Editing Exterior algebra (section)

== Algebraic properties ==

=== Alternating product ===
The exterior product is by construction [[Alternating algebra|''alternating'']] on elements of {{tmath|V}}, which means that <math> x \wedge x = 0 </math> for all <math> x \in V, </math> by the above construction.  It follows that the product is also [[anticommutative]] on elements of {{tmath|V}}, for supposing that {{tmath|x, y \in V}},
: <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>
hence
: <math> x \wedge y = -(y \wedge x). </math>

More generally, if <math>\sigma</math> is a [[permutation group|permutation]] of the integers {{tmath|[1, \dots, k]}}, and {{tmath|x_1}}, {{tmath|x_2}}, ..., {{tmath|x_k}} are elements of {{tmath|V}}, it follows that
: <math>
x_{\sigma(1)} \wedge x_{\sigma(2)} \wedge \cdots \wedge x_{\sigma(k)}
= \sgn(\sigma)x_1 \wedge x_2 \wedge \cdots \wedge x_k,
</math>
where <math>\sgn(\sigma)</math> is the [[signature of a permutation|signature of the permutation]] {{tmath|\sigma}}.<ref>A proof of this can be found in more generality in {{harvtxt|Bourbaki|1989}}.</ref>

In particular, if <math>x_i = x_j</math> for some {{tmath|i \ne j}}, then the following generalization of the alternating property also holds:
: <math> x_{1} \wedge x_{2} \wedge \cdots \wedge x_{k} = 0. </math>

Together with the distributive property of the exterior product, one further generalization is that a necessary and sufficient condition for <math> \{ x_{1}, x_{2}, \dots, x_{k} \} </math> to be a linearly dependent set of vectors is that
: <math> x_{1} \wedge x_{2} \wedge \cdots \wedge x_{k} = 0. </math>

=== Exterior power ===
The {{math|''k''}}th '''exterior power''' of {{tmath|V}}, denoted {{tmath|{\textstyle\bigwedge}^{\!k}(V)}}, is the [[vector subspace]] of {{tmath|{\textstyle\bigwedge}(V)}} [[linear span|spanned]] by elements of the form
: <math>x_1 \wedge x_2 \wedge \cdots \wedge x_k,\quad x_i \in V, i=1,2, \dots, k .</math>

If {{tmath|\alpha \in {\textstyle\bigwedge}^{\!k}(V)}}, then <math>\alpha</math> is said to be a '''[[p-vector|{{math|''k''}}-vector]]'''.  If, furthermore, <math>\alpha</math> can be expressed as an exterior product of <math>k</math> elements of {{tmath|V}}, then <math>\alpha</math> is said to be '''decomposable''' (or '''simple''', by some authors; or a '''blade''', by others).  Although decomposable {{tmath|k}}-vectors span {{tmath|{\textstyle\bigwedge}^{\!k}(V)}}, not every element of <math>{\textstyle\bigwedge}^{\!k}(V)</math> is decomposable.  For example, given {{tmath|\mathbf{R}^4}} with a basis {{tmath|1= \{ e_1, e_2, e_3, e_4 \} }}, the following 2-vector is not decomposable:
: <math> \alpha = e_1 \wedge e_2 + e_3 \wedge e_4. </math>

==== Basis and dimension ====
If the [[dimension (linear algebra)|dimension]] of <math>V</math> is <math>n</math> and <math>\{e_1,\dots,e_n\}</math> is a [[basis (linear algebra)|basis]] for <math>V</math>, then the set
: <math>
\{\,e_{i_1} \wedge e_{i_2} \wedge \cdots \wedge e_{i_k} ~
\big| ~~ 1 \le i_1 < i_2 < \cdots < i_k \le n \,\}
</math>
is a basis for {{tmath|{\textstyle\bigwedge}^{\!k}(V)}}.  The reason is the following: given any exterior product of the form
: <math> v_1 \wedge \cdots \wedge v_k , </math>
every vector <math>v_j</math> can be written as a [[linear combination]] of the basis vectors {{tmath|e_i}}; 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 {{mvar|k}}-vectors can be computed as the [[minor (linear algebra)|minor]]s of the [[matrix (mathematics)|matrix]] that describes the vectors <math>v_j</math> in terms of the basis {{tmath|e_i}}.

By counting the basis elements, the dimension of <math>{\textstyle\bigwedge}^{\!k}(V)</math> is equal to a [[binomial coefficient]]:
: <math> \dim {\textstyle\bigwedge}^{\!k}(V) = \binom{n}{k} ,</math>
where {{tmath|n}} is the dimension of the ''vectors'', and {{tmath|k}} is the number of vectors in the product.  The binomial coefficient produces the correct result, even for exceptional cases; in particular, <math>{\textstyle\bigwedge}^{\!k}(V) = \{ 0 \}</math> for {{tmath|k > n}}.

Any element of the exterior algebra can be written as a sum of [[p-vector|{{math|''k''}}-vector]]s.  Hence, as a vector space the exterior algebra is a [[Direct sum of modules|direct sum]]
: <math>
{\textstyle\bigwedge}(V)
= {\textstyle\bigwedge}^{\!0}(V)
  \oplus {\textstyle\bigwedge}^{\!1}(V)
  \oplus {\textstyle\bigwedge}^{\!2}(V)
  \oplus \cdots \oplus {\textstyle\bigwedge}^{\!n}(V)
</math>
(where, by convention, {{tmath|1={\textstyle\bigwedge}^{\!0}(V) = K}}, the [[field (mathematics)|field]] underlying {{tmath|V}}, and {{tmath|1={\textstyle\bigwedge}^{\!1}(V) = V}}), and therefore its dimension is equal to the sum of the binomial coefficients, which is {{tmath|2^n}}.

==== Rank of a ''k''-vector ====
If {{tmath|\alpha \in {\textstyle\bigwedge}^{\!k}(V)}}, then it is possible to express <math>\alpha</math> as a linear combination of decomposable [[p-vector|{{math|''k''}}-vector]]s:
: <math> \alpha = \alpha^{(1)} + \alpha^{(2)} + \cdots + \alpha^{(s)} </math>
where each <math>\alpha^{(i)}</math> is decomposable, say
: <math>
\alpha^{(i)}
= \alpha^{(i)}_1 \wedge \cdots \wedge \alpha^{(i)}_k,\quad i 
= 1,2,\ldots, s.
</math>

The '''rank''' of the {{math|''k''}}-vector <math>\alpha</math> is the minimal number of decomposable {{math|''k''}}-vectors in such an expansion of {{tmath|\alpha}}.  This is similar to the notion of [[tensor rank]].

Rank is particularly important in the study of 2-vectors {{harv|Sternberg|1964|loc=§III.6}} {{harv|Bryant|Chern|Gardner|Goldschmidt|1991}}.  The rank of a 2-vector <math>\alpha</math> can be identified with half the [[rank of a matrix|rank of the matrix]] of coefficients of <math>\alpha</math> in a basis.  Thus if <math>e_i</math> is a basis for {{tmath|V}}, then <math>\alpha</math> can be expressed uniquely as
: <math> \alpha = \sum_{i,j}a_{ij}e_i \wedge e_j </math>
where <math>a_{ij} = -a_{ji}</math> (the matrix of coefficients is [[skew-symmetric matrix|skew-symmetric]]).  The rank of the matrix <math>a_{ij}</math> is therefore even, and is twice the rank of the form <math>\alpha</math>.

In characteristic 0, the 2-vector <math>\alpha</math> has rank <math>p</math> if and only if
: <math>
\underset{p}{\underbrace{\alpha \wedge \cdots \wedge \alpha}} \neq 0 \ 
</math> and <math>
\ \underset{p+1}{\underbrace{\alpha \wedge \cdots \wedge \alpha}} = 0.
</math>

=== Graded structure ===
The exterior product of a {{math|''k''}}-vector with a {{math|''p''}}-vector is a <math>(k + p)</math>-vector, once again invoking bilinearity.  As a consequence, the direct sum decomposition of the preceding section
: <math>
{\textstyle\bigwedge}(V) = {\textstyle\bigwedge}^{\!0}(V) \oplus {\textstyle\bigwedge}^{\!1}(V) \oplus {\textstyle\bigwedge}^{\!2}(V) \oplus \cdots \oplus {\textstyle\bigwedge}^{\!n}(V)
</math>
gives the exterior algebra the additional structure of a [[graded algebra]], that is
: <math>
{\textstyle\bigwedge}^{\!k}(V)
\wedge {\textstyle\bigwedge}^{\!p}(V)
\sub {\textstyle\bigwedge}^{\!k+p}(V).
</math>

Moreover, if {{math|''K''}} is the base field, we have
: <math>
{\textstyle\bigwedge}^{\!0}(V) = K
</math> and <math>
{\textstyle\bigwedge}^{\!1}(V) = V.
</math>

The exterior product is graded anticommutative, meaning that if <math>\alpha \in {\textstyle\bigwedge}^{\!k}(V) </math> and {{tmath|\beta \in {\textstyle\bigwedge}^{\!p}(V)}}, then
: <math> \alpha \wedge \beta = (-1)^{kp}\beta \wedge \alpha. </math>

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).

=== Universal property ===
Let {{math|''V''}} be a vector space over the field {{math|''K''}}.  Informally, multiplication in <math> {\textstyle\bigwedge}(V) </math> is performed by manipulating symbols and imposing a [[distributive law]], an [[associative law]], and using the identity <math> v \wedge v = 0 </math> for {{math|''v'' ∈ ''V''}}.  Formally, <math> {\textstyle\bigwedge}(V) </math> is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative {{math|''K''}}-algebra containing {{math|''V''}} with alternating multiplication on {{math|''V''}} must contain a homomorphic image of {{tmath|{\textstyle\bigwedge}(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>

<div style="margin-left: 2em; margin-right: 2em">
Given any unital associative {{math|''K''}}-algebra {{math|''A''}} and any {{math|''K''}}-[[linear map]] <math> j : V \to A </math> such that <math> j(v)j(v) = 0 </math> for every {{math|''v''}} in {{math|''V''}}, then there exists ''precisely one'' unital [[algebra homomorphism]] <math>f : {\textstyle\bigwedge}(V)\to A </math> such that {{math|1=''j''(''v'') = ''f''(''i''(''v''))}} for all {{math|''v''}} in {{math|''V''}} (here {{math|''i''}} is the natural inclusion of {{math|''V''}} in {{tmath|{\textstyle\bigwedge}(V)}}, see above).
</div>

[[File:ExteriorAlgebra-01.svg|center|150px|Universal property of the exterior algebra]]

To construct the most general algebra that contains {{math|''V''}} and whose multiplication is alternating on {{math|''V''}}, it is natural to start with the most general associative algebra that contains {{math|''V''}}, the [[tensor algebra]] {{math|''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]] {{math|''I''}} in {{math|''T''(''V'')}} generated by all elements of the form {{math|''v'' ⊗ ''v''}} for {{math|''v''}} in {{math|''V''}}, and define <math> {\textstyle\bigwedge}(V) </math> as the quotient
: <math> {\textstyle\bigwedge}(V) = T(V)\,/\,I</math>
(and use {{math|∧}} as the symbol for multiplication in {{tmath|{\textstyle\bigwedge}(V)}}).  It is then straightforward to show that <math>{\textstyle\bigwedge}(V)</math> contains {{math|''V''}} and satisfies the above universal property.

As a consequence of this construction, the operation of assigning to a vector space {{math|''V''}} its exterior algebra <math>{\textstyle\bigwedge}(V)</math> is a [[functor]] from the [[category (mathematics)|category]] of vector spaces to the category of algebras.

Rather than defining <math>{\textstyle\bigwedge}(V)</math> first and then identifying the exterior powers <math>{\textstyle\bigwedge}^{\!k}(V)</math> as certain subspaces, one may alternatively define the spaces <math>{\textstyle\bigwedge}^{\!k}(V)</math> first and then combine them to form the algebra {{tmath|{\textstyle\bigwedge}(V)}}.  This approach is often used in differential geometry and is described in the next section.

=== Generalizations ===
Given a [[commutative ring]] <math>R</math> and an <math>R</math>-[[module (mathematics)|module]] {{tmath|M}}, we can define the exterior algebra <math>{\textstyle\bigwedge}(M)</math> just as above, as a suitable quotient of the tensor algebra {{tmath|\mathrm{T}(M)}}.  It will satisfy the analogous universal property.  Many of the properties of <math>{\textstyle\bigwedge}(M)</math> also require that <math>M</math> be a [[projective module]].  Where finite dimensionality is used, the properties further require that <math>M</math> be [[finitely generated module|finitely generated]] and projective.  Generalizations to the most common situations can be found in {{harvtxt|Bourbaki|1989}}.

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.