Complex conjugate representation

In mathematics, if Template:Math is a group and Template:Math is a representation of it over the complex vector space Template:Math, then the complex conjugate representation Template:Math is defined over the complex conjugate vector space Template:Math as follows:

Template:Math is the conjugate of Template:Math for all Template:Math in Template:Math.

Template:Math is also a representation, as one may check explicitly.

If Template:Math is a real Lie algebra and Template:Math is a representation of it over the vector space Template:Math, then the conjugate representation Template:Math is defined over the conjugate vector space Template:Math as follows:

Template:Math is the conjugate of Template:Math for all Template:Math in Template:Math.<ref>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.</ref>

Template:Math 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 Template:Math and Template:Math.

If <math>\mathfrak{g}</math> is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),

Template:Math is the conjugate of Template:Math for all Template:Math in Template:Math

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

See also

• Dual representation

Notes

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