Exterior algebra

In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. It is denoted by Λ(V) or Λ(V) and its multiplication, known as the wedge product or the exterior product, is written as ∧. The wedge product is associative and bilinear; its essential property is that it is alternating on V:

<math>v\wedge v = 0<math> for all vectors <math>v\in V<math>

which entails

<math>u\wedge v = - v\wedge u<math> for all vectors <math>u,v\in V<math>, and
<math>v_1\wedge v_2\wedge\cdots \wedge v_k = 0<math> whenever <math>v_1,\ldots,v_k\in V<math> are linearly dependent.

Note that these three properties are only valid for the vectors in V, not for all elements of the algebra Λ(V).

The exterior algebra is in fact the "most general" algebra with these properties. This means that all equations that hold in the exterior algebra follow from the above properties alone. This generality of Λ(V) is formally expressed by a certain universal property, see below.

Elements of the form v1v2∧…∧vk with v1,…,vk in V are called k-vectors. The subspace of Λ(V) generated by all k-vectors is known as the k-th exterior power of V and denoted by Λk(V). The exterior algebra can be written as the direct sum of each of the k-th powers:

<math>\Lambda(V) = \bigoplus_{k=0}^{\infty} \Lambda^k V<math>

The exterior product has the important property that the product of a k-vector and an l-vector is a k+l-vector. Thus the exterior algebra forms a graded algebra where the grade is given by k. These k-vectors have geometric interpretations: the 2-vector uv represents the oriented parallelogram with sides u and v, while the 3-vector uvw represents the oriented parallelepiped with edges u, v, and w.

Exterior powers find their main application in differential geometry, where they are used to define differential forms. As a consequence, there is a natural wedge product for differential forms. All of these concepts go back to Hermann Grassmann.


Basis and dimension

If the dimension of V is n and {e1,...,en} is a basis of V, then the set

<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>

is a basis for the k-th exterior power Λk(V). The reason is the following: given any wedge product of the form

<math>v_1\wedge\cdots\wedge v_k<math>

then every vector vj can be written as a linear combination of the basis vectors ei; using the bilinearity of the wedge product, this can be expanded to a linear combination of wedge products of those basis vectors. Any wedge product in which the same basis vector appears more than once is zero; any wedge product in which the basis vectors don't appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis k-vectors can be computed as the minors of the matrix that describes the vectors vj in terms of the basis ei.

Counting the basis elements, we see that the dimension of Λk(V) is n choose k. In particular, Λk(V) = {0} for k > n.

The exterior algebra is a graded algebra as the direct sum

<math>\Lambda(V) = \Lambda^0(V)\oplus \Lambda^1(V) \oplus \Lambda^2(V) \oplus \cdots \oplus \Lambda^n(V)<math>

(where we set Λ0(V) = K and Λ1(V) = V), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2n.

Universal property and construction

Let V be a vector space over the field K (which in most applications will be the field of real numbers). The fact that Λ(V) is the "most general" unital associative K-algebra containing V with an alternating multiplication on V can be expressed formally by the following universal property:

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

Missing image
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 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 vv for v in V, and define Λ(V) as the quotient

Λ(V) = T(V)/I

(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.

Rather than defining Λ(V) first and then identifying the exterior powers Λk(V) as certain subspaces, one may alternatively define the spaces Λk(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.

Anti-symmetric operators and exterior powers

Given two vector spaces V and X, an anti-symmetric operator from Vk to X is a multilinear map

f: VkX

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

f(v1,...,vk) = 0.

The most famous example is the determinant, an anti-symmetric operator from (Kn)n to K.

The map

w: Vk → Λk(V)

which associates to k vectors from V their wedge product, i.e. their corresponding k-vector, is also anti-symmetric. In fact, this map is the "most general" anti-symmetric operator defined on Vk: given any other anti-symmetric operator f : VkX, there exists a unique linear map φ: Λk(V) → X with f = φ o w. This universal property characterizes the space Λk(V) and can serve as its definition.

The set of all anti-symmetric maps from Vk to the base field K is a vector space, as the sum of two such maps, or the multiplication of such a map with a scalar, is again anti-symmetric. If V has finite dimension n, then this space can be identified with Λk(V*), where V* denotes the dual space of V. In particular, the dimension of the space of anti-symmetric maps from Vk to K is n choose k.

Under this identification, and if the base field is R or C, the wedge product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : VkK and η : VmK are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their wedge product is the sum of the numbers of their variables. It is defined as follows:

<math>\omega\wedge\eta=\frac{(k+m)!}{k!\,m!}{\rm 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 permutations of its variables:

<math>{\rm Alt}(\omega)(x_1,\ldots,x_k)=\frac{1}{k!}\sum_{\sigma\in S_k}{\rm sgn}(\sigma)\,\omega(x_{\sigma(1)},\ldots,x_{\sigma(k)})<math>

NB. There are few books where wedge product is defined as

<math>\omega\wedge\eta={\rm Alt}(\omega\otimes\eta)<math>

Index notation

In the index notation, used primarily by physicists,

<math>(\omega\wedge\eta)_{a_1 \cdots a_{k+m}}=\frac{1}{k!m!}\epsilon_{a_1 \cdots a_{k+m}}^{b_1 \cdots b_k c_1 \cdots c_m} \omega_{b_1 \cdots b_k} \eta_{c_1 \cdots c_m}<math>

Differential forms

Let M be a differentiable manifold. A "differential k-form" ω is a section of Λk(TM)*, the k-th exterior power of the cotangent bundle of M. Equivalently, ω is a smooth function on M which assigns to each point x of M an element of Λk(TxM)*. Roughly speaking, differential forms are globalized versions of cotangent vectors. Differential forms are important tools in differential geometry, where, among other things, they are used to define de Rham cohomology and Alexander-Spanier cohomology.


Given a commutative ring R and an R-module M, we can define the exterior algebra Λ(M) just as above, as a suitable quotient of the tensor algebra T(M). It will satisfy the analogous universal property.

Physical applications

Grassmann algebras have some important applications in physics where they are used to model various concepts related to fermions and supersymmetry.

See also: superspace, superalgebra, supergroup

Related topics

es:Producto exterior fr:Produit extrieur ru:Внешняя алгебра


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