COMPLEX CONJUGATE REPRESENTATION

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 $mathfrak\{g\}$ 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 $mathfrak\{g\}$.^[1]
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 $mathfrak\{g\}$ 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 $mathfrak\{g\}$
For a unitary representation, the dual representation and the conjugate representation coincide.

Contents

See also

Notes

See also

★ dual representation

Notes

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.