Wedge product

In mathematics, the wedge product, also known as exterior product, is an anti-symmetrisation (alternation) of the tensor product. The wedge product is a distributive associative multiplication of skew-symmetric multilinear maps which is anti-commutative for maps with odd number of variables and commutative otherwise. As in the case of tensor products, the number of variables of the wedge of two maps is the sum of the numbers of their variables:

Definition:

<math>\omega\wedge\eta=\frac{(k+m)!}{k!\,m!}{\rm Alt}(\omega\otimes\eta)</math>

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 The average of the values over all the permutations of its variables:

<math>{\rm Alt}(\omega)=\frac{1}{k!}\sum_{\sigma\in\Sigma_k}{\rm sgn}(\sigma)\,\omega[\sigma(x_1),...\sigma(x_k)]</math>

The wedge product makes pointwise sense for differential forms.

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 <math>\Lambda^k V</math>. If dim V=n, then dim <math>\Lambda^k V</math> is n-choose-k.

Example: Let <math>V^*</math> be the dual space of V, i.e. space of all linear maps from V to R. The second exterior power <math>\Lambda^2 V^*</math> 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.

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.