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| In [[mathematics]], the notion of '''alternatization''' or '''alternatisation''' is used to pass from any map to an alternating map.
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| Let <math>S</math> be a set and <math>A</math> an [[abelian group]]. Given a map <math>\alpha: S \times S \to A</math>, <math>\alpha</math> is termed an '''alternating map''' if <math>\alpha(s,s) = 0</math> for all <math>s \in S</math> and <math>\alpha(s,t) + \alpha(t,s) = 0</math> for all <math>s,t \in S</math>.
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| The alternatization of a general map <math>\alpha: S \times S \to A</math> is the map <math>(x,y) \mapsto \alpha(x,y) - \alpha(y,x)</math>.
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| The alternatization of an alternating map is simply its double, while the alternatization of a [[symmetric map]] is zero.
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| The alternatization of a [[bilinear map]] is bilinear. There may be non-bilinear maps whose alternatization 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.
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| [[Category:Algebra]]
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| {{Algebra-stub}}
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| {{Math-stub}}
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Latest revision as of 21:09, 29 August 2014
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