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Alternating multilinear map

Multilinear map that is 0 whenever arguments are linearly dependent

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated February 4, 2026

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.

01Definition

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

  1. whenever there exists {\textstyle 1\leq i\leq n-1 such that x_{i}=x_{i+1} then f(x_{1},\ldots ,x_{n})=0.
  2. whenever there exists {\textstyle 1\leq i\neq j\leq n such that x_{i}=x_{j} then f(x_{1},\ldots ,x_{n})=0.

02Vector spaces

Let V,W be vector spaces over the same field. Then a multilinear map of the form f:V^{n}\to W is alternating if it satisfies the following condition:

03Example

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.

04Properties

If any component x_{i} of an alternating multilinear map is replaced by x_{i}+cx_{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 f(\dots ,x_{i},x_{i+1},\dots )=-f(\dots ,x_{i+1},x_{i},\dots )\quad {\text{ for any }}1\leq i\leq n-1, or equivalently, 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}, where \mathrm {S} _{n} denotes the permutation group of degree n and \operatorname {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.

05Alternatization

Given a multilinear map of the form f:V^{n}\to W, the alternating multilinear map g:V^{n}\to W defined by g(x_{1},\ldots ,x_{n})\mathrel {:=} \sum _{\sigma \in S_{n}}\operatorname {sgn} (\sigma )f(x_{\sigma (1)},\ldots ,x_{\sigma (n)}) is said to be the alternatization of f.

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.
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Sources and credits

This article is adapted from the Wikipedia article “Alternating multilinear map”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Alternating algebra

In mathematics, an alternating algebra is a Z-graded algebra for which xy =deg(x)deg(y)yx for all nonzero homogeneous elements x and y (i.e. it is an anticommutative algebra) and has the further property that x2 = 0 (nilpotence) for every homogeneous element x of odd degree.

Bilinear map

In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.

Exterior algebra

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