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| In [[mathematics]], an '''adjoint bundle''' <ref>{{cite journal|title=Higher order Utiyama-like theorem|author=J. Janyška|journal=Reports on Mathematical Physics|volume=58|year=2006|pages=93–118}} [cf. page 96]</ref> <ref>{{citation|last1 = Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operators in differential geometry|year = 1993|publisher = Springer-Verlag}} page 161 and page 400
| | Andrew Simcox is the title his mothers and fathers gave him and he completely loves this title. Playing badminton is a thing that he is totally addicted to. I've always loved living in Alaska. Invoicing is my occupation.<br><br>My web-site; [http://kjhkkb.net/xe/notice/374835 love psychics] |
| </ref> is a [[vector bundle]] naturally associated to any [[principal bundle]]. The fibers of the adjoint bundle carry a [[Lie algebra]] structure making the adjoint bundle into an [[algebra bundle]]. Adjoint bundles have important applications in the theory of [[connection (mathematics)|connections]] as well as in [[gauge theory]].
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| ==Formal definition==
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| Let ''G'' be a [[Lie group]] with [[Lie algebra]] <math>\mathfrak g</math>, and let ''P'' be a [[principal bundle|principal ''G''-bundle]] over a [[smooth manifold]] ''M''. Let
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| :<math>\mathrm{Ad}: G\to\mathrm{Aut}(\mathfrak g)\sub\mathrm{GL}(\mathfrak g)</math>
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| be the [[adjoint representation]] of ''G''. The '''adjoint bundle''' of ''P'' is the [[associated bundle]]
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| :<math>\mathrm{Ad}_P = P\times_{\mathrm{Ad}}\mathfrak g</math>
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| The adjoint bundle is also commonly denoted by <math>\mathfrak g_P</math>. Explicitly, elements of the adjoint bundle are [[equivalence class]]es of pairs [''p'', ''x''] for ''p'' ∈ ''P'' and ''x'' ∈ <math>\mathfrak g</math> such that
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| :<math>[p\cdot g,x] = [p,\mathrm{Ad}_{g^{-1}}(x)]</math>
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| for all ''g'' ∈ ''G''. Since the [[structure group]] of the adjoint bundle consists of Lie algebra [[automorphism]]s, the fibers naturally carry a Lie algebra structure making the adjoint bundle into a bundle of Lie algebras over ''M''.
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| ==Properties==
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| [[Vector-valued differential form|Differential forms]] on ''M'' with values in Ad<sub>''P''</sub> are in one-to-one correspondence with [[tensorial form|horizontal, ''G''-equivariant]] [[Lie algebra-valued form]]s on ''P''. A prime example is the [[curvature form|curvature]] of any [[connection (principal bundle)|connection]] on ''P'' which may be regarded as a 2-form on ''M'' with values in Ad<sub>''P''</sub>.
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| The space of sections of the adjoint bundle is naturally an (infinite-dimensional) Lie algebra. It may be regarded as the Lie algebra of the infinite-dimensional Lie group of [[gauge transformation]]s of ''P'' which can be thought of as sections of the bundle ''P'' ×<sub>Ψ</sub> ''G'' where Ψ is the action of ''G'' on itself by [[conjugation (group theory)|conjugation]].
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = [[Foundations of Differential Geometry]]|volume=Vol. 1| publisher=[[Wiley Interscience]] | year=1996|edition=New|isbn=0-471-15733-3}}
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| * {{citation|last1 = Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operators in differential geometry|year = 1993|publisher = Springer-Verlag}} page 161 and page 400
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| [[Category:Vector bundles]]
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| [[Category:Lie algebras]]
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Andrew Simcox is the title his mothers and fathers gave him and he completely loves this title. Playing badminton is a thing that he is totally addicted to. I've always loved living in Alaska. Invoicing is my occupation.
My web-site; love psychics