Complex conjugate representation
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In mathematics, if GScript error: No such module "Check for unknown parameters". is a group and ΠScript error: No such module "Check for unknown parameters". is a representation of it over the complex vector space VScript error: No such module "Check for unknown parameters"., then the complex conjugate representation ΠScript error: No such module "Check for unknown parameters". is defined over the complex conjugate vector space VScript error: No such module "Check for unknown parameters". as follows:
- Π(g)Script error: No such module "Check for unknown parameters". is the conjugate of Π(g)Script error: No such module "Check for unknown parameters". for all gScript error: No such module "Check for unknown parameters". in GScript error: No such module "Check for unknown parameters"..
ΠScript error: No such module "Check for unknown parameters". is also a representation, as one may check explicitly.
If gScript error: No such module "Check for unknown parameters". is a real Lie algebra and πScript error: No such module "Check for unknown parameters". is a representation of it over the vector space VScript error: No such module "Check for unknown parameters"., then the conjugate representation πScript error: No such module "Check for unknown parameters". is defined over the conjugate vector space VScript error: No such module "Check for unknown parameters". as follows:
- π(X)Script error: No such module "Check for unknown parameters". is the conjugate of π(X)Script error: No such module "Check for unknown parameters". for all XScript error: No such module "Check for unknown parameters". in gScript error: No such module "Check for unknown parameters"..[1]
πScript error: No such module "Check for unknown parameters". 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)Script error: No such module "Check for unknown parameters". and Spin(p, q)Script error: No such module "Check for unknown parameters"..
If is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket),
- π(X)Script error: No such module "Check for unknown parameters". is the conjugate of −π(X*)Script error: No such module "Check for unknown parameters". for all XScript error: No such module "Check for unknown parameters". in gScript error: No such module "Check for unknown parameters".
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.