Alternatization

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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 \times S \to A, α is termed an alternating map if α(s,s) = 0 for all s \in S and α(s,t) + α(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.