Complex conjugate representation explained

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 .^[1]

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

ak{g} 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

1. This is the mathematicians' convention. Physicists use a different convention where the Lie bracket of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition.