Alternating multilinear map

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 ${\displaystyle x_1,\dots ,x_n}$ are linearly dependent then ${\displaystyle f(x_1,\dots ,x_n)=0}$.

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 ${\displaystyle x_i+c{x}_j}$ for any ${\displaystyle j\ne i}$ and ${\displaystyle c}$ in the base ring ${\displaystyle R}$, then the value of that map is not changed.

Every alternating multilinear map is antisymmetric, meaning that $${\displaystyle f(\dots ,x_i,x_{i+1},\dots )=-f(\dots ,x_{i+1},x_i,\dots )\text{ for any }1\le i\le n-1,}$$ or equivalently, $${\displaystyle f(x_{\sigma (1)},\dots ,x_{\sigma (n)})=(\mathrm{sgn}\sigma )f(x_1,\dots ,x_n)\text{ for any }\sigma \in {\mathrm{S}}_n,}$$ where ${\displaystyle {\mathrm{S}}_n}$ denotes the permutation group of degree ${\displaystyle n}$ and ${\displaystyle \mathrm{sgn}\sigma }$ is the sign of ${\displaystyle \sigma }$. 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 $${\displaystyle g(x_1,\dots ,x_n):=\sum _{\sigma \in S_n}\mathrm{sgn}(\sigma )f(x_{\sigma (1)},\dots ,x_{\sigma (n)})}$$ 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.