Vacuum solution (general relativity): Difference between revisions
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{{about|a mathematical concept|the psychological concept|Dual representation (psychology)}} | |||
In [[mathematics]], if ''G'' is a [[group (mathematics)|group]] and ρ is a [[linear representation]] of it on the real [[vector space]] ''V'', then the '''dual representation''' {{Overline|ρ}} is defined over the [[dual vector space]] {{Overline|''V''}} as follows:<ref>Lecture 1 of {{Fulton-Harris}}</ref> | |||
:{{Overline|ρ}}(''g'') is the [[transpose of a linear map|transpose]] of ρ(''g''<sup>−1</sup>) | |||
for all ''g'' in ''G''. Then {{Overline|ρ}} is also a representation, as may be checked explicitly. The dual representation is also known as the '''contragredient representation'''. For a complex vector space, take <math> \overline{\rho}(g) := \rho(g)^{\dagger}</math> | |||
If <math>\mathfrak{g}</math> is a [[Lie algebra]] and ρ is a representation of it over the vector space ''V'', then the dual representation {{Overline|ρ}} is defined over the dual vector space {{Overline|''V''}} as follows:<ref>Lecture 8 of {{Fulton-Harris}}</ref> | |||
:{{Overline|ρ}}(''u'') is the transpose of −ρ(u) for all u in <math>\mathfrak{g}</math>. | |||
:{{Overline|ρ}} is also a representation, as can be explicitly checked. | |||
For a [[unitary representation]], the conjugate representation and the dual representation coincide, up to equivalence of representations. | |||
==Generalization== | |||
A general ring [[Module (mathematics)|module]] does not admit a dual representation. Modules of [[Hopf algebra]]s do, however. | |||
==See also== | |||
* [[Complex conjugate representation]] | |||
* [[Kirillov Character Formula]] | |||
==References== | |||
<references/> | |||
[[Category:Representation theory of groups]] | |||
Revision as of 21:59, 21 March 2013
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. In mathematics, if G is a group and ρ is a linear representation of it on the real vector space V, then the dual representation Template:Overline is defined over the dual vector space Template:Overline as follows:[1]
- Template:Overline(g) is the transpose of ρ(g−1)
for all g in G. Then Template:Overline is also a representation, as may be checked explicitly. The dual representation is also known as the contragredient representation. For a complex vector space, take
If is a Lie algebra and ρ is a representation of it over the vector space V, then the dual representation Template:Overline is defined over the dual vector space Template:Overline as follows:[2]
- Template:Overline(u) is the transpose of −ρ(u) for all u in .
- Template:Overline is also a representation, as can be explicitly checked.
For a unitary representation, the conjugate representation and the dual representation coincide, up to equivalence of representations.
Generalization
A general ring module does not admit a dual representation. Modules of Hopf algebras do, however.
See also
References
- ↑ Lecture 1 of Template:Fulton-Harris
- ↑ Lecture 8 of Template:Fulton-Harris