In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of its arguments is equal. This generalizes directly to a module over a commutative ring. The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map of which all arguments belong to the same space.

Definition

Let R be a commutative ring and V, W be modules over R. A multilinear map of the form is said to be alternating if it satisfies the following equivalent conditions:

Vector spaces

Let V, W be vector spaces over the same field. Then a multilinear map of the form is alternating if it satisfies the following condition:

Example

In a Lie algebra, the Lie bracket is an alternating bilinear map. The determinant of a matrix is a multilinear alternating map of the rows or columns of the matrix.

Properties

If any component x_i of an alternating multilinear map is replaced by x_i + c x_j for any j \neq i and c in the base ring R, then the value of that map is not changed. Every alternating multilinear map is antisymmetric, meaning that or equivalently, where denotes the permutation group of degree n and \sgn\sigma is the sign of \sigma. If n! is a unit in the base ring R, then every antisymmetric n-multilinear form is alternating.

Alternatization

Given a multilinear map of the form the alternating multilinear map defined by is said to be the alternatization of f. Properties