Definition as a quotient of the tensor algebra

Let ${\displaystyle V}$ be a vector space over a field. Let ${\displaystyle \mathrm {T} (V)}$ be the tensor algebra ${\displaystyle V}$. As a vector space, ${\displaystyle \mathrm {T} (V)}$ is spanned by symbols ${\displaystyle v_{1}\otimes \dots \otimes v_{k}}$ for ${\displaystyle k\geq 0}$ and ${\displaystyle v_{i}\in V}$, and the only relations are those specifying these objects be linear in each variable ${\displaystyle v_{i}}$.

Definition. The exterior algebra ${\displaystyle {\textstyle \bigwedge }(V)}$ of the vector space ${\displaystyle V}$ is the quotient

${\displaystyle {\textstyle \bigwedge }(V)\ =\ \mathrm {T} (V)\,/\,\langle v\otimes v~|~v\in V\rangle ,}$

that is, the quotient of the tensor algebra ${\displaystyle \mathrm {T} (V)}$ by the two-sided ideal that is generated by elements of the form ${\displaystyle v\otimes v}$.

Denote the equivalence class of ${\displaystyle v_{1}\otimes \dots \otimes v_{k}}$ in the quotient ${\displaystyle {\textstyle \bigwedge }(V)}$ as ${\displaystyle v_{1}\wedge \dots \wedge v_{k}}$.

${\displaystyle {\textstyle \bigwedge }(V)}$ inherits the structure of a graded algebra from ${\displaystyle \mathrm {T} (V)}$. An element of ${\displaystyle {\textstyle \bigwedge }(V)}$ may be written (non-uniquely) as a finite sum

${\displaystyle v_{1}\wedge \dots \wedge v_{k_{1}}+w_{1}\wedge \dots \wedge w_{k_{2}}+\dots +z_{1}\wedge \dots \wedge z_{k_{n}},}$

where each letter is an element of ${\displaystyle V}$, and ${\displaystyle k_{i}=k}$, ${\displaystyle n\geq 0}$. Such an element is called a k-vector if ${\displaystyle k_{i}=k}$ for all ${\displaystyle i}$, in which case its rank is the minimum ${\displaystyle n}$ for which it can be written in this form. ${\displaystyle k}$-vectors of rank 1, i.e. of the form

${\displaystyle v_{1}\wedge \dots \wedge v_{k},}$

are called k-blades.

Definition in terms of a basis

Let ${\displaystyle e_{1},\dots ,e_{n}}$ be a basis of ${\displaystyle V}$.

Definition. The exterior algebra ${\displaystyle {\textstyle \bigwedge }(V)}$ of the vector space ${\displaystyle V}$ is the vector space with basis formal symbols

${\displaystyle e_{i_{1}}\wedge \dots \wedge e_{i_{a}},}$

where ${\displaystyle a\geq 0}$ and ${\displaystyle i_{1}<\dots <i_{a}}$ is a strictly increasing sequence. The exterior product is

${\displaystyle (e_{i_{1}}\wedge \dots \wedge e_{i_{a}})\wedge (e_{j_{1}}\wedge \dots \wedge e_{j_{b}})=(0{\text{ or }}\pm e_{k_{1}}\wedge \dots \wedge e_{k_{a+b}}),}$

where in the first case the sequences ${\displaystyle i}$ and ${\displaystyle j}$ have an element in common. If they do not, ${\displaystyle k}$ is formed by taking ${\displaystyle i_{1}\dots i_{a}j_{1}\dots j_{b}}$ and permuting elements until the sequence is increasing, each swap of adjacent elements flipping the sign ${\displaystyle \pm }$.

Areas in the plane

The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices.

The Cartesian plane ${\displaystyle \mathbf {R} ^{2}}$ is a real vector space equipped with a basis consisting of a pair of unit vectors

${\displaystyle {\mathbf {e} }_{1}={\begin{bmatrix}1\\0\end{bmatrix}},\quad {\mathbf {e} }_{2}={\begin{bmatrix}0\\1\end{bmatrix}},}$ with the orientation ${\displaystyle \mathbf {e} _{1}\times \mathbf {e} _{2}}$ and with the metric ${\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}.}$

Suppose that

${\displaystyle \mathbf {v} ={\begin{bmatrix}a\\b\end{bmatrix}}=a\mathbf {e} _{1}+b\mathbf {e} _{2},\quad \mathbf {w} ={\begin{bmatrix}c\\d\end{bmatrix}}=c\mathbf {e} _{1}+d\mathbf {e} _{2}}$

are a pair of given vectors in ${\displaystyle \mathbf {R} ^{2}}$, written in components. There is a unique parallelogram having ${\displaystyle \mathbf {v} }$ and ${\displaystyle \mathbf {w} }$ as two of its sides. The area of this parallelogram is given by the standard determinant formula:

${\displaystyle {\text{Area}}={\Bigl |}\det {\begin{bmatrix}\mathbf {v} &\mathbf {w} \end{bmatrix}}{\Bigr |}={\Biggl |}\det {\begin{bmatrix}a&c\\b&d\end{bmatrix}}{\Biggr |}=\left|ad-bc\right|.}$

Consider now the exterior product of ${\displaystyle \mathbf {v} }$ and ${\displaystyle \mathbf {w} }$:

${\displaystyle {\begin{aligned}\mathbf {v} \wedge \mathbf {w} &=(a\mathbf {e} _{1}+b\mathbf {e} _{2})\wedge (c\mathbf {e} _{1}+d\mathbf {e} _{2})\\&=ac\mathbf {e} _{1}\wedge \mathbf {e} _{1}+ad\mathbf {e} _{1}\wedge \mathbf {e} _{2}+bc\mathbf {e} _{2}\wedge \mathbf {e} _{1}+bd\mathbf {e} _{2}\wedge \mathbf {e} _{2}\\&=\left(ad-bc\right)\mathbf {e} _{1}\wedge \mathbf {e} _{2}\end{aligned}}}$

where the first step uses the distributive law for the exterior product, and the last uses the fact that the exterior product is an alternating map, and in particular ${\displaystyle \mathbf {e} _{2}\wedge \mathbf {e} _{1}=-(\mathbf {e} _{1}\wedge \mathbf {e} _{2}).}$ (The fact that the exterior product is an alternating map also forces ${\displaystyle \mathbf {e} _{1}\wedge \mathbf {e} _{1}=\mathbf {e} _{2}\wedge \mathbf {e} _{2}=0.}$) Note that the coefficient in this last expression is precisely the determinant of the matrix [v w]. The fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the signed area of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation.

The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if A(v, w) denotes the signed area of the parallelogram of which the pair of vectors v and w form two adjacent sides, then A must satisfy the following properties:

1. A(rv, sw) = rsA(v, w) for any real numbers r and s, 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).
2. A(v, v) = 0, since the area of the degenerate parallelogram determined by v (i.e., a line segment) is zero.
3. A(w, v) = −A(v, w), since interchanging the roles of v and w reverses the orientation of the parallelogram.
4. A(v + rw, w) = A(v, w) for any real number r, since adding a multiple of w to v affects neither the base nor the height of the parallelogram and consequently preserves its area.
5. A(e₁, e₂) = 1, since the area of the unit square is one.

The cross product (blue vector) in relation to the exterior product (light blue parallelogram). The length of the cross product is to the length of the parallel unit vector (red) as the size of the exterior product is to the size of the reference parallelogram (light red).

With the exception of the last property, the exterior product of two vectors satisfies the same 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 chosen parallelogram in a parallel plane (here, the one with sides e₁ and e₂). In other words, the exterior product provides a basis-independent formulation of area.^[2]

Oriented areas in space

The power of linearity of the cross product, implicitly, of the exterior product, is seen in a simple geometric property, not that obvious: the total signed/oriented area of a non self overlapping polyhedron is zero. This is just the simplest linear algebra, if using exterior product, but quite a lot of work, using metric geometry:

${\displaystyle (\mathbf {u} -\mathbf {v} )\wedge (\mathbf {w} -\mathbf {v} )+\mathbf {u} \wedge \mathbf {v} +\mathbf {w} \wedge \mathbf {u} +\mathbf {v} \wedge \mathbf {w} =0}$

Each blade in the sum above is (twice) the oriented (either all inward, or all outward) surface area of the four faces of a tetrahedron with concurrent edges along vectors ${\displaystyle \mathbf {u} ,\mathbf {v} ,\mathbf {w} }$. Iterating this property for each and all tetrahedrons that build a non self overlapping polyhedron, we get the general statement.

Oriented volume in affine space

The natural setting for (oriented) ${\displaystyle k}$-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 ${\displaystyle \mathbb {A} }$ is an affine space over the vector space ${\displaystyle V}$, and a (simplex) collection of ordered ${\displaystyle k+1}$ points ${\displaystyle A_{0},A_{1},...,A_{k}}$, we can define its oriented ${\displaystyle k}$-dimensional volume as the exterior product of vectors ${\displaystyle A_{0}A_{1}\wedge A_{0}A_{2}\wedge \cdots \wedge A_{0}A_{k}={}}$ ${\displaystyle (-1)^{j}A_{j}A_{0}\wedge A_{j}A_{1}\wedge A_{j}A_{2}\wedge \cdots \wedge A_{j}A_{k}}$ (using concatenation ${\displaystyle PQ}$ to mean the displacement vector from point ${\displaystyle P}$ to ${\displaystyle Q}$); if the order of the points is changed, the oriented volume changes by a sign, according to the parity of the permutation. In ${\displaystyle n}$-dimensional space, the volume of any ${\displaystyle n}$-dimensional simplex is a scalar multiple of any other.

The sum of the ${\displaystyle (k-1)}$-dimensional oriented areas of the boundary simplexes of a ${\displaystyle k}$-dimensional simplex is zero, as for the sum of vectors around a triangle or the oriented triangles bounding the tetrahedron in the previous section. This remains true if the simplex is "flat" (is contained in an ${\displaystyle (k-1)}$-dimensional affine subspace, as 4 points in a plane).

Orientation defined by an ordered set of vectors.

Reversed orientation corresponds to negating the exterior product.

Geometric interpretation of grade n elements in a real exterior algebra for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-parallelotope, n-ellipsoid); with magnitude (hypervolume), and orientation defined by that of its (n − 1)-dimensional boundary and on which side the interior is.^[3][4]

The vector space structure on ${\displaystyle {\textstyle \bigwedge }(V)}$ generalises addition of vectors in ${\displaystyle V}$: we have ${\displaystyle (u_{1}+u_{2})\wedge v=u_{1}\wedge v+u_{2}\wedge v}$ and similarly a k-blade ${\displaystyle v_{1}\wedge \dots \wedge v_{k}}$ is linear in each factor.

Cross and triple products

Take a 3-dimensional vector space, e.g. ${\displaystyle \mathbf {R} ^{3}}$, and choose a basis ${\displaystyle (\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3})}$. The exterior product of a pair of vectors

${\displaystyle \mathbf {u} =u_{1}\mathbf {e} _{1}+u_{2}\mathbf {e} _{2}+u_{3}\mathbf {e} _{3}\quad {\text{ and }}\quad \mathbf {v} =v_{1}\mathbf {e} _{1}+v_{2}\mathbf {e} _{2}+v_{3}\mathbf {e} _{3}}$

is

${\displaystyle \mathbf {u} \wedge \mathbf {v} =(u_{1}v_{2}-u_{2}v_{1})(\mathbf {e} _{1}\wedge \mathbf {e} _{2})+(u_{2}v_{3}-u_{3}v_{2})(\mathbf {e} _{2}\wedge \mathbf {e} _{3})+(u_{3}v_{1}-u_{1}v_{3})(\mathbf {e} _{3}\wedge \mathbf {e} _{1}).}$

If we use the basis ${\displaystyle (\mathbf {e} _{1}\wedge \mathbf {e} _{2},\mathbf {e} _{2}\wedge \mathbf {e} _{3},\mathbf {e} _{3}\wedge \mathbf {e} _{1})}$ to identify ${\displaystyle {\textstyle \bigwedge }^{\!2}(\mathbf {R} ^{3})}$ with ${\displaystyle \mathbf {R} ^{3}}$, this becomes the usual formula for cross product of the two vectors.

Bringing in a third vector

${\displaystyle \mathbf {w} =w_{1}\mathbf {e} _{1}+w_{2}\mathbf {e} _{2}+w_{3}\mathbf {e} _{3},}$

the exterior product of three vectors is

${\displaystyle \mathbf {u} \wedge \mathbf {v} \wedge \mathbf {w} =(u_{1}v_{2}w_{3}+u_{2}v_{3}w_{1}+u_{3}v_{1}w_{2}-u_{1}v_{3}w_{2}-u_{2}v_{1}w_{3}-u_{3}v_{2}w_{1})(\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3}).}$

Using the basis ${\displaystyle (\mathbf {e} _{1}\wedge \mathbf {e} _{2}\wedge \mathbf {e} _{3})}$ to identify ${\displaystyle {\textstyle \bigwedge }^{\!3}(\mathbf {R} ^{3})}$ with ${\displaystyle \mathbf {R} }$, this becomes the usual formula for triple product of the three vectors.

Electromagnetic field

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

Adding the Lorentz metric and an orientation provides the Hodge star operator ${\displaystyle \star }$ and thus makes it possible to define ${\displaystyle J={\star }d{\star }F}$ or the equivalent tensor divergence ${\displaystyle J^{i}=F_{,j}^{ij}=F_{;j}^{ij}}$ where ${\displaystyle F^{ij}=g^{ik}g^{jl}F_{kl}.}$

Plücker embedding

The wedge product was introduced originally as an algebraic construction to study k-dimensional subspaces of ${\displaystyle \mathbf {R} _{n}}$, which forms a manifold ${\displaystyle \operatorname {Gr} (k,n)}$, which is called the Grassmannian. It lives inside the projectivisation of the space of multivectors

${\displaystyle \operatorname {Gr} (k,n)\ \subseteq \ \mathbf {P} ({\textstyle \bigwedge }^{\!k}(\mathbf {R} ^{n})).}$

This map is called the Plücker embedding, and its image is defined by the Plücker relations.

Differential forms

The definition of the exterior algebra can be extended to any module over a commutative ring, or more generally a sheaf of modules over a ringed space. For instance, differential forms over a manifold ${\displaystyle X}$

${\displaystyle {\textstyle \bigwedge }_{{\mathcal {O}}_{X}}(\Omega _{X}^{1})}$

is the exterior algebra on the sheaf of one-forms ${\displaystyle \Omega _{X}}$, which is a sheaf of modules over the sheaf of functions ${\displaystyle {\mathcal {O}}_{X}}$.

Alternating product

The exterior product is by construction alternating on elements of ${\displaystyle V}$, which means that ${\displaystyle x\wedge x=0}$ for all ${\displaystyle x\in V,}$ by the above construction. It follows that the product is also anticommutative on elements of ${\displaystyle V}$, for supposing that ${\displaystyle x,y\in V}$,

${\displaystyle 0=(x+y)\wedge (x+y)=x\wedge x+x\wedge y+y\wedge x+y\wedge y=x\wedge y+y\wedge x}$

hence

${\displaystyle x\wedge y=-(y\wedge x).}$

More generally, if ${\displaystyle \sigma }$ is a permutation of the integers ${\displaystyle [1,\dots ,k]}$, and ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$, ..., ${\displaystyle x_{p}}$ are elements of ${\displaystyle V}$, it follows that

${\displaystyle x_{\sigma (1)}\wedge x_{\sigma (2)}\wedge \cdots \wedge x_{\sigma (k)}=\operatorname {sgn}(\sigma )x_{1}\wedge x_{2}\wedge \cdots \wedge x_{k},}$

where ${\displaystyle \operatorname {sgn}(\sigma )}$ is the signature of the permutation ${\displaystyle \sigma }$.^[5]

In particular, if ${\displaystyle x_{i}=x_{j}}$ for some ${\displaystyle i\neq j}$, then the following generalization of the alternating property also holds:

${\displaystyle x_{1}\wedge x_{2}\wedge \cdots \wedge x_{k}=0.}$

Together with the distributive property of the exterior product, one further generalization is that if and only if ${\displaystyle \{x_{1},x_{2},\dots ,x_{k}\}}$ is a linearly dependent set of vectors, then

${\displaystyle x_{1}\wedge x_{2}\wedge \cdots \wedge x_{k}=0.}$

Exterior power

The kth exterior power of ${\displaystyle V}$, denoted ${\displaystyle {\textstyle \bigwedge }^{\!k}(V)}$, is the vector subspace of ${\displaystyle {\textstyle \bigwedge }(V)}$ spanned by elements of the form

${\displaystyle x_{1}\wedge x_{2}\wedge \cdots \wedge x_{k},\quad x_{i}\in V,i=1,2,\dots ,k.}$

If ${\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}$, then ${\displaystyle \alpha }$ is said to be a k-vector. If, furthermore, ${\displaystyle \alpha }$ can be expressed as an exterior product of ${\displaystyle k}$ elements of ${\displaystyle V}$, then ${\displaystyle \alpha }$ is said to be decomposable (or simple, by some authors; or a blade, by others). Although decomposable ${\displaystyle k}$-vectors span ${\displaystyle {\textstyle \bigwedge }^{\!k}(V)}$, not every element of ${\displaystyle {\textstyle \bigwedge }^{\!k}(V)}$ is decomposable. For example, in ${\displaystyle \mathbf {R} ^{4}}$, the following 2-vector is not decomposable:

${\displaystyle \alpha =e_{1}\wedge e_{2}+e_{3}\wedge e_{4}.}$

(This is a symplectic form, since ${\displaystyle \alpha \wedge \alpha \neq 0}$.^[6]) Note that being decomposable is crucial property in projective geometry and it is the outcome of the Plücker constraints (see also Plücker coordinates).

Graded structure

The exterior product of a k-vector with a p-vector is a ${\displaystyle (k+p)}$-vector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section

${\displaystyle {\textstyle \bigwedge }(V)={\textstyle \bigwedge }^{\!0}(V)\oplus {\textstyle \bigwedge }^{\!1}(V)\oplus {\textstyle \bigwedge }^{\!2}(V)\oplus \cdots \oplus {\textstyle \bigwedge }^{\!n}(V)}$

gives the exterior algebra the additional structure of a graded algebra, that is

${\displaystyle {\textstyle \bigwedge }^{\!k}(V)\wedge {\textstyle \bigwedge }^{\!p}(V)\subset {\textstyle \bigwedge }^{\!k+p}(V).}$

Moreover, if K is the base field, we have

${\displaystyle {\textstyle \bigwedge }^{\!0}(V)=K}$ and ${\displaystyle {\textstyle \bigwedge }^{\!1}(V)=V.}$

The exterior product is graded anticommutative, meaning that if ${\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}$ and ${\displaystyle \beta \in {\textstyle \bigwedge }^{\!p}(V)}$, then

${\displaystyle \alpha \wedge \beta =(-1)^{kp}\beta \wedge \alpha .}$

In addition to studying the graded structure on the exterior algebra, 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 V be a vector space over the field K. Informally, multiplication in ${\displaystyle {\textstyle \bigwedge }(V)}$ is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity ${\displaystyle v\wedge v=0}$ for v ∈ V. Formally, ${\displaystyle {\textstyle \bigwedge }(V)}$ 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 ${\displaystyle {\textstyle \bigwedge }(V)}$. In other words, the exterior algebra has the following universal property:^[7]

Universal property of the exterior algebra

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 associative algebra that contains V, the tensor algebra T(V), and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided ideal I in T(V) generated by all elements of the form v ⊗ v for v in V, and define ${\displaystyle {\textstyle \bigwedge }(V)}$ as the quotient

${\displaystyle {\textstyle \bigwedge }(V)=T(V)\,/\,I}$

(and use ∧ as the symbol for multiplication in ${\displaystyle {\textstyle \bigwedge }(V)}$). It is then straightforward to show that ${\displaystyle {\textstyle \bigwedge }(V)}$ contains V and satisfies the above universal property.

As a consequence of this construction, the operation of assigning to a vector space V its exterior algebra ${\displaystyle {\textstyle \bigwedge }(V)}$ is a functor from the category of vector spaces to the category of algebras.

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

Generalizations

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

Exterior algebras of vector bundles 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 sheaves of modules.

Index notation

Suppose that V has finite dimension n, and that a basis e₁, ..., e_n of V is given. Then any alternating tensor t ∈ A^r(V) ⊂ T^r(V) can be written in index notation as

${\displaystyle t=t^{i_{1}i_{2}\cdots i_{r}}\,{\mathbf {e} }_{i_{1}}\otimes {\mathbf {e} }_{i_{2}}\otimes \cdots \otimes {\mathbf {e} }_{i_{r}},}$

where t^{i₁⋅⋅⋅i_r} is completely antisymmetric in its indices.

The exterior product of two alternating tensors t and s of ranks r and p is given by

${\displaystyle t~{\widehat {\otimes }}~s={\frac {1}{(r+p)!}}\sum _{\sigma \in {\mathfrak {S}}_{r+p}}\operatorname {sgn} (\sigma )t^{i_{\sigma (1)}\cdots i_{\sigma (r)}}s^{i_{\sigma (r+1)}\cdots i_{\sigma (r+p)}}{\mathbf {e} }_{i_{1}}\otimes {\mathbf {e} }_{i_{2}}\otimes \cdots \otimes {\mathbf {e} }_{i_{r+p}}.}$

The components of this tensor are precisely the skew part of the components of the tensor product s ⊗ t, denoted by square brackets on the indices:

${\displaystyle (t~{\widehat {\otimes }}~s)^{i_{1}\cdots i_{r+p}}=t^{[i_{1}\cdots i_{r}}s^{i_{r+1}\cdots i_{r+p}]}.}$

The interior product may also be described in index notation as follows. Let ${\displaystyle t=t^{i_{0}i_{1}\cdots i_{r-1}}}$ be an antisymmetric tensor of rank ${\displaystyle r}$. Then, for α ∈ V^∗, ${\displaystyle \iota _{\alpha }t}$ is an alternating tensor of rank ${\displaystyle r-1}$, given by

${\displaystyle (i_{\alpha }t)^{i_{1}\cdots i_{r-1}}=r\sum _{j=0}^{n}\alpha _{j}t^{ji_{1}\cdots i_{r-1}}.}$

where n is the dimension of V.

Alternating operators

Given two vector spaces V and X and a natural number k, an alternating operator from V^k to X is a multilinear map

${\displaystyle f:V^{k}\to X}$

such that whenever v₁, ..., v_k are linearly dependent vectors in V, then

${\displaystyle f(v_{1},\ldots ,v_{k})=0.}$

The map

${\displaystyle w:V^{k}\to {\textstyle \bigwedge }^{\!k}(V),}$

which associates to ${\displaystyle k}$ vectors from ${\displaystyle V}$ their exterior product, i.e. their corresponding ${\displaystyle k}$-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on ${\displaystyle V^{k};}$ given any other alternating operator ${\displaystyle f:V^{k}\rightarrow X,}$ there exists a unique linear map ${\displaystyle \phi$ :{\textstyle \bigwedge }^{\!k}(V)\rightarrow X} with ${\displaystyle f=\phi \circ w.}$ This universal property characterizes the space ${\displaystyle {\textstyle \bigwedge }^{\!k}(V)}$ and can serve as its definition.

Alternating multilinear forms

Geometric interpretation for the exterior product of n 1-forms (ε, η, ω) to obtain an n-form ("mesh" of coordinate surfaces, here planes),^[3] for n = 1, 2, 3. The "circulations" show orientation.^[9][10]

The above discussion specializes to the case when ${\displaystyle X=K}$, the base field. In this case an alternating multilinear function

${\displaystyle f:V^{k}\to K}$

is called an alternating multilinear form. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the 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 ${\displaystyle k}$ on ${\displaystyle V}$ is naturally isomorphic with the dual vector space ${\displaystyle {\bigl (}{\textstyle \bigwedge }^{\!k}(V){\bigr )}^{*}}$. If ${\displaystyle V}$ is finite-dimensional, then the latter is naturally isomorphic to ${\displaystyle {\textstyle \bigwedge }^{\!k}\left(V^{*}\right)}$. In particular, if ${\displaystyle V}$ is ${\displaystyle n}$-dimensional, the dimension of the space of alternating maps from ${\displaystyle V^{k}}$ to ${\displaystyle K}$ is the binomial coefficient ${\displaystyle \textstyle {\binom {n}{k}}}$.

Under such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : V^k → K and η : V^m → K are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables. Depending on the choice of identification of elements of exterior power with multilinear forms, the exterior product is defined as

${\displaystyle \omega \wedge \eta =\operatorname {Alt} (\omega \otimes \eta )}$

or as

${\displaystyle \omega {\dot {\wedge }}\eta ={\frac {(k+m)!}{k!\,m!}}\operatorname {Alt} (\omega \otimes \eta ),}$

where, if the characteristic of the base field ${\displaystyle K}$ is 0, the alternation Alt of a multilinear map is defined to be the average of the sign-adjusted values over all the permutations of its variables:

${\displaystyle \operatorname {Alt} (\omega )(x_{1},\ldots ,x_{k})={\frac {1}{k!}}\sum _{\sigma \in S_{k}}\operatorname {sgn} (\sigma )\,\omega (x_{\sigma (1)},\ldots ,x_{\sigma (k)}).}$

When the field ${\displaystyle K}$ has finite characteristic, an equivalent version of the second expression without any factorials or any constants is well-defined:

${\displaystyle {\omega {\dot {\wedge }}\eta (x_{1},\ldots ,x_{k+m})}=\sum _{\sigma \in \mathrm {Sh} _{k,m}}\operatorname {sgn} (\sigma )\,\omega (x_{\sigma (1)},\ldots ,x_{\sigma (k)})\,\eta (x_{\sigma (k+1)},\ldots ,x_{\sigma (k+m)}),}$

where here Sh_k,m ⊂ S_k+m is the subset of (k, m) shuffles: permutations 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). As this might look very specific and fine tuned, an equivalent raw version is to sum in the above formula over permutations in left cosets of S_k+m/S_k x S_m.

Interior product

Suppose that ${\displaystyle V}$ is finite-dimensional. If ${\displaystyle V^{*}}$ denotes the dual space to the vector space ${\displaystyle V}$, then for each ${\displaystyle \alpha \in V^{*}}$, it is possible to define an antiderivation on the algebra ${\displaystyle {\textstyle \bigwedge }(V)}$,

${\displaystyle \iota _{\alpha }:{\textstyle \bigwedge }^{\!k}(V)\rightarrow {\textstyle \bigwedge }^{\!k-1}(V).}$

This derivation is called the interior product with ${\displaystyle \alpha }$, or sometimes the insertion operator, or contraction by ${\displaystyle \alpha }$.

Suppose that ${\displaystyle w\in {\textstyle \bigwedge }^{\!k}(V)}$. Then ${\displaystyle w}$ is a multilinear mapping of ${\displaystyle V^{*}}$ to ${\displaystyle K}$, so it is defined by its values on the k-fold Cartesian product ${\displaystyle V^{*}\times V^{*}\times \dots \times V^{*}}$. If u₁, u₂, ..., u_k−1 are ${\displaystyle k-1}$ elements of ${\displaystyle V^{*}}$, then define

${\displaystyle (\iota _{\alpha }w)(u_{1},u_{2},\ldots ,u_{k-1})=w(\alpha ,u_{1},u_{2},\ldots ,u_{k-1}).}$

Additionally, let ${\displaystyle \iota _{\alpha }f=0}$ whenever ${\displaystyle f}$ is a pure scalar (i.e., belonging to ${\displaystyle {\textstyle \bigwedge }^{\!0}(V)}$).

Axiomatic characterization and properties

The interior product satisfies the following properties:

1. For each ${\displaystyle k}$ and each ${\displaystyle \alpha \in V^{*}}$,

$${\displaystyle i_{\alpha }:{\textstyle \bigwedge }^{\!k}(V)\rightarrow {\textstyle \bigwedge }^{\!k-1}(V).}$$

(By convention, ${\displaystyle {1}}$.)

1. If ${\displaystyle v}$ is an element of ${\displaystyle V}$ (${\displaystyle ={\textstyle \bigwedge }^{\!1}(V)}$), then ${\displaystyle i_{\alpha }v=\alpha (v)}$ is the dual pairing between elements of ${\displaystyle V}$ and elements of ${\displaystyle V^{*}}$.
2. For each ${\displaystyle \alpha \in V^{*}}$, ${\displaystyle \iota _{\alpha }}$ is a graded derivation of degree −1:

${\displaystyle \iota _{\alpha }(a\wedge b)=(\iota _{\alpha }a)\wedge b+(-1)^{\deg a}a\wedge (\iota _{\alpha }b).}$

These three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.

Further properties of the interior product include:

• ${\displaystyle \iota _{\alpha }\circ \iota _{\alpha }=0.}$
• ${\displaystyle \iota _{\alpha }\circ \iota _{\beta }=-\iota _{\beta }\circ \iota _{\alpha }.}$

Hodge duality

Suppose that ${\displaystyle V}$ has finite dimension ${\displaystyle V}$. Then the interior product induces a canonical isomorphism of vector spaces

${\displaystyle {\textstyle \bigwedge }^{\!k}(V^{*})\otimes {\textstyle \bigwedge }^{\!n}(V)\to {\textstyle \bigwedge }^{\!n-k}(V)}$

by the recursive definition

${\displaystyle \iota _{\alpha \wedge \beta }=\iota _{\beta }\circ \iota _{\alpha }.}$