Alternatization

In mathematics, the notion of alternatization is used to pass from any map to an alternating map.

Let $S$ be a set and $A$ an Abelian group. Given a map $alpha:\; S\; imes\; S\; o\; A$, $alpha$ is termed an alternating map if $alpha(s,s)\; =\; 0$ for all $s\; in\; S$ and $alpha(s,t)\; +\; alpha(t,s)\; =\; 0$ for all $s,t\; in\; S$.

The alternatization of a general (not necessarily alternating) map $alpha:\; S\; imes\; S\; o\; A$ is the map $(x,y)\; mapsto\; alpha(x,y)\; -\; alpha(y,x)$.

The alternatization of an alternating map is simply its double, while the alternatization of a symmetric map is zero.

The alternatization of a bilinear map is bilinear. There may be non-bilinear maps whose alternatization is also 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.