Complex conjugate representation

In mathematics, if $G$ is a group and $Π$ is a representation of it over the complex vector space $V$ , then the complex conjugate representation $Π$ is defined over the complex conjugate vector space $V$ as follows:

Π (g)

is the conjugate of

Π(g)

for all

g

in

G

.

$Π$ is also a representation, as one may check explicitly.

If $g$ is a real Lie algebra and $π$ is a representation of it over the vector space $V$ , then the conjugate representation $π$ is defined over the conjugate vector space $V$ as follows:

π (X)

is the conjugate of

π(X)

for all

X

in

g

.^[1]

$π$ is also a representation, as one 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 ${\mathfrak {g}}$ is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),

π (X)

is the conjugate of

-π(X *)

for all

X

in

g

For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.

Notes

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

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

[1]

See also

Notes