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In mathematics, if is a group and is a representation of it over the complex vector space , then the complex conjugate representation is defined over the complex conjugate vector space as follows: : is the conjugate of for all in . is also a representation, as you may check explicitly. If is a real Lie algebra and is a representation of it over the vector space , then the conjugate representation is defined over the conjugate vector space as follows: : is the conjugate of for all in .〔This is the mathematicians' convention. Physicists use a different convention where the Lie bracket of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition.〕 is also a representation, as you may check explicitly. If two real Lie algebras have the same complexification, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See spinor for some examples associated with spinor representations of the spin groups and . If is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), : is the conjugate of for all in For a unitary representation, the dual representation and the conjugate representation coincide. ==See also== *Dual 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「complex conjugate representation」の詳細全文を読む スポンサード リンク
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