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In mathematics, if {{math|''G''}} is a group and {{math|Π}} is a representation of it over the complex vector space {{math|''V''}}, then the '''complex conjugate representation''' {{math|{{overline|Π}}}} is defined over the complex conjugate vector space {{math|{{overline|V}}}} as follows:
:{{math|{{overline|Π}}(''g'')}} is the conjugate of {{math|Π(''g'')}} for all {{math|''g''}} in {{math|''G''}}.
{{math|{{overline|Π}}}} is also a representation, as one may check explicitly.
If {{math|'''g'''}} is a real Lie algebra and {{math|π}} is a representation of it over the vector space {{math|''V''}}, then the conjugate representation {{math|{{overline|π}}}} is defined over the conjugate vector space {{math|{{overline|''V''}}}} as follows:
:{{math|{{overline|π}}(''X'')}} is the conjugate of {{math|π(''X'')}} for all {{math|''X''}} in {{math|'''g'''}}.<ref>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.</ref>
{{math|{{overline|π}}}} is also a representation, as one 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 {{math|Spin(''p'' + ''q'')}} and {{math|Spin(''p'', ''q'')}}.
If <math>\mathfrak{g}</math> is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),
:{{math|{{overline|π}}(''X'')}} is the conjugate of {{math|−π(''X''*)}} for all {{math|''X''}} in {{math|'''g'''}}
For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.
==See also==
==Notes== <references/>
Category:Representation theory of groups