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 space (e.g., a bilinear form or a multilinear form) that is zero whenever any two adjacent arguments are equal.

The notion of alternatization (or alternatisation in British English) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.

Definition

An $n$ -multilinear map of the form $f\colon V\times \cdots \times V\to W$ is said to be alternating if $f(x_{1},\ldots ,x_{n})=0$ whenever there exists ${\textstyle 1\leq i\leq n-1}$ such that $x_{i}=x_{i+1}.$ ^[1]^[2]

Example

In a Lie algebra, the multiplication is an alternating bilinear map called the Lie bracket.

Properties

If any distinct pair of components of an alternating multilinear map are equal, then such a map is zero:^[1]^[3]

f(x_{1},\ldots ,x_{i},\ldots ,x_{j},\ldots ,x_{n})=0

whenever there exist

i

and

j

such that

i<j

and

x_{i}=x_{j}.

If the components of an alternating multilinear map are linearly dependent, then such a map is zero.
If any component x_i of an alternating multilinear map is replaced by x_i + c x_j for any j ≠ i and c in the base ring R, then the value of that map is not changed.^[3]
Every alternating multilinear map is antisymmetric.^[4]
If n! is a unit in the base ring R, then every antisymmetric n-multilinear form is alternating.

Alternatization

Given an $n$ -multilinear map of the form $f\colon V\times \cdots \times V\to W$ , the alternating multilinear map $g\colon V\times \cdots \times V\to W$ defined by $g(x_{1},\ldots ,x_{n}):=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )f(x_{\sigma (1)},\ldots ,x_{\sigma (n)})$ 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.

Notes

1 2 Lang 2002, pp. 511-512.
↑ Bourbaki 2007, p. A III.80, §4.
1 2 Dummit & Foote 2004, p. 436.
↑ Rotman 1995, p. 235.

References

Bourbaki, N. (2007). Eléments de mathématique. 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. 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. 148 (4th ed.). Springer. ISBN 0-387-94285-8. OCLC 30028913.

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