In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

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

Let be a set and an Abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

. Given a map , is termed an alternating map if for all and for all .

The alternatization of a general (not necessarily alternating) map is the map .

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

Cocycle

A cocycle refers to any one of the following:*A closed cochain in algebraic topology is called a cocycle.*A particular type of map in an autonomous dynamical system; see Oseledec theorem....

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.