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In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (''V'', ''ω'') which preserves the symplectic form ''ω''. Here ''ω'' is a nondegenerate skew symmetric bilinear form : where F is the field of scalars. A representation of a group ''G'' preserves ''ω'' if : for all ''g'' in ''G'' and ''v'', ''w'' in ''V'', whereas a representation of a Lie algebra g preserves ''ω'' if : for all ''ξ'' in g and ''v'', ''w'' in ''V''. Thus a representation of ''G'' or g is equivalently a group or Lie algebra homomorphism from ''G'' or g to the symplectic group Sp(''V'',''ω'') or its Lie algebra sp(''V'',''ω'') If G is a compact group (for example, a finite group), and F is the field of complex numbers, then by introducing a compatible unitary structure (which exists by an averaging argument), one can show that any complex symplectic representation is a quaternionic representation. Quaternionic representations of finite or compact groups are often called symplectic representations, and may be identified using the Frobenius-Schur indicator. ==References== *. 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「Symplectic representation」の詳細全文を読む スポンサード リンク
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