complex conjugate representation

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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}}.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.

{{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 $\backslash mathfrak\{g\}$ 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.