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.
Let be vector spaces over the same field. Then a multilinear map of the form is alternating if it satisfies the following condition:
- if are linearly dependent then .
If any component of an alternating multilinear map is replaced by for any and in the base ring , 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 and is the sign of .
If is a unit in the base ring , then every antisymmetric -multilinear form is alternating.
Given a multilinear map of the form the alternating multilinear map defined by
is said to be the alternatization of .
Properties
- The alternatization of an -multilinear alternating map is 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.