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 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