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{{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]] | |||
Revision as of 18:11, 11 November 2013
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising (if for all , then ). 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 algebras and generalized Kac–Moody algebras 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 and also has the following property of its adjoint representation: the weight eigenspaces of restricted to diagonalize the representation, and the eigenspace of the zero weight vector is . (So, the centralizer of coincides with .) The non-zero weights are called the roots, and the corresponding eigenspaces are called root spaces, and are all 1-dimensional.
If 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 is the centralizer of a maximal toral Lie subalgebra of ; 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 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 is semisimple, then the adjoint representation presents as a linear Lie algebra, so that a subalgebra of 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 has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity. Hence, must have a nonzero semisimple element.
Examples
- Any nilpotent Lie algebra is its own Cartan subalgebra.
- A Cartan subalgebra of the Lie algebra of n×n matrices over a field is the algebra of all diagonal matrices.
- The Lie algebra sl2(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 sl2n(C) of 2n by 2n matrices of trace 0 has a Cartan subalgebra of rank 2n−1 but has a maximal abelian subalgebra of dimension n2 consisting of all matrices of the form 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
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Over non-algebraically closed fields, not all Cartan subalgebras are conjugate. An important class are splitting Cartan subalgebras: if a Lie algebra admits a splitting Cartan subalgebra then it is called splittable, and the pair 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
References
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
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Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
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