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Undid revision 642185579 by 109.104.166.38 (talk) – WP pages are not the place to advertise related tools

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	~~In [[mathematics]], a '''Cartan subalgebra''', often abbreviated as '''CSA''', is a [[nilpotent Lie algebra\|nilpotent]] [[subalgebra]] <math>\mathfrak{h}</math> of a [[Lie algebra]]~~ <~~math~~>~~\mathfrak{g}~~<~~/math~~> ~~that~~ is [~~[self-normalising]] (if <math>[X,Y] \in \mathfrak{h}<~~/~~math> for all <math>X \in \mathfrak{h}<~~/~~math>, then <math>Y \in \mathfrak{h}</math>)~~. ~~They were introduced by [[Élie Cartan~~]~~] in~~ his ~~doctoral thesis.~~

	~~== Existence and uniqueness ==~~

	~~Cartan subalgebras exist for finite dimensional Lie algebras whenever~~ the ~~base field is infinite~~. If the ~~field is algebraically closed~~ of ~~characteristic 0 and the algebra is finite dimensional then all Cartan subalgebras are conjugate under automorphisms of the Lie algebra, and in particular are all isomorphic~~.

	~~[[Kac–Moody algebra]]s and [[generalized Kac–Moody algebra]]s also have Cartan subalgebras~~.

	~~== Properties ==~~

	~~A Cartan subalgebra of a finite dimensional [[semisimple Lie algebra]] over an [[algebraically closed field]~~] of ~~characteristic 0 is [[abelian Lie algebra\|abelian]]~~ and
	~~also has the following property of its~~ [~~[adjoint representation of a Lie algebra\|adjoint representation]]~~: ~~the [[weight (representation theory)\|weight]] [[eigenspace]]s of <math>\mathfrak{g}<~~/~~math> [[restricted representation\|restricted~~]~~] to <math>\mathfrak{h}</math> diagonalize the representation~~, ~~and the eigenspace of the zero weight vector~~ is ~~<math>\mathfrak{h}</math>. (So,~~ the ~~centralizer of <math>\mathfrak{h}</math> coincides with <math>\mathfrak{h}</math>~~.) The ~~non-zero weights are called~~ the ~~'''[[root system\|roots]]''',~~ and the ~~corresponding eigenspaces are called '''[[root system\|root spaces]]'''~~, ~~and are all 1-dimensional.~~

	~~If <math>\mathfrak{g}</math> is a [[linear Lie algebra]] (a Lie subalgebra of~~ the ~~Lie algebra of endomorphisms~~ of a ~~finite-dimensional vector space ''V'') over an algebraically closed field, then any Cartan subalgebra of~~ <~~math~~>~~\mathfrak{g}~~<~~/math~~> ~~is the [[centralizer]]~~ of ~~a maximal [[toral Lie algebra\|toral Lie subalgebra]] of <math>\mathfrak{g}</math>; that~~ is~~, a subalgebra consisting entirely~~ of elements ~~which are diagonalizable as endomorphisms of ''V'' which is maximal~~ in ~~the sense that it is not properly included in any other such subalgebra~~. ~~If <math>\mathfrak{g}</math> is semisimple and the field has characteristic zero~~, ~~then a maximal toral subalgebra is self-normalizing, and so is equal~~ to ~~the associated Cartan subalgebra. If in addition <math>\mathfrak g</math> is semisimple~~, ~~then~~ the [~~[Adjoint representation of a Lie group\|adjoint representation]] presents <math>\mathfrak g<~~/~~math> as a linear Lie algebra, so that a subalgebra of <math>\mathfrak g<~~/~~math> is Cartan if and only if it is a maximal toral subalgebra~~. ~~An advantage of this approach is that it is trivial to show the existence of such a subalgebra~~. ~~In fact, if <math>\mathfrak g</math> has only nilpotent elements, then it is [[nilpotent Lie algebra\|nilpotent~~]~~] ([[Engel's theorem]])~~, ~~but then its [[Killing form]] is identically zero~~, ~~contradicting semisimplicity. Hence~~, ~~<math>\mathfrak g</math> must have a nonzero semisimple element~~.

	~~== Examples ==~~

	*Any nilpotent Lie algebra is its own Cartan subalgebra.
	*A Cartan subalgebra of the Lie algebra of [[square matrix\|''n''×''n'' matrices]] over a ~~field is the algebra of all diagonal matrices~~.
	*The Lie algebra sl<~~sub~~>2<~~/sub~~>~~('''R''')~~ of ~~2 by 2 matrices of trace 0 has two non-conjugate Cartan subalgebras.~~
	*The dimension of a Cartan subalgebra is not in ~~general the maximal dimension~~ of ~~an abelian subalgebra~~, ~~even for complex simple Lie algebras~~. ~~For example~~, ~~the Lie algebra ''sl''<sub>2''n''</sub>('''C''') of 2''n'' by 2''n'' matrices of trace 0 has~~ a ~~Cartan subalgebra~~ of ~~rank 2''n''−1 but has~~ a ~~maximal abelian subalgebra of dimension ''n''<sup>2</sup> consisting of all matrices~~ of the ~~form <math>{0\ A\choose 0\ 0}</math> with ''A'' any ''n'' by ''n'' matrix.~~ ~~One can directly see this abelian subalgebra is not a Cartan subalgebra, since it is contained~~ in ~~the nilpotent algebra of strictly upper triangular matrices (which is also not a Cartan subalgebra since it is normalized by diagonal matrices).~~

	~~== Splitting Cartan subalgebra ==~~
	~~{{main\|Splitting Cartan subalgebra}}~~

	~~Over non~~-~~algebraically closed fields, not all Cartan subalgebras are conjugate~~. ~~An important class are [[splitting Cartan subalgebra]]s: if a Lie algebra admits a splitting Cartan subalgebra~~ <~~math~~>~~\mathfrak{h}~~<~~/math~~> ~~then it~~ is ~~called ''splittable,'' and the pair <math>~~(~~\mathfrak{g},\mathfrak{h})<~~/~~math> is called a [[split Lie algebra]]; over an algebraically closed field every semisimple Lie algebra is splittable~~. Any two splitting Cartan algebras are conjugate, and they fulfill a similar function to Cartan algebras in semisimple Lie algebras over algebraically closed fields, so split semisimple Lie algebras (indeed, split reductive Lie algebras) share many properties with semisimple Lie algebras over algebraically closed fields.

	~~Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however~~.

	~~== See also ==~~
	*[[Cartan subgroup]]
	*[[Carter subgroup]]

	~~== References ==~~
	*{{Citation \| last1=Borel \| first1=Armand \| author1-link=Armand Borel \| title=Linear algebraic groups \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=2nd \| series=Graduate Texts in Mathematics \| isbn=978-0-387-97370-8 \| mr=1102012 \| year=1991 \| volume=126}}
	*{{Citation \| last1=Jacobson \| first1=Nathan \| author1-link=Nathan Jacobson \| title=Lie algebras \| publisher=[[Dover Publications]] \| location=New York \| isbn=978-0-486-63832-4 \| mr=559927 \| year=1979}}
	*{{Citation \| last1=Humphreys \| first1=James E. \| title=Introduction to Lie Algebras and Representation Theory \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-90053-7 \| year=1972}}
	*{{springer\|id=C/c020550\|first=V.~~L.\|last= Popov\|authorlink=Vladimir L. Popov}}~~

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

en>Quondum: /* External links */ removing link to simple PHP password generator. The outputs appear to be completely predictable, so definitely not strong.

2013-11-11T17:11:53Z

External links: removing link to simple PHP password generator. The outputs appear to be completely predictable, so definitely not strong.

← Older revision		Revision as of 19:11, 11 November 2013
Line 1:		Line 1:
	~~h?tel sol cayo santa maria all inclusive~~ ~~cayo santa~~ <br><br>~~mariaThe issues on which AWID operates run the gamut from spiritual fundamentalisms to funding for women's legal rights work. In essence, any concern that touches~~ <br><br><br>~~on women's rights~~ in ~~development~~ <br><br>~~falls~~ in ~~our concentrate. Internally~~, ~~we divide our do the job into strategic initiatives and programs -~~ <br><br>~~Demanding~~ and ~~Resisting Spiritual Fundamentalisms~~, ~~Where~~ is the ~~Revenue for Women's Legal rights?, Making~~ [~~https~~://~~Www~~.~~flickr.com~~/~~search~~/~~?q=Feminist+Movements Feminist Movements~~] and ~~Companies, Women~~'~~s Rights Facts~~, ~~Influencing Development Actors~~ and ~~Exercise for Women~~'~~s Rights~~, ~~Younger Feminist~~ <br><br>~~Activist System and~~ the ~~AWID Forum.. En Espagne, la finale de l opposant l l~~ a ~~rassembl fifteen,5 hundreds~~ of ~~thousands~~ <br><br>~~de t sur la cha Tele 5~~, ~~meilleure viewers de tous les temps pour tn requin pas cher ce pays. Le soccer y est le activity le~~ as well as regard o il repr 59% du volume total de activity visionn la t C le cas dans l de l : en Allemagne (forty four%) ou au Royaume nike air max Uni (34%) o il est talonn par les Jeux Olympiques de Londres. ~~La chaussure nike pas cher nike shox c de cl des Jeux Olympiques de Londres r d la meilleure viewers de l tous genres confondus~~, ~~fin ao 2012~~, ~~avec 24~~,~~5 millions de t sur BBC1 (78~~,~~7% de portion d En France~~, ~~m cas de figure chaussure nike : le soccer~~, ~~en t des sports activities les plus regard la t~~ (~~26%) est suivi de pr par les JO (twenty five%~~)~~.. Pont-Aven. Modeste bourg au Moyen-age puis petite ville commer?ante connue~~, ~~jusqu'au XIXme sicle~~, ~~essentiellement pour ses moulins et son port~~, ~~Pont-Aven attirera par son charme unique, partir de 1864, maillot psg les artistes [http:~~/~~/Photobucket~~.~~com/images/amoureux amoureux~~] ~~de la nature maillot de foot et de la lumire~~. ~~Ce fut le maillot foot dbut d~~'~~un changement radical pour la vie du village~~. ~~If you happen to be~~ even ~~now jonesing~~ for ~~soccer food~~, ~~test reducing~~ the ~~hurt. Consuming the pet dog~~ minus the ~~bun will conserve maillot de foot pas cher about five PointsPlus values~~. ~~Have only 1 air jordan 6 slice~~ of ~~pizza and you can cut six PointsPlus values~~.. ~~En ce qui concerne les th de ces affrontement rugueux, les traditionnels terrains de foot font put des ar ferm permettant de~~ s'~~aider des murs pour progresser m s~~'~~il faut avouer que cet element-l air jordan n~~'~~est pas vraiment d dans le jeu. Not possible d~~' ~~aux coups de boutoir de votre adversaire~~, ~~il faut foncer t baiss dans le beat, tout en restant concentr air jordan four au most~~. ~~En dehors des tirs traditionnels avec B~~, ~~des passes avec A~~, ~~des lobs avec Z et des d avec le adhere du Nunchuck~~, ~~le jeu poss quelques subtilit absentes du leading volet~~, ~~r air jordan femme aux demandes l r une diversit du gameplay plut maussade~~. ~~Bref~~, ~~cibl trs grand public~~, ~~ce jeu de football ne sduira que les air jordan pas cher joueurs occasionnels qui veulent faire trembler les filets air jordan three une dernire fois avec Zinedine Zidane~~. ~~Pour les execs, les vrais, vous pouvez rester sur Pro Evolution Soccer five et attendre gentiment le sixime opus en Octobre~~. ~~Sur ce, bons matchs la tlvision~~.~~<br><br>Here is more info regarding~~ [~~http~~:~~//tinyurl.com/odudrnx jordan shoes sale~~] ~~stop by our own web site.~~		{{Lie groups \|Semi-simple}}

			In [[mathematics]], a '''Cartan subalgebra''', often abbreviated as '''CSA''', is a [[nilpotent Lie algebra\|nilpotent]] [[subalgebra]] <math>\mathfrak{h}</math> of a [[Lie algebra]] <math>\mathfrak{g}</math> that is [[self-normalising]] (if <math>[X,Y] \in \mathfrak{h}</math> for all <math>X \in \mathfrak{h}</math>, then <math>Y \in \mathfrak{h}</math>). They were introduced by [[Élie Cartan]] in his doctoral thesis.

			== Existence and uniqueness ==

			Cartan subalgebras exist for finite dimensional Lie algebras whenever the base field is infinite. If the field is algebraically closed of characteristic 0 and the algebra is finite dimensional then all Cartan subalgebras are conjugate under automorphisms of the Lie algebra, and in particular are all isomorphic.

			[[Kac–Moody algebra]]s and [[generalized Kac–Moody algebra]]s also have Cartan subalgebras.

			== Properties ==

			A Cartan subalgebra of a finite dimensional [[semisimple Lie algebra]] over an [[algebraically closed field]] of characteristic 0 is [[abelian Lie algebra\|abelian]] and
			also has the following property of its [[adjoint representation of a Lie algebra\|adjoint representation]]: the [[weight (representation theory)\|weight]] [[eigenspace]]s of <math>\mathfrak{g}</math> [[restricted representation\|restricted]] to <math>\mathfrak{h}</math> diagonalize the representation, and the eigenspace of the zero weight vector is <math>\mathfrak{h}</math>. (So, the centralizer of <math>\mathfrak{h}</math> coincides with <math>\mathfrak{h}</math>.) The non-zero weights are called the '''[[root system\|roots]]''', and the corresponding eigenspaces are called '''[[root system\|root spaces]]''', and are all 1-dimensional.

			If <math>\mathfrak{g}</math> is a [[linear Lie algebra]] (a Lie subalgebra of the Lie algebra of endomorphisms of a finite-dimensional vector space ''V'') over an algebraically closed field, then any Cartan subalgebra of <math>\mathfrak{g}</math> is the [[centralizer]] of a maximal [[toral Lie algebra\|toral Lie subalgebra]] of <math>\mathfrak{g}</math>; that is, a subalgebra consisting entirely of elements which are diagonalizable as endomorphisms of ''V'' which is maximal in the sense that it is not properly included in any other such subalgebra. If <math>\mathfrak{g}</math> is semisimple and the field has characteristic zero, then a maximal toral subalgebra is self-normalizing, and so is equal to the associated Cartan subalgebra. If in addition <math>\mathfrak g</math> is semisimple, then the [[Adjoint representation of a Lie group\|adjoint representation]] presents <math>\mathfrak g</math> as a linear Lie algebra, so that a subalgebra of <math>\mathfrak g</math> is Cartan if and only if it is a maximal toral subalgebra. An advantage of this approach is that it is trivial to show the existence of such a subalgebra. In fact, if <math>\mathfrak g</math> has only nilpotent elements, then it is [[nilpotent Lie algebra\|nilpotent]] ([[Engel's theorem]]), but then its [[Killing form]] is identically zero, contradicting semisimplicity. Hence, <math>\mathfrak g</math> must have a nonzero semisimple element.

			== Examples ==

			*Any nilpotent Lie algebra is its own Cartan subalgebra.
			*A Cartan subalgebra of the Lie algebra of [[square matrix\|''n''×''n'' matrices]] over a field is the algebra of all diagonal matrices.
			*The Lie algebra sl<sub>2</sub>('''R''') of 2 by 2 matrices of trace 0 has two non-conjugate Cartan subalgebras.
			*The dimension of a Cartan subalgebra is not in general the maximal dimension of an abelian subalgebra, even for complex simple Lie algebras. For example, the Lie algebra ''sl''<sub>2''n''</sub>('''C''') of 2''n'' by 2''n'' matrices of trace 0 has a Cartan subalgebra of rank 2''n''−1 but has a maximal abelian subalgebra of dimension ''n''<sup>2</sup> consisting of all matrices of the form <math>{0\ A\choose 0\ 0}</math> with ''A'' any ''n'' by ''n'' matrix. One can directly see this abelian subalgebra is not a Cartan subalgebra, since it is contained in the nilpotent algebra of strictly upper triangular matrices (which is also not a Cartan subalgebra since it is normalized by diagonal matrices).

			== Splitting Cartan subalgebra ==
			{{main\|Splitting Cartan subalgebra}}

			Over non-algebraically closed fields, not all Cartan subalgebras are conjugate. An important class are [[splitting Cartan subalgebra]]s: if a Lie algebra admits a splitting Cartan subalgebra <math>\mathfrak{h}</math> then it is called ''splittable,'' and the pair <math>(\mathfrak{g},\mathfrak{h})</math> is called a [[split Lie algebra]]; over an algebraically closed field every semisimple Lie algebra is splittable. Any two splitting Cartan algebras are conjugate, and they fulfill a similar function to Cartan algebras in semisimple Lie algebras over algebraically closed fields, so split semisimple Lie algebras (indeed, split reductive Lie algebras) share many properties with semisimple Lie algebras over algebraically closed fields.

			Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however.

			== See also ==
			*[[Cartan subgroup]]
			*[[Carter subgroup]]

			== References ==
			*{{Citation \| last1=Borel \| first1=Armand \| author1-link=Armand Borel \| title=Linear algebraic groups \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=2nd \| series=Graduate Texts in Mathematics \| isbn=978-0-387-97370-8 \| mr=1102012 \| year=1991 \| volume=126}}
			*{{Citation \| last1=Jacobson \| first1=Nathan \| author1-link=Nathan Jacobson \| title=Lie algebras \| publisher=[[Dover Publications]] \| location=New York \| isbn=978-0-486-63832-4 \| mr=559927 \| year=1979}}
			*{{Citation \| last1=Humphreys \| first1=James E. \| title=Introduction to Lie Algebras and Representation Theory \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-90053-7 \| year=1972}}
			*{{springer\|id=C/c020550\|first=V.L.\|last= Popov\|authorlink=Vladimir L. Popov}}

			[[Category:Lie algebras]]

en>DavidCary: convert external link to internal reference

2012-08-10T18:35:19Z

convert external link to internal reference

New page

h?tel sol cayo santa maria all inclusive cayo santa mariaThe issues on which AWID operates run the gamut from spiritual fundamentalisms to funding for women's legal rights work. In essence, any concern that touches on women's rights in development falls in our concentrate. Internally, we divide our do the job into strategic initiatives and programs - Demanding and Resisting Spiritual Fundamentalisms, Where is the Revenue for Women's Legal rights?, Making [https://Www.flickr.com/search/?q=Feminist+Movements Feminist Movements] and Companies, Women's Rights Facts, Influencing Development Actors and Exercise for Women's Rights, Younger Feminist Activist System and the AWID Forum.. En Espagne, la finale de l opposant l l a rassembl fifteen,5 hundreds of thousands de t sur la cha Tele 5, meilleure viewers de tous les temps pour tn requin pas cher ce pays. Le soccer y est le activity le as well as regard o il repr 59% du volume total de activity visionn la t C le cas dans l de l : en Allemagne (forty four%) ou au Royaume nike air max Uni (34%) o il est talonn par les Jeux Olympiques de Londres. La chaussure nike pas cher nike shox c de cl des Jeux Olympiques de Londres r d la meilleure viewers de l tous genres confondus, fin ao 2012, avec 24,5 millions de t sur BBC1 (78,7% de portion d En France, m cas de figure chaussure nike : le soccer, en t des sports activities les plus regard la t (26%) est suivi de pr par les JO (twenty five%).. Pont-Aven. Modeste bourg au Moyen-age puis petite ville commer?ante connue, jusqu'au XIXme sicle, essentiellement pour ses moulins et son port, Pont-Aven attirera par son charme unique, partir de 1864, maillot psg les artistes [http://Photobucket.com/images/amoureux amoureux] de la nature maillot de foot et de la lumire. Ce fut le maillot foot dbut d'un changement radical pour la vie du village. If you happen to be even now jonesing for soccer food, test reducing the hurt. Consuming the pet dog minus the bun will conserve maillot de foot pas cher about five PointsPlus values. Have only 1 air jordan 6 slice of pizza and you can cut six PointsPlus values.. En ce qui concerne les th de ces affrontement rugueux, les traditionnels terrains de foot font put des ar ferm permettant de s'aider des murs pour progresser m s'il faut avouer que cet element-l air jordan n'est pas vraiment d dans le jeu. Not possible d' aux coups de boutoir de votre adversaire, il faut foncer t baiss dans le beat, tout en restant concentr air jordan four au most. En dehors des tirs traditionnels avec B, des passes avec A, des lobs avec Z et des d avec le adhere du Nunchuck, le jeu poss quelques subtilit absentes du leading volet, r air jordan femme aux demandes l r une diversit du gameplay plut maussade. Bref, cibl trs grand public, ce jeu de football ne sduira que les air jordan pas cher joueurs occasionnels qui veulent faire trembler les filets air jordan three une dernire fois avec Zinedine Zidane. Pour les execs, les vrais, vous pouvez rester sur Pro Evolution Soccer five et attendre gentiment le sixime opus en Octobre. Sur ce, bons matchs la tlvision. Here is more info regarding [http://tinyurl.com/odudrnx jordan shoes sale] stop by our own web site.