In mathematics, if is a group and is a representation of it over the complex vector space , then the complex conjugate representation is defined over the complex conjugate vector space as follows: is the conjugate of for all in . is also a representation, as one may check explicitly. If is a real Lie algebra and is a representation of it over the vector space , then the conjugate representation is defined over the conjugate vector space as follows: is the conjugate of for all in . 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 and . If is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), is the conjugate of for all in 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 Representation theory of groups