Static universe - Revision history

130.102.158.18 at 02:13, 18 December 2014

2014-12-18T02:13:43Z

← Older revision		Revision as of 04:13, 18 December 2014
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	~~In mathematics~~, '~~''Cartan's criterion''' gives conditions for a [[Lie algebra]] in characteristic 0 to be~~ ~~[[solvable Lie algebra\|solvable]]~~, ~~which implies a related criterion for the Lie algebra to be~~ [~~[semisimple Lie algebra\|semisimple]]. It is based on the notion of the [[Killing form]], a [[symmetric bilinear form]] on <math>\mathfrak{g}</math> defined by the formula~~		I would like to introduce myself to you, I am Jayson Simcox but I don't like when people authentic psychic readings, [http://Www.Skullrocker.com/blogs/post/10991 skullrocker.com], use my complete name. I've always cherished living in Kentucky [http://165.132.39.93/xe/visitors/372912 telephone psychic] but now I'm considering other options. The preferred hobby for him and his kids is fashion and he'll be starting something else alongside with it. Distributing production is where my primary income arrives from and it's some thing I truly enjoy.<br><br>Check out my web page; [http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13352970 psychic love readings]
	: ~~<math>K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),<~~/~~math>~~
	~~where tr denotes the [[Trace (linear algebra)\|trace of a linear operator]]. The criterion was introduced by {{harvs\|txt\|authorlink=Élie Cartan\|first=Élie\|last= Cartan\|year=1894}}.~~

	~~== Cartan's criterion for solvability ==~~

	~~Cartan's criterion for solvability states:~~
	:''A Lie subalgebra <math>\mathfrak{g}</math> of endomorphisms of a finite dimensional vector space over a [[field (mathematics)\|field]] of [[characteristic zero]] is solvable if and only if <math>Tr(ab)=0</~~math> whenever <math>a\in\mathfrak{g},b\in[\mathfrak{g},\mathfrak{g}]~~.~~</math>''~~

	The fact that <math>Tr(ab)=0</math> in the solvable case follows immediately from [[Lie–Kolchin theorem\|Lie's theorem]] that solvable Lie algebras in characteristic 0 can be put in upper triangular form.

	~~Applying Cartan's criterion to the adjoint representation gives:~~
	~~:''A finite-dimensional Lie algebra <math>\mathfrak{g}<~~/~~math> over a [[field (mathematics)\|field]] of [[characteristic zero]] is solvable if and only if <math>K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0<~~/~~math> (where K is the Killing form).''~~

	~~== Cartan's criterion for semisimplicity ==~~

	~~Cartan's criterion for semisimplicity states:~~
	~~: ''A finite-dimensional Lie algebra <math>\mathfrak{g}<~~/~~math> over a [[field (mathematics)\|field]] of [[characteristic zero]] is semisimple if and only if the Killing form is [[degenerate form\|non-degenerate]]~~.''

	~~{{harvtxt\|Dieudonné\|1953}} gave a very short proof that if a finite dimensional Lie algebra (in any characteristic) has a [[Quadratic Lie algebra\|non-degenerate invariant bilinear form]~~] ~~and no non-zero abelian ideals, and in particular if its Killing form is non-degenerate, then it is a sum of simple Lie algebras.~~

	~~Conversely~~, ~~it follows easily from Cartan's criterion for solvability that a semisimple algebra (in characteristic 0) has a non-degenerate Killing form~~.

	~~==Examples==~~

	~~Cartan~~'~~s criteria fail~~ in ~~characteristic ''p''>0; for example~~:
	*the Lie algebra SL<sub>''p''</~~sub>(''k'') is simple if ''k'' has characteristic not 2 and has vanishing Killing form, though it does have a nonzero invariant bilinear form given by (''a'',''b'') = Tr(''ab'')~~.
	*the Lie algebra with basis ''a''<sub>''n''</sub> for ''n''∈'''Z'''/~~''p'''''Z''' and bracket [''a''<sub>''i''<~~/~~sub>,''a''<sub>''j''<~~/~~sub>~~] ~~= (''i''−''j'')''a''<sub>''i''+''j''</sub> is simple for ''p''>2~~ but ~~has no nonzero invariant bilinear form.~~
	*If ''k'' has characteristic 2 then the semidirect product gl<sub>2</sub>('~~'k'')~~.~~''k''<sup>2</sup> is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl<sub>2</sub>(''k'').''k''<sup>2</sup>.~~

	~~If a finite dimensional Lie algebra is nilpotent, then the Killing form is identically zero (~~and more generally the Killing form vanishes on any nilpotent ideal). The converse is false: there are non-nilpotent Lie algebras whose Killing form vanishes. An example is given by the semidirect product of an abelian Lie algebra ''V'' with a 1-dimensional Lie algebra acting on ''V'' as an endomorphism ''b'' such that ''b'' is ~~not nilpotent~~ and ~~Tr(~~'~~'b''<sup>2</sup>)=0.~~

	In characteristic 0, every reductive Lie algebra (one that is a sum of abelian and simple Lie algebras) has a non-degenerate invariant symmetric bilinear form. However the converse is false: a Lie algebra with ~~a non-degenerate invariant symmetric bilinear form need not be a sum of simple and abelian Lie algebras~~. ~~A typical counterexample~~ is ~~''G'' = ''L''[''t'']/''t''<sup>''n''</sup>''L''[''t'']~~ where ~~''n''>1, ''L'' is a simple complex Lie algebra with a bilinear form (,),~~ and ~~the bilinear form on ''G'' is given by taking the coefficient of ''t'~~'<~~sup~~>~~''n''−1~~<~~/sup~~> ~~of the '''C'''~~[~~''t'']-valued bilinear form on ''G'' induced by the form on ''L''. The bilinear form is non-degenerate, but the Lie algebra is not a sum of simple and abelian Lie algebras.~~

	~~== References ==~~

	*{{Citation \| last1=Cartan \| first1=Élie \| title=Sur la structure des groupes de transformations finis et continus \| url=http://~~books~~.~~google~~.com/~~books~~?id=~~JY8LAAAAYAAJ \| publisher~~=~~Nony \| series=Thesis \| year=1894}}~~
	*{{Citation \| last1=Dieudonné \| first1=Jean \| author1-link=Jean Dieudonné \| title=On semi-simple Lie algebras \| jstor=2031832 \| mr=0059262 \| year=1953 \| journal=[[Proceedings of the American Mathematical Society]] \| issn=0002-9939 \| volume=4 \| pages=931–932}}
	*{{Citation \| last1=Serre \| first1=Jean-Pierre \| author1-link=Jean-Pierre Serre \| title=Lie algebras and Lie groups \| origyear=1964 \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Mathematics \| isbn=978-3-540-55008-2 \| doi=10.1007/978-3-540-70634-2 \| mr=2179691 \| year=2006 \| volume=1500}}

	~~== See also~~ ==
	* [[Modular Lie algebra]]

	~~[[Category:Lie algebras]~~]

en>AstroHistorian: /* Einstein's universe */ more detail and accuracy

2014-01-07T13:18:33Z

Einstein's universe: more detail and accuracy

← Older revision		Revision as of 15:18, 7 January 2014
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	The ~~author~~'s ~~title~~ is ~~Christy~~. ~~Invoicing~~ is ~~my occupation~~. ~~The favorite hobby~~ for ~~him~~ and ~~his kids~~ is ~~fashion~~ and he'~~ll be starting something else alongside~~ with it. I'~~ve usually loved living in Alaska~~.<br><br>~~my web~~-~~site~~: ~~free psychic readings~~ - [http://~~www.Chk~~.~~woobi~~.~~co.kr/xe~~/?~~document_srl~~=~~346069 just click~~ the ~~up coming article~~] -		In mathematics, '''Cartan's criterion''' gives conditions for a [[Lie algebra]] in characteristic 0 to be [[solvable Lie algebra\|solvable]], which implies a related criterion for the Lie algebra to be [[semisimple Lie algebra\|semisimple]]. It is based on the notion of the [[Killing form]], a [[symmetric bilinear form]] on <math>\mathfrak{g}</math> defined by the formula
			: <math>K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),</math>
			where tr denotes the [[Trace (linear algebra)\|trace of a linear operator]]. The criterion was introduced by {{harvs\|txt\|authorlink=Élie Cartan\|first=Élie\|last= Cartan\|year=1894}}.

			== Cartan's criterion for solvability ==

			Cartan's criterion for solvability states:
			:''A Lie subalgebra <math>\mathfrak{g}</math> of endomorphisms of a finite dimensional vector space over a [[field (mathematics)\|field]] of [[characteristic zero]] is solvable if and only if <math>Tr(ab)=0</math> whenever <math>a\in\mathfrak{g},b\in[\mathfrak{g},\mathfrak{g}].</math>''

			The fact that <math>Tr(ab)=0</math> in the solvable case follows immediately from [[Lie–Kolchin theorem\|Lie's theorem]] that solvable Lie algebras in characteristic 0 can be put in upper triangular form.

			Applying Cartan's criterion to the adjoint representation gives:
			:''A finite-dimensional Lie algebra <math>\mathfrak{g}</math> over a [[field (mathematics)\|field]] of [[characteristic zero]] is solvable if and only if <math>K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0</math> (where K is the Killing form).''

			== Cartan's criterion for semisimplicity ==

			Cartan's criterion for semisimplicity states:
			: ''A finite-dimensional Lie algebra <math>\mathfrak{g}</math> over a [[field (mathematics)\|field]] of [[characteristic zero]] is semisimple if and only if the Killing form is [[degenerate form\|non-degenerate]].''

			{{harvtxt\|Dieudonné\|1953}} gave a very short proof that if a finite dimensional Lie algebra (in any characteristic) has a [[Quadratic Lie algebra\|non-degenerate invariant bilinear form]] and no non-zero abelian ideals, and in particular if its Killing form is non-degenerate, then it is a sum of simple Lie algebras.

			Conversely, it follows easily from Cartan's criterion for solvability that a semisimple algebra (in characteristic 0) has a non-degenerate Killing form.

			==Examples==

			Cartan's criteria fail in characteristic ''p''>0; for example:
			*the Lie algebra SL<sub>''p''</sub>(''k'') is simple if ''k'' has characteristic not 2 and has vanishing Killing form, though it does have a nonzero invariant bilinear form given by (''a'',''b'') = Tr(''ab'').
			*the Lie algebra with basis ''a''<sub>''n''</sub> for ''n''∈'''Z'''/''p'''''Z''' and bracket [''a''<sub>''i''</sub>,''a''<sub>''j''</sub>] = (''i''−''j'')''a''<sub>''i''+''j''</sub> is simple for ''p''>2 but has no nonzero invariant bilinear form.
			*If ''k'' has characteristic 2 then the semidirect product gl<sub>2</sub>(''k'').''k''<sup>2</sup> is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl<sub>2</sub>(''k'').''k''<sup>2</sup>.

			If a finite dimensional Lie algebra is nilpotent, then the Killing form is identically zero (and more generally the Killing form vanishes on any nilpotent ideal). The converse is false: there are non-nilpotent Lie algebras whose Killing form vanishes. An example is given by the semidirect product of an abelian Lie algebra ''V'' with a 1-dimensional Lie algebra acting on ''V'' as an endomorphism ''b'' such that ''b'' is not nilpotent and Tr(''b''<sup>2</sup>)=0.

			In characteristic 0, every reductive Lie algebra (one that is a sum of abelian and simple Lie algebras) has a non-degenerate invariant symmetric bilinear form. However the converse is false: a Lie algebra with a non-degenerate invariant symmetric bilinear form need not be a sum of simple and abelian Lie algebras. A typical counterexample is ''G'' = ''L''[''t'']/''t''<sup>''n''</sup>''L''[''t''] where ''n''>1, ''L'' is a simple complex Lie algebra with a bilinear form (,), and the bilinear form on ''G'' is given by taking the coefficient of ''t''<sup>''n''−1</sup> of the '''C'''[''t'']-valued bilinear form on ''G'' induced by the form on ''L''. The bilinear form is non-degenerate, but the Lie algebra is not a sum of simple and abelian Lie algebras.

			== References ==

			*{{Citation \| last1=Cartan \| first1=Élie \| title=Sur la structure des groupes de transformations finis et continus \| url=http://books.google.com/books?id=JY8LAAAAYAAJ \| publisher=Nony \| series=Thesis \| year=1894}}
			*{{Citation \| last1=Dieudonné \| first1=Jean \| author1-link=Jean Dieudonné \| title=On semi-simple Lie algebras \| jstor=2031832 \| mr=0059262 \| year=1953 \| journal=[[Proceedings of the American Mathematical Society]] \| issn=0002-9939 \| volume=4 \| pages=931–932}}
			*{{Citation \| last1=Serre \| first1=Jean-Pierre \| author1-link=Jean-Pierre Serre \| title=Lie algebras and Lie groups \| origyear=1964 \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Mathematics \| isbn=978-3-540-55008-2 \| doi=10.1007/978-3-540-70634-2 \| mr=2179691 \| year=2006 \| volume=1500}}

			== See also ==
			* [[Modular Lie algebra]]

			[[Category:Lie algebras]]

en>Allens: Revert of good faith edit(s) by 69.139.0.23 using STiki

2012-05-29T11:42:57Z

Revert of good faith edit(s) by 69.139.0.23 using STiki

New page

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