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 ${\displaystyle R}$  be a commutative ring and ${\displaystyle V,W}$  be modules over ${\displaystyle R}$ . A multilinear map of the form ${\displaystyle f\colon V^{n}\to W}$  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 ${\displaystyle x_{i}=x_{i+1}}$  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]

Vector spaces

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

• 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 antisymmetric,^[4] meaning that^[1]

$${\displaystyle f(\dots ,x_{i},x_{i+1},\dots )=-f(\dots ,x_{i+1},x_{i},\dots )\quad {\text{ for any }}1\leq i\leq n-1,}$$

 

or equivalently,

$${\displaystyle f(x_{\sigma (1)},\dots ,x_{\sigma (n)})=(\operatorname {sgn} \sigma )f(x_{1},\dots ,x_{n})\quad {\text{ for any }}\sigma \in S_{n},}$$

 

where ${\displaystyle S_{n}}$ 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

$${\displaystyle g(x_{1},\ldots ,x_{n})\mathrel {:=} \sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )f(x_{\sigma (1)},\ldots ,x_{\sigma (n)})}$$

 

is said to be the alternatization of ${\displaystyle f.}$ 

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.
5. Tu (2011). An Introduction to Manifolds. Springer-Verlag New York. p. 23. ISBN 978-1-4419-7400-6.

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.