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en>Monkbot: /* References */Task 6g: add |script-title=; replace {{xx icon}} with |language= in CS1 citations; clean up language icons;

2014-11-18T12:15:48Z

References: Task 6g: add |script-title=; replace {{xx icon}} with |language= in CS1 citations; clean up language icons;

← Older revision		Revision as of 14:15, 18 November 2014
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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]].

	~~==Formal definition==~~

	~~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~~
	~~:<math>\mathrm{Ad}: G\to\mathrm{Aut}(\mathfrak g)\sub\mathrm{GL}(\mathfrak g)</math>~~
	~~be the [[adjoint representation]] of ''G''. The '''adjoint bundle''' of ''P''~~ is ~~the [[associated bundle]]~~
	~~:<math>\mathrm{Ad}_P = P\times_{\mathrm{Ad}}\mathfrak g</math>~~
	~~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~~
	~~:<math>[p\cdot g,x] = [p,\mathrm{Ad}_{g^{-1}}(x)]</math>~~
	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''.

	~~==Properties==~~

	~~[[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~~>.

	~~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]].~~

	~~==Notes==~~
	~~{{Reflist}}~~

	~~==References==~~
	* {{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}}
	* {{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~~

	~~[[Category:Vector bundles]]~~
	~~[[Category:Lie algebras]~~]

en>David Eppstein: /* References */ found one of the russian papers online at mathnet.ru (freely available, but untranslated)

2013-12-02T05:52:38Z

References: found one of the russian papers online at mathnet.ru (freely available, but untranslated)

← Older revision		Revision as of 07:52, 2 December 2013
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	I [http://~~help~~.~~ksu~~.~~edu~~.sa/~~node/65129 psychic phone readings~~] ~~would like~~ to ~~introduce myself to you~~, ~~I am Andrew~~ and ~~my wife doesn~~'~~t like it at all~~. ~~Distributing production is where my main income arrives from and it~~'~~s some thing I really enjoy~~. ~~To climb~~ is ~~something she would never give up. For many years he's been living good psychic -~~ [~~http~~://~~ltreme~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~127790/info/ ltreme~~.~~com~~] - in ~~Alaska and he doesn~~'~~t plan on changing it~~.<br><br>~~Stop~~ by ~~my web page clairvoyants~~ - [http://~~srncomm~~.~~com~~/~~blog~~/~~2014~~/~~08/25/relieve~~-~~that-stress-find-a-new-hobby/ mouse click the up coming website~~ page],		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
			</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]].

			==Formal definition==

			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
			:<math>\mathrm{Ad}: G\to\mathrm{Aut}(\mathfrak g)\sub\mathrm{GL}(\mathfrak g)</math>
			be the [[adjoint representation]] of ''G''. The '''adjoint bundle''' of ''P'' is the [[associated bundle]]
			:<math>\mathrm{Ad}_P = P\times_{\mathrm{Ad}}\mathfrak g</math>
			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
			:<math>[p\cdot g,x] = [p,\mathrm{Ad}_{g^{-1}}(x)]</math>
			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''.

			==Properties==

			[[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>.

			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]].

			==Notes==
			{{Reflist}}

			==References==
			* {{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}}
			* {{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

			[[Category:Vector bundles]]
			[[Category:Lie algebras]]

en>David Eppstein: /* References */ authorlink Gordon Royle

2012-08-15T19:09:13Z

References: authorlink Gordon Royle

New page

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