Editing Complex conjugate representation

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In [[mathematics]], if {{math|''G''}} is a [[group (mathematics)|group]] and {{math|Π}} is a [[representation theory|representation]] of it over the [[complex number|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 linear map|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 number|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 vector fields|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 group]]s {{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|&minus;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==

*[[Dual representation]]

==Notes==
<references/>

[[Category:Representation theory of groups]]