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: The term group can refer to several concepts: Look up Group in Wiktionary, the free dictionary In music, a group is another term for band or other musical ensemble. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

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: In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

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

(This is the mathematician's 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.) A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

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). In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ... In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ...

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

See also dual representation.

For a unitary representation, the dual representation and the conjugate representation coincides. In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group...