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 (e.g., 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

A multilinear map of the form ${\displaystyle f\colon V^{n}\to W}$ is said to be alternating if it satisfies any of the following equivalent conditions:

1. whenever there exists ${\textstyle 1\leq i\leq n-1}$ such that ${\displaystyle x_{i}=x_{i+1}}$ then ${\displaystyle f(x_{1},\ldots ,x_{n})=0}$.[1][2]
2. whenever there exists ${\textstyle 1\leq i\neq j\leq n}$ such that ${\displaystyle x_{i}=x_{j}}$ then ${\displaystyle f(x_{1},\ldots ,x_{n})=0}$.[1][3]
3. if ${\displaystyle x_{1},\ldots ,x_{n}}$ are linearly dependent then ${\displaystyle f(x_{1},\ldots ,x_{n})=0}$.

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

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

Notes

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

References

• Bourbaki, N. (2007). Eléments de mathématique. Algèbre Chapitres 1 à 3 (reprint ed.). Springer.CS1 maint: ref=harv (link)
• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley.CS1 maint: ref=harv (link)
• Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)