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 some variants of British English) 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 $f\colon V^{n}\to W$ 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 $f(x_{1},\ldots ,x_{n})=0$.
3. if $x_{1},\ldots ,x_{n}$ are linearly dependent then $f(x_{1},\ldots ,x_{n})=0$.
4. for any $1\leq i\leq n-1$, $f(\dots ,x_{i},x_{i+1},\dots )=-f(\dots ,x_{i+1},x_{i},\dots )$.
5. if for any $\sigma \in S_{n}$, $f(x_{\sigma (1)},\dots ,x_{\sigma (n)})=(\operatorname {sgn} \sigma )f(x_{1},\dots ,x_{n})$, where $S_{n}$denotes the permutation group of order $n$and $\operatorname {sgn} \sigma$the sign of $\sigma$.

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.
• Every alternating multilinear map is antisymmetric.
• 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 $f\colon V^{n}\to W$, the alternating multilinear map $g\colon V^{n}\to W$ 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.

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. {{cite book}}: Invalid |ref=harv (help)
• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. {{cite book}}: Invalid |ref=harv (help)
• Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673. {{cite book}}: Invalid |ref=harv (help)
• 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. {{cite book}}: Invalid |ref=harv (help)