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 its arguments is equal. This generalizes directly to a module over a commutative ring.

The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map of which all arguments belong to the same space.

Definition

Let ${\displaystyle R}$ be a commutative ring and ${\displaystyle V}$, ${\displaystyle W}$ be modules over ${\displaystyle R}$. A multilinear map of the form ${\displaystyle f:V^{n}\to W}$ 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 ${\displaystyle f:V^{n}\to W}$ is alternating if 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,

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

where ${\displaystyle \mathrm {S} _{n}}$ denotes the permutation group of degree ${\displaystyle n}$ 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 ${\displaystyle n}$-multilinear alternating map is ${\displaystyle n!}$ 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, A III.80, §4
3. Dummit & Foote 2004, p. 436
4. Rotman 1995, p. 235
5. Tu 2011, p. 23

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.
• Tu, Loring W. (2011). An Introduction to Manifolds. Springer-Verlag New York. ISBN   978-1-4419-7400-6.