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[]

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 ${\displaystyle x_{i}}$ of an alternating multilinear map is replaced by ${\displaystyle x_{i}+cx_{j}}$ for any ${\displaystyle j\neq i}$ and ${\displaystyle c}$ in the base ring ${\displaystyle R,}$ then the value of that map is not changed.^[3]

Every alternating multilinear map is ,^[4] meaning that^[1]

or equivalently, where ${\displaystyle S_{n}}$denotes the permutation group of order ${\displaystyle n}$ and ${\displaystyle \operatorname {sgn} \sigma }$ is the sign of ${\displaystyle \sigma .}$^[5]

If ${\displaystyle n!}$ is a unit in the base ring ${\displaystyle R,}$ then every antisymmetric ${\displaystyle n}$-multilinear form is alternating.

Alternatization[]

Given a multilinear map of the form ${\displaystyle f:V^{n}\to W,}$ the alternating multilinear map ${\displaystyle g:V^{n}\to W}$ defined by

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[]

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.