Alternatization

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

Let $S$ be a set and $A$ an Abelian group. Given a map $\alpha: S \times S \to A$ , $\alpha$ is termed an alternating map if $\alpha(s,s) = 0$ for all $s \in S$ and $\alpha(s,t) %2B \alpha(t,s) = 0$ for all $s,t \in S$ .

The alternatization of a general (not necessarily alternating) map $\alpha: S \times S \to A$ 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.