Complex conjugate representation

In mathematics, if $G$ is a group and $Π$ is a representation of it over the complex vector space $V$ , then the complex conjugate representation $Π$ is defined over the complex conjugate vector space $V$ as follows:

$Π$ is also a representation, as one may check explicitly.

If $g$ is a real Lie algebra and $π$ is a representation of it over the vector space $V$ , then the conjugate representation $π$ is defined over the conjugate vector space $V$ as follows:

$π$ 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 $Spin(p + q)$ and $Spin(p, q)$ .

If ${\mathfrak {g}}$ is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),

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