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| | In [[mathematics]], if ''G'' is a [[group (mathematics)|group]] and ρ is a [[representation theory|representation]] of it over the [[complex number|complex]] [[vector space]] V, then the '''complex conjugate representation''' ρ* is defined over the [[conjugate vector space]] V* as follows: |
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| | :ρ*(''g'') is the [[conjugate linear map|conjugate]] of ρ(''g'') for all ''g'' in ''G''. |
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| | ρ* is also a representation, as you may check explicitly. |
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| | If <math>\mathfrak{g}</math> is a [[real number|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: |
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| | :ρ*(''u'') is the conjugate of ρ(''u'') for all ''u'' in <math>\mathfrak{g}</math>.<ref>This is the mathematicians' convention. Physicists use a different convention where the [[Lie bracket of vector fields|Lie bracket]] of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition.</ref> |
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| | ρ* is also a representation, as you may check explicitly. |
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| | 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 group]]s Spin(p+q) and Spin(p,q). |
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| | If <math>\mathfrak{g}</math> is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), |
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| | :ρ*(''u'') is the conjugate of −ρ(''u''*) for all ''u'' in <math>\mathfrak{g}</math> |
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| | For a [[unitary representation]], the dual representation and the conjugate representation coincide. |
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| | ==See also== |
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| | *[[Dual representation]] |
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| | ==Notes== |
| | <references/> |
| | [[Category:Representation theory of groups]] |
Revision as of 22:27, 16 January 2014
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 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 .[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 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
For a unitary representation, the dual representation and the conjugate representation coincide.
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.