Complex conjugate representation

In mathematics, if G is a group and ρ is a representation of it over the complex vector space V, then the complex conjugate representation ρ* is defined over the conjugate vector space V* as follows:

ρ*(g) is the conjugate of ρ(g) for all g in G.

ρ* is also a representation, as you may check explicitly.

If $\mathfrak{g}$ is a real Lie algebra and ρ is a representation of it over the vector space V, then the conjugate representation ρ* is defined over the conjugate vector space V* as follows:

ρ*(u) is the conjugate of ρ(u) for all u in $\mathfrak{g}$.^[1]

ρ* is also a representation, as you 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 Spin(p+q) and Spin(p,q).

If $\mathfrak{g}$ is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),

ρ*(u) is the conjugate of −ρ(u*) for all u in $\mathfrak{g}$

For a unitary representation, the dual representation and the conjugate representation coincide.

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.