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 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 .
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 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
For a unitary representation, the dual representation and the conjugate representation coincide.
Famous quotes containing the word complex:
“In the case of all other sciences, arts, skills, and crafts, everyone is convinced that a complex and laborious programme of learning and practice is necessary for competence. Yet when it comes to philosophy, there seems to be a currently prevailing prejudice to the effect that, although not everyone who has eyes and fingers, and is given leather and last, is at once in a position to make shoes, everyone nevertheless immediately understands how to philosophize.”
—Georg Wilhelm Friedrich Hegel (1770–1831)