Exterior algebra

Grassmann's theory

The algebraic theory goes back to Hermann Grassmann. His method of constructing algebraic structures used generators and relations and is not manifestly basis independent. Grassmann used only real algebras, i.e. algebras whose scalars are real numbers (He made no distinction between real numbers and real-valued functions, which however changes algebra theory drastically.) However, we follow this attitude here and give definitions for some of his products:

; product (general definition) : A product is a linear map from the tensor product of a space with itself into a linear space. 'Note:' such a product is left- and right-distributive, but may not be unital or associative.

; exterior product (wedge product) : Let {e_i} be a basis of a vector space V. An exterior product of two such generators is defined by demanding the following computational rules (relations):

• e_i ∧ e_j = e_ij = −e_ji if and only if i <> j.
• e_i ∧ e_j = 0 if i = j.
• extend this process recursively to products of higher grade via e_i ∧ e_j..k = e_ij..k = −e_ji..k etc.

Note, that the product takes values in a new space V∧V (doubly indexed) which is a factor space of the tensor product V × V. The product is associative by definition and alternating, i.e. it vanishes if two indices are equal. A short combinatorial calculation shows that one finds from n basis vectors 2ⁿ linear independent products. They build the vector space V^∧ underlying a Grassmann algebra (see below). The exterior product is extended to the whole space V^∧ by bilinearity.

The Grassmann algebra is a graded algebra. We define the grade of scalars to be zero and the grade of basis vectors to be 1. The grade of a non-zero product of generators counts the number of generators. The space of a Grassmann algebra can therefore be decomposed into a direct sum of homogeneous subspaces of definite grade, i.e. the space spanned by all products having exactly k generators:
V^∧ = V^∧0 (+) V^∧1 (+) ... (+) V^∧n
where V^∧0 = R is identified with R, the real numbers.

; interior product : not yet ...

; regressive product : not yet ...

Modern theory

Definition:

where k and m are the numbers of variables for each of the two skew-symmetric functions and alternation of a map is defined to be the signed average of the values over all the permutations of its variables:

Wedge product of spaces, exterior powers

The wedge product of two vector spaces may be identified with the subspace of their tensor product generated by the skew-symmetric tensors. (This definition, though, works only over fields of characteristic zero. In algebraic work one may need an alternate definition, based on a universal property. This means taking an appropriate quotient of the tensor product, instead - of the same dimension. The difference is harmless for real and complex vector spaces.)

The wedge product of a vector space V with itself k times is called its k-th exterior power and is denoted . If dim V=n, then dim is n-choose-k.

Example: Let be the dual space of V, i.e. space of all linear maps from V to R. The second exterior power is the space of all skew-symmetric bilinear maps from VxV to R.

Definition in generality

The definition of an anti-symmetric multilinear operator is an operator m: Vⁿ -> X such that if there is a linear dependence between its arguments, the result is 0. Note that the addition of anti-symmetric operators, or multiplying one by a scalar, is still anti-symmetric -- so the anti-symmetric multilinear operators on Vⁿ form a vector space.

The most famous example of an anti-symmetric operator is the determinant.

The nth wedge space W, for a module V over a commutative ring R, together with the anti-symmetric linear wedge operator w: Vⁿ -> W is such that for every n-linear anti-symmetric operator m: Vⁿ -> X there exists a unique linear operator l: W -> X such that m = l o w. The wedge is unique up to a unique isomorphism.

One way of defining the wedge space constructively is by dividing the Tensor space by the subspace generated by all the tensors of n-tuples which are linearily dependent.

The dimension of the kth wedge space for a free module of dimension n is n! / (k!(n-k)!). In particular, that means that up to a constant, there is a single anti-symmetric functional with the arity of the dimension of the space. Also note that every linear functional is anti-symmetric.

Note that the wedge operator commutes with the ^* operator. In other words, we can define a wedge on functionals such that the result is an anti-symmetric multilinear functional. In general, we can define the wedge of an n-linear anti-symmetric functional and an m-linear anti-symmetric functional to be an (n+m)-linear anti-symmetric functional. Since it turns out that this operation is associative, we can also define the power of an anti-symmetric linear functional.

Grassmann algebras

A abstract Grassmann algebra (also known as an exterior algebra) is a unital associative algebra K generated by a set, S subject to the relation χξ+ξχ=0 for any χ,ξ in S. This definition amounts to saying that the generators are anti-commuting quantities (and otherwise 'as general as possible); it should be modified in case K has characteristic 2.

The construction of such an algebra comes from the wedge product: take the vector space V that has S as basis, and the direct sum of all the exterior powers of V, using wedge product in each graded piece. If S is finite of cardinality n, the Grassmann algebra has as basis one wedge product for each subset of S, and each product made up by wedging elements of S with repeats is equal to 0.

For physics applications see:

• Fermion
• Supersymmetry
• Superspace
• Superalgebra
• Supergroup
• Berezin integral

Differential forms

When dealing with differentiable manifolds, we define an "n-form" to be a function from the manifold to the n-th wedge of the cotangent bundle. Such a form will be said to be differentiable if, when applied to n differentiable vector fields, the result is a differentiable function. See p-form. The wedge product makes pointwise sense for differential forms.

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