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:
“We must open our eyes and see that modern civilization has become so complex and the lives of civilized men so interwoven with the lives of other men in other countries as to make it impossible to be in this world and out of it.”
—Franklin D. Roosevelt (1882–1945)