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.
is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide.
This also holds for pseudounitary representations.