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 $\overline{Π}$ is defined over the complex conjugate vector space $\overline{V}$ as follows:


 * $\overline{Π}(g)$ is the conjugate of $Π(g)$ for all $g$ in $G$.

$\overline{Π}$ 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 $\overline{π}$ is defined over the conjugate vector space $\overline{V}$ as follows:


 * $\overline{π}(X)$ is the conjugate of $π(X)$ for all $X$ in $g$.

$\overline{π}$ 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 $Spin(p + q)$ 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),


 * $\overline{π}(X)$ is the conjugate of $&minus;π(X*)$ for all $X$ in $g$

For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.