Английская Википедия:Complex conjugate representation
In mathematics, if Шаблон:Math is a group and Шаблон:Math is a representation of it over the complex vector space Шаблон:Math, then the complex conjugate representation Шаблон:Math is defined over the complex conjugate vector space Шаблон:Math as follows:
- Шаблон:Math is the conjugate of Шаблон:Math for all Шаблон:Math in Шаблон:Math.
Шаблон:Math is also a representation, as one may check explicitly.
If Шаблон:Math is a real Lie algebra and Шаблон:Math is a representation of it over the vector space Шаблон:Math, then the conjugate representation Шаблон:Math is defined over the conjugate vector space Шаблон:Math as follows:
- Шаблон:Math is the conjugate of Шаблон:Math for all Шаблон:Math in Шаблон:Math.[1]
Шаблон: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 Шаблон:Math and Шаблон:Math.
If <math>\mathfrak{g}</math> is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),
- Шаблон:Math is the conjugate of Шаблон:Math for all Шаблон:Math in Шаблон:Math
For a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.
See also
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.