Complex conjugate representation

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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] 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].

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

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

Please note that 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 Spin(p+q) and Spin(p,q).

If [\mathfrak] 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]