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An affine representation of a topological (Lie) group ''G'' on an affine space ''A'' is a continuous (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'') upon the Euclidean space E''n''. 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 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 * Projective representation 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Affine representation」の詳細全文を読む スポンサード リンク
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