Alternating multilinear map

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.

be a commutative ring and

A multilinear map of the form

is said to be alternating if it satisfies the following equivalent conditions: Let

be vector spaces over the same field.

Then a multilinear map of the form

is alternating if it satisfies the following condition: 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.

of an alternating multilinear map is replaced by

in the base ring

, then the value of that map is not changed.

[3] Every alternating multilinear map is antisymmetric,[4] meaning that[1]

or equivalently,

σ ( 1 )

σ ( n )

) = ( sgn ⁡ σ ) f (

σ ∈

{\displaystyle f(x_{\sigma (1)},\dots ,x_{\sigma (n)})=(\operatorname {sgn} \sigma )f(x_{1},\dots ,x_{n})\quad {\text{ for any }}\sigma \in \mathrm {S} _{n},}

denotes the permutation group of degree

sgn ⁡ σ

{\displaystyle \operatorname {sgn} \sigma }

is a unit in the base ring

-multilinear form is alternating.

Given a multilinear map of the form

the alternating multilinear map

sgn ⁡ ( σ ) f (