname{Alt}(\omega\otimes\eta)</math>
| |
| | |
| where the alternation Alt of a multilinear map is defined to be the signed average of the 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>
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| | |
| This definition of the exterior product is well-defined even if the [[field (mathematics)|field]] ''K'' has [[characteristic of a field|finite characteristic]], if
| |
| one considers an equivalent version of the above that does not use factorials or any constants:
| |
| | |
| :<math>{\omega \wedge \eta(x_1,\ldots,x_{k+m})} = \sum_{\sigma \in 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>
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| | |
| where here {{nowrap|1=Sh<sub>''k'',''m''</sub> ⊂ ''S''<sub>''k''+''m''</sub>}} is the subset of [[(p,q) shuffle|(''k'',''m'') shuffles]]: [[permutation]]s σ of the set {1,2,…,''k'' + ''m''} such that σ(1) < σ(2) < … < σ(''k''), and σ(''k'' + 1) < σ(''k'' + 2) < … < σ(''k'' + ''m'').
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| ===Bialgebra structure===
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| In formal terms, there is a correspondence between the graded dual of the graded algebra Λ(''V'') and alternating multilinear forms on ''V''. The exterior 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 {{nowrap|1=Δ : Λ(''V'') → Λ(''V'') ⊗ Λ(''V'')}} given on decomposable elements by
| |
| :<math>\Delta(x_1\wedge\dots\wedge x_k) = \sum_{p=0}^k \sum_{\sigma\in Sh_{p,k-p}} \operatorname{sgn}(\sigma) (x_{\sigma(1)}\wedge\dots\wedge x_{\sigma(p)})\otimes (x_{\sigma(p+1)}\wedge\dots\wedge x_{\sigma(k)}).</math>
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| For example,
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| | |
| :<math>\Delta(x_1) = 1 \otimes x_1 + x_1 \otimes 1,</math>
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| | |
| :<math>
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| \Delta(x_1 \wedge x_2) = 1 \otimes (x_1 \wedge x_2) + x_1 \otimes x_2 - x_2 \otimes x_1 + (x_1 \wedge x_2) \otimes 1.
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| </math>
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| This extends by linearity to an operation defined on the whole 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\dots\wedge x_k) = (\alpha\otimes\beta)\left(\Delta(x_1\wedge\dots\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 ε is the counit, as defined presently).
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| The '''counit''' is the homomorphism {{nowrap|1=ε : Λ(''V'') → ''K''}} which 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.
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| With an '''antipode''' defined on homogeneous elements by {{nowrap|1=''S''(''x'') = (−1)<sup>deg ''x''</sup>''x''}}, the exterior algebra is furthermore a [[Hopf algebra]].<ref>Indeed, the exterior algebra of ''V'' is the [[Universal enveloping algebra|enveloping algebra]] of the abelian [[Lie superalgebra]] structure on ''V''.</ref>
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| ===Interior product===
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| {{See also|Interior product}}
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| Suppose that ''V'' is finite-dimensional. If ''V''<sup>*</sup> denotes the [[dual space]] to the vector space ''V'', then for each {{nobreak|α ∈ ''V''<sup>*</sup>}}, it is possible to define an [[derivation (abstract algebra)|antiderivation]] on the algebra Λ(''V''),
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| :<math>i_\alpha:\Lambda^k V\rightarrow\Lambda^{k-1}V.</math>
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| This derivation is called the '''interior product''' with α, or sometimes the '''insertion operator''', or '''contraction''' by α.
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| Suppose that {{nobreak|'''w''' ∈ Λ<sup>''k''</sup>''V''}}. Then '''w''' is a multilinear mapping of ''V''<sup>*</sup> to ''K'', so it is defined by its values on the ''k''-fold [[Cartesian product]] ''V''<sup>*</sup> × ''V''<sup>*</sup> × ... × ''V''<sup>*</sup>. If ''u''<sub>1</sub>, ''u''<sub>2</sub>, ..., ''u''<sub>''k''−1</sub> are ''k'' − 1 elements of ''V''<sup>*</sup>, then define
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| :<math>(i_\alpha {\bold w})(u_1,u_2\dots,u_{k-1})={\bold w}(\alpha,u_1,u_2,\dots, u_{k-1}).</math>
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| Additionally, let ''i''<sub>α</sub>''f'' = 0 whenever ''f'' is a pure scalar (i.e., belonging to Λ<sup>0</sup>''V'').
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| ====Axiomatic characterization and properties====
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| The interior product satisfies the following properties:
| |
| | |
| # For each ''k'' and each α ∈ V<sup>*</sup>,
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| #::<math>i_\alpha:\Lambda^kV\rightarrow \Lambda^{k-1}V.</math>
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| #:(By convention, Λ<sup>−1</sup> = {0}.)
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| # If ''v'' is an element of ''V'' (= Λ<sup>1</sup>''V''), then ''i''<sub>α</sub>''v'' = α(''v'') is the dual pairing between elements of ''V'' and elements of ''V''<sup>*</sup>.
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| # For each α ∈ ''V''<sup>*</sup>, ''i''<sub>α</sup> is a [[graded derivation]] of degree −1:
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| #::<math>i_\alpha (a\wedge b) = (i_\alpha a)\wedge b + (-1)^{\deg a}a\wedge (i_\alpha b).</math>
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| In fact, these three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.
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| Further properties of the interior product include:
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| :* <math>i_\alpha\circ i_\alpha = 0.</math>
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| :* <math>i_\alpha\circ i_\beta = -i_\beta\circ i_\alpha.</math>
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| <!--
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| It may be worth saying something about this, but has already been mentioned in passing above.
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| ===The duality isomorphism===
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| In general, there are two different kinds of alternating structures defined via duality:
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| * The structure of alternating multilinear forms on Λ(''V''). The space of all such forms is the graded dual Λ(''V'')<sup>*</sup>, and the product of such forms dualizes the coproduct on the exterior algebra.
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| * The exterior algebra of the dual vector space Λ(''V''<sup>*</sup>).
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| If ''V'' is finite-dimensional, then these two exterior algebras are naturally isomorphic.
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| -->
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| ===Hodge duality===
| |
| {{main|Hodge dual}}
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| Suppose that ''V'' has finite dimension ''n''. Then the interior product induces a canonical isomorphism of vector spaces
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| :<math>\Lambda^k(V^*) \otimes \Lambda^n(V) \to \Lambda^{n-k}(V). \, </math>
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| In the geometrical setting, a non-zero element of the top exterior power Λ<sup>''n''</sup>(''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
| |
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| :<math> \alpha \in \Lambda^k(V^*) \mapsto i_\alpha\sigma \in \Lambda^{n-k}(V). \, </math>
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| If, in addition to a volume form, the vector space ''V'' is equipped with an [[inner product]] identifying ''V'' with ''V''<sup>*</sup>, then the resulting isomorphism is called the '''Hodge dual''' (or more commonly the '''Hodge star operator''')
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| :<math>* : \Lambda^k(V) \rightarrow \Lambda^{n-k}(V). \, </math>
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| The composite of <math>*</math> with itself maps Λ<sup>''k''</sup>(''V'') → Λ<sup>''k''</sup>(''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 an exterior product of an [[orthonormal basis]] of ''V''. In this case,
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| :<math>*\circ * : \Lambda^k(V) \to \Lambda^k(V) = (-1)^{k(n-k) + q}I</math>
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| where ''I'' is the identity, and the inner product has [[metric signature]] (''p'',''q'') — ''p'' plusses and ''q'' minuses.
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| ===Inner product===
| |
| For ''V'' a finite-dimensional space, an [[inner product]] on ''V'' defines an isomorphism of ''V'' with ''V''<sup>∗</sup>, and so also an isomorphism of Λ<sup>''k''</sup>''V'' with (Λ<sup>''k''</sup>''V'')<sup>∗</sup>. The pairing between these two spaces also takes the form of an inner product. On decomposable ''k''-vectors,
| |
| :<math>\left\langle v_1\wedge\cdots\wedge v_k, w_1\wedge\cdots\wedge w_k\right\rangle = \det(\langle v_i,w_j\rangle),</math>
| |
| the determinant of the matrix of inner products. In the special case ''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]] (⟨''v''<sub>''i''</sub>, ''v''<sub>''j''</sub>⟩). This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on Λ<sup>''k''</sup>''V''. If ''e''<sub>''i''</sub>, ''i''=1,2,...,''n'', form an [[orthonormal basis]] of ''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 Λ<sup>''k''</sup>(''V'').
| |
| | |
| With respect to the inner product, exterior multiplication and the interior product are mutually adjoint. Specifically, for '''v''' ∈ Λ<sup>''k''−1</sup>(''V''), '''w''' ∈ Λ<sup>''k''</sup>(''V''), and ''x'' ∈ ''V'',
| |
| :<math>\langle x\wedge\mathbf{v}, \mathbf{w}\rangle = \langle \mathbf{v}, i_{x^\flat}\mathbf{w}\rangle</math>
| |
| where ''x''<sup>♭</sup> ∈ ''V''<sup>*</sup> is the linear functional defined by
| |
| :<math>x^\flat(y) = \langle x, y\rangle</math>
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| for all {{nowrap|1=''y'' ∈ ''V''}}. This property completely characterizes the inner product on the exterior algebra.
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| | |
| ==Functoriality==
| |
| Suppose that ''V'' and ''W'' are a pair of vector spaces and {{nowrap|1=''f'' : ''V'' → ''W''}} is a [[linear transformation]]. Then, by the universal construction, there exists a unique homomorphism of graded algebras
| |
| | |
| :<math>\Lambda(f) : \Lambda(V)\rightarrow \Lambda(W)</math>
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| such that
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| :<math>\Lambda(f)|_{\Lambda^1(V)} = f : V=\Lambda^1(V)\rightarrow W=\Lambda^1(W).</math>
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| | |
| In particular, Λ(''f'') preserves homogeneous degree. The ''k''-graded components of Λ(''f'') are given on decomposable elements by
| |
| :<math>\Lambda(f)(x_1\wedge \dots \wedge x_k) = f(x_1)\wedge\dots\wedge f(x_k).</math>
| |
| | |
| Let
| |
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| :<math>\Lambda^k(f) = \Lambda(f)_{\Lambda^k(V)} : \Lambda^k(V) \rightarrow \Lambda^k(W).</math>
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| | |
| The components of the transformation Λ(''k'') relative to a basis of ''V'' and ''W'' is the matrix of {{nowrap|1=''k'' × ''k''}} minors of ''f''. In particular, if {{nowrap|1=''V'' = ''W''}} and ''V'' is of finite dimension ''n'', then Λ<sup>''n''</sup>(''f'') is a mapping of a one-dimensional vector space Λ<sup>''n''</sup> to itself, and is therefore given by a scalar: the [[determinant]] of ''f''.
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| ===Exactness===
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| If
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| :<math>0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0</math>
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| is a [[short exact sequence]] of vector spaces, then
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| :<math>0\to \Lambda^1(U)\wedge\Lambda(V) \to \Lambda(V)\rightarrow \Lambda(W)\rightarrow 0</math>
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| is an exact sequence of graded vector spaces<ref>This part of the statement also holds in greater generality if ''V'' and ''W'' are modules over a commutative ring: That Λ converts epimorphisms to epimorphisms. See {{harvtxt|Bourbaki|1989|loc=Proposition 3, III.7.2}}.</ref> as is
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| :<math>0\to \Lambda(U)\to\Lambda(V).</math><ref>This statement generalizes only to the case where ''V'' and ''W'' are projective modules over a commutative ring. Otherwise, it is generally not the case that Λ converts monomorphisms to monomorphisms. See {{harvtxt|Bourbaki|1989|loc=Corollary to Proposition 12, III.7.9}}.</ref>
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| | |
| === Direct sums ===
| |
| In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:
| |
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| :<math>\Lambda(V\oplus W)= \Lambda(V)\otimes\Lambda(W).</math>
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| This is a graded isomorphism; i.e.,
| |
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| :<math>\Lambda^k(V\oplus W)= \bigoplus_{p+q=k} \Lambda^p(V)\otimes\Lambda^q(W).</math>
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| Slightly more generally, if
| |
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| :<math>0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0</math>
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| is a [[short exact sequence]] of vector spaces then Λ''<sup>k</sup>(V)'' has a [[Filtration (mathematics)|filtration]]
| |
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| :<math>0 = F^0 \subseteq F^1 \subseteq \dotsb \subseteq F^k \subseteq F^{k+1} = \Lambda^k(V)</math>
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| | |
| with quotients :<math>F^{p+1}/F^p = \Lambda^{k-p}(U) \otimes \Lambda^p(W)</math>. In particular, if ''U'' is 1-dimensional then
| |
| | |
| :<math>0\rightarrow U \otimes \Lambda^{k-1}(W) \rightarrow \Lambda^k(V)\rightarrow \Lambda^k(W)\rightarrow 0</math>
| |
| | |
| is exact, and if ''W'' is 1-dimensional then
| |
| | |
| :<math>0\rightarrow \Lambda^k(U) \rightarrow \Lambda^k(V)\rightarrow \Lambda^{k-1}(U) \otimes W\rightarrow 0</math>
| |
| | |
| is exact.<ref>Such a filtration also holds for [[vector bundle]]s, and projective modules over a commutative ring. This is thus more general than the result quoted above for direct sums, since not every short exact sequence splits in other [[abelian category|abelian categories]].</ref>
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| | |
| ==The alternating tensor algebra==
| |
| If ''K'' is a field of characteristic 0,<ref>See {{harvtxt|Bourbaki|1989|loc=III.7.5}} for generalizations.</ref> then the exterior algebra of a vector space ''V'' can be canonically identified with the vector subspace of T(''V'') consisting of [[antisymmetric tensor]]s. Recall that the exterior algebra is the quotient of T(''V'') by the ideal ''I'' generated by ''x'' ⊗ ''x''.
| |
| | |
| Let T<sup>r</sup>(''V'') be the space of homogeneous tensors of degree ''r''. This is spanned by decomposable tensors
| |
| | |
| :<math>v_1\otimes\dots\otimes v_r,\quad v_i\in V.</math>
| |
| | |
| The '''antisymmetrization''' (or sometimes the '''skew-symmetrization''') of a decomposable tensor is defined by
| |
| | |
| :<math>\operatorname{Alt}(v_1\otimes\dots\otimes v_r) = \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} \operatorname{sgn}(\sigma) v_{\sigma(1)}\otimes\dots\otimes v_{\sigma(r)}</math>
| |
| | |
| 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 <math>\widehat{\otimes}</math> defined by
| |
| | |
| :<math>t \widehat{\otimes} s = \operatorname{Alt}(t\otimes s).</math>
| |
| | |
| 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
| |
| | |
| :<math>A(V)\cong \Lambda(V).</math>
| |
| | |
| ===Index notation===
| |
| Suppose that ''V'' has finite dimension ''n'', and that a basis '''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub> of ''V'' is given. then any alternating tensor {{nowrap|1=''t'' ∈ A<sup>''r''</sup>(''V'') ⊂ ''T''<sup>''r''</sup>(''V'')}} can be written in [[index notation]] as
| |
| | |
| :<math>t = t^{i_1i_2\dots i_r}\, {\mathbf e}_{i_1}\otimes {\mathbf e}_{i_2}\otimes\dots\otimes {\mathbf e}_{i_r}</math>
| |
| | |
| where ''t''<sup>''i''<sub>1</sub> ... ''i''<sub>''r''</sub></sup> is [[antisymmetric tensor|completely antisymmetric]] in its indices.
| |
| | |
| The exterior product of two alternating tensors ''t'' and ''s'' of ranks ''r'' and ''p'' is given by
| |
| | |
| :<math>t\widehat{\otimes} s = \frac{1}{(r+p)!}\sum_{\sigma\in {\mathfrak S}_{r+p}}\operatorname{sgn}(\sigma)t^{i_{\sigma(1)}\dots i_{\sigma(r)}}s^{i_{\sigma(r+1)}\dots i_{\sigma(r+p)}} {\mathbf e}_{i_1}\otimes {\mathbf e}_{i_2}\otimes\dots\otimes {\mathbf e}_{i_{r+p}}.</math>
| |
| | |
| The components of this tensor are precisely the skew part of the components of the tensor product {{nowrap|1=''s'' ⊗ ''t''}}, denoted by square brackets on the indices:
| |
| | |
| :<math>(t\widehat{\otimes} s)^{i_1\dots i_{r+p}} = t^{[i_1\dots i_r}s^{i_{r+1}\dots i_{r+p}]}.</math>
| |
| | |
| <!--For the interior product-->
| |
| The interior product may also be described in index notation as follows. Let <math>t = t^{i_0i_1\dots i_{r-1}}</math> be an antisymmetric tensor of rank ''r''. Then, for α ∈ ''V''<sup>*</sup>, ''i''<sub>α</sub>'''t''' is an alternating tensor of rank ''r'' − 1, given by
| |
| | |
| :<math>(i_\alpha t)^{i_1\dots i_{r-1}}=r\sum_{j=0}^n\alpha_j t^{ji_1\dots i_{r-1}}.</math>
| |
| | |
| where ''n'' is the dimension of ''V''.
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| | |
| ==Applications==
| |
| | |
| ===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 magnitude of the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix. This suggests that the determinant can be ''defined'' in terms of the exterior product of the column vectors. Likewise, the {{nowrap|1=''k''×''k''}} minors of a matrix can be defined by looking at the exterior products of column vectors chosen ''k'' at a time. These ideas can be extended not just to matrices but to [[linear transformation]]s as well: the magnitude of the determinant of a linear transformation is the factor by which it scales the 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.
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| ===Linear geometry===
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| The decomposable ''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 ''u'' and ''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 ''u'', ''v'', and ''w''.
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| ===Projective geometry===
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| Decomposable ''k''-vectors in Λ<sup>''k''</sup>''V'' correspond to weighted ''k''-dimensional [[linear subspace]]s of ''V''. In particular, the [[Grassmannian]] of ''k''-dimensional subspaces of ''V'', denoted ''Gr''<sub>k</sub>(''V''), can be naturally identified with an [[algebraic variety|algebraic subvariety]] of the [[projective space]] '''P'''(Λ<sup>k</sup>''V''). This is called the [[Plücker embedding]].
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| ===Differential geometry===
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| The exterior algebra has notable applications in [[differential geometry]], where it is used to define [[differential form]]s. 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 ''k'' is a [[linear functional]] on the ''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.
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| In particular, the [[exterior derivative]] gives the exterior algebra of differential forms on a manifold the structure of a [[differential 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.
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| ===Representation theory===
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| 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]].
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| ===Physics===
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| 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)]].
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| ===Lie algebra homology===
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| Let ''L'' be a Lie algebra over a field ''K'', then it is possible to define the structure of a [[chain complex]] on the exterior algebra of ''L''. This is a ''K''-linear mapping
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| :<math>\partial : \Lambda^{p+1}L\to\Lambda^pL</math>
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| defined on decomposable elements by
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| :<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>
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| The [[Jacobi identity]] holds if and only if ∂∂ = 0, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra ''L'' to be a Lie algebra. Moreover, in that case Λ''L'' is a [[chain complex]] with boundary operator ∂. The [[homology theory|homology]] associated to this complex is the [[Lie algebra homology]].
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| ===Homological algebra===
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| The exterior algebra is the main ingredient in the construction of the [[Koszul complex]], a fundamental object in [[homological algebra]].
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| == History ==
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| The exterior algebra was first introduced by [[Hermann Grassmann]] in 1844 under the blanket term of ''Ausdehnungslehre'', or ''Theory of Extension''.<ref>{{harvcoltxt|Kannenberg|2000}} published a translation of Grassmann's work in English; he translated ''Ausdehnungslehre'' as ''Extension Theory''.</ref>
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| 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]]. [[Adhémar Jean Claude Barré de Saint-Venant|Saint-Venant]] also published similar ideas of exterior calculus for which he claimed priority over Grassmann.<ref>J Itard, Biography in Dictionary of Scientific Biography (New York 1970–1990).</ref>
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| 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.<ref>Authors have in the past referred to this calculus variously as the ''calculus of extension'' ({{harvnb|Whitehead|1898}}; {{harvnb|Forder|1941}}), or ''extensive algebra'' {{harv|Clifford|1878}}, and recently as ''extended vector algebra'' {{harv|Browne|2007}}.</ref>
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| 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.
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| The import of this new theory of vectors and [[multivector]]s was lost to mid 19th century mathematicians,<ref>{{harvnb|Bourbaki|1989|p=661}}.</ref>
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| 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]], [[Élie Cartan]], and [[Gaston Darboux]]) who applied Grassmann's ideas to the calculus of [[differential form]]s.
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| 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.
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| ==See also==
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| *[[Symmetric algebra]], the symmetric analog
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| *[[Clifford algebra]], a [[quantization (physics)|quantum deformation]] of the exterior algebra by a [[quadratic form]]
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| *[[Weyl algebra]], a [[:Category:Mathematical quantization|quantum deformation]] of the symmetric algebra by a [[symplectic form]]
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| *[[Multilinear algebra]]
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| *[[Tensor algebra]]
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| *[[Geometric algebra]]
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| *[[Koszul complex]]
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| ===Mathematical references===
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| * {{citation|last1=Bishop|first1=R.|last2=Goldberg|first2=S.I.|title=Tensor analysis on manifolds|publisher=Dover|year=1980|isbn=0-486-64039-6}}
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| :: 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.
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| * {{citation|first = Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki | title = Elements of mathematics, Algebra I| publisher = Springer-Verlag | year = 1989|isbn=3-540-64243-9}}
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| :: 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.
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| * {{citation|first1=R.L.|last1=Bryant|first2=S.S.|last2=Chern|authorlink2=Shiing-Shen Chern|first3=R.B.|last3=Gardner|first4=H.L.|last4=Goldschmidt|first5=P.A.|last5=Griffiths|authorlink5=Philip A. Griffiths|title=Exterior differential systems|publisher=Springer-Verlag|year=1991}}
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| :: This book contains applications of exterior algebras to problems in [[partial differential equations]]. Rank and related concepts are developed in the early chapters.
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| * {{citation|first1=S.|last1=Mac Lane|authorlink1=Saunders Mac Lane|authorlink2=Garrett Birkhoff|last2=Birkhoff|first2=G.|title=Algebra|publisher=AMS Chelsea|year=1999|isbn=0-8218-1646-2}}
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| :: Chapter XVI sections 6–10 give a more elementary account of the exterior algebra, including duality, determinants and minors, and alternating forms.
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| * {{citation|authorlink=Shlomo Sternberg|last=Sternberg|first=Shlomo|title=Lectures on Differential Geometry|publisher=Prentice Hall|year=1964}}
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| :: Contains a classical treatment of the exterior algebra as alternating tensors, and applications to differential geometry.
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| ===Historical references===
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| * {{Citation|first = Nicolas|last = Bourbaki | chapter=Historical note on chapters II and III|title = Elements of mathematics, Algebra I| publisher = Springer-Verlag | year = 1989}}
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| * {{citation|last = Clifford|first=W.|authorlink=William Kingdon Clifford|title = Applications of Grassmann's Extensive Algebra | journal = American Journal of Mathematics | volume = 1 | issue = 4 | year = 1878 | pages = 350–358|doi = 10.2307/2369379|publisher = The Johns Hopkins University Press|jstor = 2369379}}
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| * {{citation|last = Forder|first = H. G. |title = The Calculus of Extension|year = 1941|publisher = Cambridge University Press}}
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| * {{citation|first = Hermann|last = Grassmann|authorlink=Hermann Grassmann|title = Die Lineale Ausdehnungslehre – Ein neuer Zweig der Mathematik |url = http://books.google.com/books?id=bKgAAAAAMAAJ&pg=PA1&dq=Die+Lineale+Ausdehnungslehre+ein+neuer+Zweig+der+Mathematik| year = 1844 }} (The Linear Extension Theory – A new Branch of Mathematics) [http://resolver.sub.uni-goettingen.de/purl?PPN534901565 alternative reference]
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| * {{citation|last = Kannenberg|first = Lloyd|title = Extension Theory (translation of Grassmann's ''Ausdehnungslehre'')|year = 2000|publisher = American Mathematical Society|isbn = 0-8218-2031-1}}
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| * {{citation|first = Giuseppe|last = Peano|authorlink=Giuseppe Peano | title = Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva | year = 1888 }}; {{citation|last=Kannenberg|first=Lloyd|year=1999|title=Geometric calculus: According to the Ausdehnungslehre of H. Grassmann|publisher=Birkhäuser|isbn=978-0-8176-4126-9}}.
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| * {{citation|first = Alfred North|last = Whitehead|authorlink=Alfred North Whitehead|title = a Treatise on Universal Algebra, with Applications|url = http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=01950001&seq=5|year = 1898|publisher = Cambridge}}
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| ===Other references and further reading===
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| <!--For works inessential to the article, though these may also have been referenced in passing.-->
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| * {{citation|first = J.M.|last = Browne|title = Grassmann algebra – Exploring applications of Extended Vector Algebra with Mathematica|year = 2007|publisher = [http://www.grassmannalgebra.info/grassmannalgebra/book/index.htm Published on line]}}
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| :: An introduction to the exterior algebra, and [[geometric algebra]], with a focus on applications. Also includes a history section and bibliography.
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| * {{citation|authorlink=Michael Spivak|last=Spivak|first=Michael|title=Calculus on manifolds|publisher=Addison-Wesley|year=1965|isbn=978-0-8053-9021-6}}
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| :: Includes applications of the exterior algebra to differential forms, specifically focused on [[integral|integration]] and [[Stokes's theorem]]. The notation Λ<sup>''k''</sup>''V'' in this text is used to mean the space of alternating ''k''-forms on ''V''; i.e., for Spivak Λ<sup>''k''</sup>''V'' is what this article would call Λ<sup>''k''</sup>''V''*. Spivak discusses this in Addendum 4.
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| * {{citation|authorlink=Gilbert Strang|last=Strang|first=G.|title=Introduction to linear algebra|publisher=Wellesley-Cambridge Press|year=1993|isbn=978-0-9614088-5-5}}
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| :: Includes an elementary treatment of the axiomatization of determinants as signed areas, volumes, and higher-dimensional volumes.
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| * {{springer|id=E/e037080|title=Exterior algebra|author=Onishchik, A.L.}}
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| * Wendell H. Fleming (1965) ''Functions of Several Variables'', [[Addison-Wesley]].
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| :: Chapter 6: Exterior algebra and differential calculus, pages 205–38. This textbook in [[multivariate calculus]] introduces the exterior algebra of differential forms adroitly into the calculus sequence for colleges.
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| * {{citation|first = S.|last = Winitzki|title = Linear Algebra via Exterior Products|year = 2010|publisher = [http://sites.google.com/site/winitzki/linalg Published on line]}}
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| :: An introduction to the coordinate-free approach in basic finite-dimensional linear algebra, using exterior products.
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| * {{cite book
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| | last = Shafarevich
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| | first = I. R.
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| | authorlink = Igor Shafarevich
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| | coauthors = A. O. Remizov
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| | title = Linear Algebra and Geometry
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | year = 2012
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| | url = http://www.springer.com/mathematics/algebra/book/978-3-642-30993-9
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| | isbn = 978-3-642-30993-9}}
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| :: Chapter 10: The Exterior Product and Exterior Algebras
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| * [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_grassman_and_proj._geom..pdf "The Grassmann method in projective geometry"] A compilation of English translations of three notes by Cesare Burali-Forti on the application of exterior algebra to projective geometry
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| * [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_diff._geom._following_grassmann.pdf C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"] An English translation of an early book on the geometric applications of exterior algebras
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| * [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/grassmann_-_mechanics_and_extensions.pdf "Mechanics, according to the principles of the theory of extension"] An English translation of one Grassmann's papers on the applications of exterior algebra
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| {{tensors}}
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| [[Category:Algebras]]
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| [[Category:Multilinear algebra]]
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| [[Category:Differential forms]]
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