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Affine representation
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{{one source |date=May 2024}} In [[mathematics]], an '''affine representation''' of a [[topological group|topological]] [[Lie group]] ''G'' on an [[affine space]] ''A'' is a [[continuity (topology)|continuous]] ([[smooth function|smooth]]) [[group homomorphism]] from ''G'' to the [[automorphism group]] of ''A'', the [[affine group]] Aff(''A''). Similarly, an affine representation of a [[Lie algebra]] '''g''' on ''A'' is a [[Lie algebra homomorphism]] from '''g''' to the Lie algebra '''aff'''(''A'') of the affine group of ''A''. An example is the action of the [[Euclidean group]] E(''n'') on the [[Euclidean space]] E<sup>''n''</sup>. Since the affine group in dimension ''n'' is a matrix group in dimension ''n'' + 1, an affine representation may be thought of as a particular kind of [[linear representation]]. We may ask whether a given affine representation has a [[fixed point (mathematics)|fixed point]] in the given affine space ''A''. If it does, we may take that as origin and regard ''A'' as a [[vector space]]; in that case, we actually have a linear representation in dimension ''n''. This reduction depends on a [[group cohomology]] question, in general. ==See also== * [[Group action (mathematics)|Group action]] * [[Projective representation]] ==References== * {{citation|first1=Elisabeth|last1=Remm|first2= Michel|last2= Goze|title=Affine Structures on abelian Lie Groups|arxiv=math/0105023|journal=Linear Algebra and Its Applications|volume= 360|year=2003|pages= 215–230|doi=10.1016/S0024-3795(02)00452-4}}. [[Category:Homological algebra]] [[Category:Representation theory of Lie algebras]] [[Category:Representation theory of Lie groups]] {{algebra-stub}}
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