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 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.

Definition

Let $R$ be a commutative ring and $V,W$ be modules over $R$. A multilinear map of the form $f\colon V^{n}\to W$ 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 $f(x_{1},\ldots ,x_{n})=0.$

Vector spaces

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

• if $x_{1},\ldots ,x_{n}$ are linearly dependent then $f(x_{1},\ldots ,x_{n})=0$.

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

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, $$f(x_{\sigma (1)},\dots ,x_{\sigma (n)})=(\operatorname {sgn} \sigma )f(x_{1},\dots ,x_{n})\quad {\text{ for any }}\sigma \in S_{n},$$ where $S_{n}$denotes the permutation group of order $n$ 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.

Alternatization

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.

See also

• Alternating algebra
• Bilinear map
• Exterior algebra § Alternating multilinear forms
• Map (mathematics)
• Multilinear algebra
• Multilinear map
• Multilinear form
• Symmetrization

References

• Bourbaki, N. (2007). Eléments de mathématique. Vol. Algèbre Chapitres 1 à 3 (reprint ed.). Springer.
• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley.
• Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673.
• Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics. Vol. 148 (4th ed.). Springer. ISBN 0-387-94285-8. OCLC 30028913.