In mathematics, if is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

and is a representation of it over the

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

, then the complex conjugate representation is defined over the

complex conjugate vector space In mathematics, the complex conjugate of a complex vector space V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose scalar multiplication involves conjugation of the scalars. In other ...

as follows: : is the conjugate of for all in . is also a representation, as one may check explicitly. If is a

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

Lie algebra and is a representation of it over the vector space , then the conjugate representation is defined over the conjugate vector space as follows: : is the conjugate of for all in .This is the mathematicians' convention. Physicists use a different convention where the

Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition. is also a representation, as one may check explicitly. If two real Lie algebras have the same

complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...

, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See

spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...

for some examples associated with spinor representations of the

spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a ...

s and . If

\mathfrak

is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), : is the conjugate of for all in {{math, g For a finite-dimensional

unitary representation 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 ...

, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.

Notes

Representation theory of groups

See also

Notes