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