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The | 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]] | |||
Revision as of 14:18, 7 January 2014
In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on defined by the formula
where tr denotes the trace of a linear operator. The criterion was introduced by Template:Harvs.
Cartan's criterion for solvability
Cartan's criterion for solvability states:
- A Lie subalgebra of endomorphisms of a finite dimensional vector space over a field of characteristic zero is solvable if and only if whenever
The fact that in the solvable case follows immediately from 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 over a field of characteristic zero is solvable if and only if (where K is the Killing form).
Cartan's criterion for semisimplicity
Cartan's criterion for semisimplicity states:
- A finite-dimensional Lie algebra over a field of characteristic zero is semisimple if and only if the Killing form is non-degenerate.
Template:Harvtxt gave a very short proof that if a finite dimensional Lie algebra (in any characteristic) has a 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 SLp(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 an for n∈Z/pZ and bracket [ai,aj] = (i−j)ai+j is simple for p>2 but has no nonzero invariant bilinear form.
- If k has characteristic 2 then the semidirect product gl2(k).k2 is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl2(k).k2.
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(b2)=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]/tnL[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 tn−1 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
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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.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010