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

be a commutative ring and is said to be alternating if it satisfies the following equivalent conditions:

1. then .

2. then .

Vector spaces

is alternating if it satisfies the following condition:

• are linearly dependent then .

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

in the base ring , then the value of that map is not changed.

Every alternating multilinear map is antisymmetric, meaning that denotes the permutation group is the sign of . is a unit in the base ring -multilinear form is alternating.

Alternatization

is said to be the alternatization of .

Properties

• times itself.

• The alternatization of a symmetric map is zero.

• The alternatization of a bilinear map is bilinear. Most notably, the alternatization of any cocycle is bilinear. This fact plays a crucial role in identifying the second cohomology group of a lattice with the group of alternating bilinear forms on a lattice.

See also

• Alternating algebra

• Bilinear map

• Map (mathematics)

• Multilinear algebra

• Multilinear map

• Multilinear form

• Symmetrization

Notes

References

fr:Application multilinéaire#Application alternée