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Concept
Alternating multilinear map
Summary
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 arguments is equal. More generally, the vector space may be a module over a commutative ring.
The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.
Let be a commutative ring and be modules over . A multilinear map of the form is said to be alternating if it satisfies the following equivalent conditions:
whenever there exists such that then
whenever there exists such that then
Let be vector spaces over the same field. Then a multilinear map of the form is alternating iff it satisfies the following condition:
if are linearly dependent then .
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.
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 order 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 n-multilinear alternating map is n! 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.
Official source
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