Exterior algebra

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

[v\wedge v = 0] for all vectors [v\in V]

which entails

[u\wedge v = - v\wedge u] for all vectors [u,v\in V], and

[v_1\wedge v_2\wedge\cdots \wedge v_k = 0] whenever [v_1,\ldots,v_k\in V] 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 [v_1\wedge v_2\wedge\cdots\wedge v_k] with v₁,…,v_k 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 Λ^k(V). The exterior algebra can be written as the direct sum of each of the k-th powers:

[\Lambda(V) = \bigoplus_^ \Lambda^k V]

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 [u\wedge v] represents the oriented parallelogram with sides u and v, while the 3-vector [u\wedge v\wedge w] 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 is a basis of V, then the set

[\\wedge e_\wedge\cdots\wedge e_ \mid 1\le i_1 < i_2 < \cdots < i_k \le n\}]

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

[v_1\wedge\cdots\wedge v_k]

then every vector v_j can be written as a linear combination of the basis vectors e_i; 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 v_j in terms of the basis e_i.

Counting the basis elements, we see that the dimension of Λ^k(V) is n choose k. In particular, Λ^k(V) = for k > n.

The exterior algebra is a graded algebra as the direct sum

[\Lambda(V) = \Lambda^0(V)\oplus \Lambda^1(V) \oplus \Lambda^2(V) \oplus \cdots \oplus \Lambda^n(V)]

(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.

Example: the exterior algebra of Euclidean 3-space

For vectors in R³, the exterior algebra is closely related to the cross product and triple product. Using the standard basis , the wedge product of a pair of vectors

[ \mathbf = u_1 \mathbf + u_2 \mathbf + u_3 \mathbf ]

and

[ \mathbf = v_1 \mathbf + v_2 \mathbf + v_3 \mathbf ]

is

[ \mathbf \wedge \mathbf = (u_1 v_2 - u_2 v_1) (\mathbf \wedge \mathbf) + (u_1 v_3 - u_3 v_1) (\mathbf \wedge \mathbf) + (u_2 v_3 - u_3 v_2) (\mathbf \wedge \mathbf) ]

where is the basis for the three-space Λ²(R³). This imitates the usual definition of the cross product of vectors in three dimensions.

Bringing in a third vector

[ \mathbf = w_1 \mathbf + w_2 \mathbf + w_3 \mathbf ],

the wedge product of three vectors is

[ \mathbf \wedge \mathbf \wedge \mathbf = (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 \wedge \mathbf \wedge \mathbf) ]

where i Λ j Λ k is the basis vector for the one-space Λ³(R³). This imitates the usual definition of the triple product.

The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product 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 minors of the matrix with columns v and w. 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.

The space Λ¹(R³) is R³, and the space Λ⁰(R³) is R. Direct-summing all four subspaces together yields a vector space Λ(R³) of eight-dimensional vectors

[ \mathbf = (a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8) := (a_1, a_2 \mathbf + a_3 \mathbf + a_4 \mathbf, a_5 \mathbf \wedge \mathbf + a_6 \mathbf \wedge \mathbf + a_7 \mathbf \wedge \mathbf, a_8 \mathbf \wedge \mathbf \wedge \mathbf) ].

Then given a pair of eight-dimensional vectors a and b, with a given as above and

[ \mathbf = (b_1, b_2, b_3, b_4, b_5, b_6, b_7, b_8) ],

the wedge product of a and b is (expressed as a column vector),

[ \mathbf \wedge \mathbf = \begin a_1 b_1 \\ a_1 b_2 + a_2 b_1 \\ a_1 b_3 + a_3 b_1 \\ a_1 b_4 + a_4 b_1 \\

a_1 b_5 + a_5 b_1 + a_2 b_3 - a_3 b_2 \\ a_1 b_6 + a_6 b_1 + a_2 b_4 - a_4 b_2 \\ a_1 b_7 + a_7 b_1 + a_3 b_4 - a_4 b_3 \\ a_1 b_8 + a_8 b_1 + a_2 b_7 + a_7 b_2 - a_3 b_6 - a_6 b_3 + a_4 b_5 + a_5 b_4 \end ].

It is easy to verify by inspection that the eight-dimensional wedge product has the vector (1,0,0,0,0,0,0,0) as the multiplicative unit element. It is also possible to verify by multiplying out components that this Λ(R³) algebra wedge product is associative (as well as bilinear):

[ (\mathbf \wedge \mathbf) \wedge \mathbf = \mathbf \wedge (\mathbf \wedge \mathbf) \qquad \qquad \forall \, \mathbf, \mathbf, \mathbf \isin \Lambda (\mathbf^3),]

so that the algebra is unital associative.

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:

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 v⊗v for v in V, and define Λ(V) as the quotient

Λ(V) = T(V)/I

(and use Λ 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.

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 of vector spaces to the category of algebras.

Rather than defining Λ(V) first and then identifying the exterior powers Λ^k(V) as certain subspaces, one may alternatively define the spaces Λ^k(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 V^k to X is a multilinear map

f: V^k → X

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

f(v₁,...,v_k) = 0.

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

The map

w: V^k → Λ^k(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 V^k: given any other anti-symmetric operator f : V^k → X, there exists a unique linear map φ: Λ^k(V) → X with f = φ o w. This universal property characterizes the space Λ^k(V) and can serve as its definition.

The set of all anti-symmetric maps from V^k 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 Λ^k(V^∗), where V^∗ denotes the dual space of V. In particular, the dimension of the space of anti-symmetric maps from V^k to K is the binomial coefficient

[].

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 : 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 wedge product is the sum of the numbers of their variables. It is defined as follows:

[\omega\wedge\eta=\frac(\omega\otimes\eta)]

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:

[(\omega)(x_1,\ldots,x_k)=\frac\sum_(\sigma)\,\omega(x_,\ldots,x_)]

This definition of the wedge product is well-defined even in fields having finite characteristic.

Note. Some conventions define the wedge product as

[\omega\wedge\eta=(\omega\otimes\eta).]

The exterior power

If V is a vector space of finite dimension, then the k-th exterior power of V is the vector space generated by the k-fold exterior products of elements of V, and is denoted by Λ^kV. (See above for various descriptions.) The operation assigning to each vector space V its exterior power Λ^kV is a functor on the category of finite-dimensional vector spaces. In particular, it satisfies the following properties

[\left(\bigwedge^k V\right)^*\cong\bigwedge^k(V^*).]

[\bigwedge^k(V\oplus W)= \bigoplus_\bigwedge^aV\otimes\bigwedge^b W.]

Furthermore, if

[0\to U\to V\to W\to 0]

is an exact sequence with dim(U) = a, dim(V) = a+b, and dim(W) = b, then

[\bigwedge^V=\bigwedge^aU\otimes\bigwedge^bW.]

The interior product or insertion operator

If V* denotes the dual space to the vector space V, then for each α ∈ V^*, it is possible to define an antiderivation on the algebra Λ(V),

[i_\alpha:\bigwedge^k V\rightarrow\bigwedge^V.]

Suppose that w ∈ Λ^kV. Then w is a multilinear mapping of V^* to R, so it is defined by its values on the k-fold Cartesian product V^*× V^*× ... × V^*. If u₁, u₂, ..., u_k-1 are k-1 elements of V^*, then define

[(i_\alpha )(u_1,u_2\dots,u_)=(\alpha,u_1,u_2,\dots, u_)]

Additionally, let i_αf = 0 whenever f is a pure scalar (i.e., belonging to Λ⁰V).

Index notation

Alternatively in index notation, if [\bold w = w_}] is a skew-symmetric k form in Λ^kV, then i_αw is a skew-symmetric (k-1)-form in Λ^k-1V given by

[(i_\alpha )_}=k\sum_^n\alpha^j w_}].

where n is the dimension of V.

Properties

The interior product satisfies the following properties:

1. For each k and each α ∈ V^*,
2. ::[i_\alpha:\bigwedge^kV\rightarrow \bigwedge^V.]
3. :(By convention, Λ^-1 = 0.)
4. If v is an element of V ( = Λ¹V), then i_αv = α(v) is the dual pairing between elements of V and elements of V^*.
5. For each α ∈ V^*, i_{α is a graded derivation of degree -1:}
6. ::[i_\alpha (a\wedge b) = (i_\alpha a)\wedge b + (-1)^a\wedge (i_\alpha b)].

In fact, these three properties are sufficient to characterize the interior product.

Index notation

In the index notation, used primarily by physicists,

[(\omega\wedge\eta)_}=\omega_ eta_ cdots a_]}]

where the square brackets denote the skew-symmetric part:

[\frac\sum__}\hbox(\sigma)\omega_ \cdots a_} \eta_ \cdots a_}]

Differential forms

Let M be a differentiable manifold. A differential k-form ω is a section of Λ^kT^∗M, 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 Λ^k(T_xM)^∗. 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.

Generalization

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. For a physical description see Grassmann number.

See also: superspace, superalgebra, supergroup (physics).

See also

• multilinear algebra
• tensor algebra
• symmetric algebra
• Clifford algebra

 

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