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In [[mathematics]], the '''Lie–Kolchin theorem''' is a theorem in the [[representation theory]] of [[linear algebraic group]]s; '''Lie's theorem''' is the analog for [[linear Lie algebra]]s.
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It states that if ''G'' is a [[connected space|connected]] and [[solvable group|solvable]] [[linear algebraic group]] defined over an [[algebraically closed]] [[field (mathematics)|field]] and 
 
:<math>\rho\colon G \to GL(V)</math>
 
a [[group representation|representation]] on a nonzero finite-dimensional [[vector space]] ''V'', then there is a one-dimensional linear subspace ''L'' of ''V'' such that
 
: <math>\rho(G)(L) = L.</math>
 
That is, ρ(''G'') has an invariant line ''L'', on which ''G'' therefore acts through a one-dimensional representation. This is equivalent to the statement that ''V'' contains a nonzero vector ''v'' that is a common (simultaneous) eigenvector for all <math> \rho(g), \,\, g \in G </math>.
 
Because every (nonzero finite-dimensional) representation of ''G'' has a one-dimensional invariant subspace according to the Lie–Kolchin theorem, every [[irreducible]] finite-dimensional representation of a connected and solvable linear algebraic group ''G'' has dimension one, which is another way to state the Lie–Kolchin theorem.
 
Lie's theorem states that any nonzero representation of a solvable Lie algebra on a finite dimensional vector space over an algebraically closed field of characteristic 0 has a one-dimensional invariant subspace.  
 
The result  for Lie algebras was proved by {{harvs|txt|authorlink=Sophus Lie|first=Sophus |last=Lie|year=1876}} and for algebraic groups was proved by {{harvs|txt|authorlink=Ellis Kolchin|first=Ellis|last= Kolchin|year=1948|loc=p.19}}.
 
The [[Borel fixed point theorem]] generalizes the Lie–Kolchin theorem.
 
== Triangularization ==
Sometimes the theorem is also referred to as the ''Lie–Kolchin triangularization theorem'' because by induction it implies that with respect to a suitable basis of ''V'' the image <math>\rho(G)</math> has a ''triangular shape''; in other words, the image group <math>\rho(G)</math> is conjugate in GL(''n'',''K'') (where ''n'' = dim ''V'') to a subgroup of the group T of [[upper triangular]] matrices, the standard [[Borel subgroup]] of GL(''n'',''K''): the image is [[simultaneously triangularizable]].
 
The theorem applies in particular to a [[Borel subgroup]] of a [[semisimple algebraic group|semisimple]] [[linear algebraic group]] ''G''.
 
==Lie's theorem==
 
Lie's theorem states that if ''V'' is a finite dimensional vector space over an algebraically closed field of characteristic 0, then for any solvable Lie algebra of endomorphisms of ''V'' there is a vector that is an eigenvector for every element of the Lie algebra.  
 
Applying this result repeatedly shows that there is a basis for ''V'' such that all elements of the Lie algebra are represented by upper triangular matrices.
This is a generalization of the result of Frobenius that [[commuting matrices]] are simultaneously upper triangularizable, as commuting matrices form an [[abelian Lie algebra]], which is a fortiori solvable.
 
A consequence of Lie's theorem  is that any finite dimensional solvable Lie algebra over a field of characteristic 0 has a nilpotent derived algebra.
 
== Counter-examples ==
 
If the field ''K'' is not algebraically closed, the theorem can fail. The standard [[unit circle]], viewed as the set of [[complex number]]s <math> \{ x+iy \in \mathbb{C} \, | \, x^2+y^2=1 \} </math> of absolute value one is a one-dimensional commutative (and therefore solvable) [[linear algebraic group]] over the real numbers which has a two-dimensional representation into the [[special orthogonal group]] SO(2) without an invariant (real) line. Here the image <math> \rho(z)</math>  of <math> z=x+iy </math> is the [[orthogonal matrix]]
 
: <math> \begin{pmatrix} x & y \\ -y & x \end{pmatrix}.</math>
 
For algebraically closed fields of characteristic ''p''>0 Lie's theorem holds provided the dimension of the representation is less than ''p'', but can fail for representations of dimension ''p''. An example is given by the 3-dimensional nilpotent Lie algebra spanned by 1, ''x'', and ''d''/''dx'' acting on the ''p''-dimensional vector space ''k''[''x'']/(''x''<sup>''p''</sup>), which has no eigenvectors. Taking the semidirect product of this 3-dimensional Lie algebra by the ''p''-dimensional representation (considered as an abelian Lie algebra) gives a solvable Lie algebra whose derived algebra is not nilpotent.
 
==References==
 
*{{eom|first=V.V.|last= Gorbatsevich|id=l/l058710}}
*{{Citation | last1=Kolchin | first1=E. R. | title=Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations | jstor=1969111 | mr=0024884 | year=1948 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=49 | pages=1–42}}
*{{Citation | last1=Lie | first1=Sophus | author1-link=Sophus Lie | title=Theorie der Transformationsgruppen. Abhandlung II | url=http://www.archive.org/details/archivformathem02sarsgoog | year=1876 | journal=Archiv for Mathematik og Naturvidenskab | volume=1 | pages=152–193}}
*[[William C. Waterhouse]], ''Introduction to Affine Group Schemes'', Graduate Texts in Mathematics vol. 66, Springer Verlag New York, 1979 (chapter 10, in particular section 10.2).
 
{{DEFAULTSORT:Lie-Kolchin theorem}}
[[Category:Lie algebras]]
[[Category:Representation theory of algebraic groups]]
[[Category:Theorems in representation theory]]

Latest revision as of 01:28, 4 April 2014

To reach selling a home, it is advisable be competent in actual property marketing, legal, financial, operational features, and other knowledge and expertise. This is essential as a result of you'll want to negotiate with more and more refined consumers. It is advisable to outperform opponents, use latest applied sciences, and keep ahead of the quick altering market.

Professional brokers are readily available if you wish to find an residence for rent in singapore In some instances, landlords will consider you more favourably in case your agent involves them than if you happen to tried to strategy them by yourself. You want to watch out, nonetheless, as you determine in your agent. Ensure that the agent you're contemplating working with is registered with the IEA – Institute of Estate Agents. While it might seem a bother to you, will probably be worth it in the long run. The IEA works by an ordinary set of rules and regulations, so you will defend yourself towards presumably going with a rogue agent who costs you more than they need to for his or her service to find you an house for rent in singapore.

Leases of personal residential properties elevated by zero.9% in third Quarter 2012, in contrast with zero.3% in the earlier quarter. The rental market is impacted by the current surge in provide of each HDB and personal flats. There was a further supply of 9,824 Ec new launch items in the pipeline. In whole, the pipeline provide of 93,799 items, including ECs, was the very best ever recorded since such information were first available in 2001. You need to know the right way to follow-up and seemingly compromise on your demands without giving the consumers any impression that you're determined to sell. In addition, you'll want to handle all closingany of which may cause the sale to fall via. House number and tidy letterbox in simple-to-see position. Quid pro quo - Agent commission

a few months ago i spoke with CASE relating to our downside (unprofessional real estate brokers). the woman advised me that agents are an issue but unfortunately it's out of their remit. nonetheless, she advised me of a semi-authorities actual property body (i forgot the identify now, sorry) that you would be able to file a complain. they apparently keep a log of all the agents. or if the scenario permits it they may reprimand the true estate firm that they work for (if certainly they belong to a company). don't count on a miracle, however.

That is precisely what occurred to me and my husband today, to not mention a very unscrupulous developer working in a really unprofessional manner. I need to share this story with everybody right here, and please move the message around particularly amongst expats communities, beware once you wish to purchase property developed by VicLand Pte Ltd and if developer's agent is ECG property. There was just one unit left for sale by developer, 03-09, a three bedroom flat. On the time my husband was out of city, and initially I appreciated what I noticed so I instructed the developer's agent and my agent we must come again with my husband in two weeks to view it once more and make a decision after ward. Complaint / Suggestions about lousy property agent Darren Ng from Dennis Wee

Singapore has a very strong property market. It has one of many strongest economies within the area, and with its talented human useful resource pool and strategic location, it's no shock that Singapore has develop into enterprise and financial hub in Southeast Asia. Singapore also has an economy that all the time seems to be able to decide itself up without any worry or concern, and rich foreigners get pleasure from making the attractive island city-state their second homes. With all this going for it, it's positively a superb time to be a real property agency in Singapore as residential units and industrial house are being quickly snapped up by proprietor-occupants and investors alike.

This website is visited by many property hunters ON A REGULAR BASIS! My objective is to attract native & overseas patrons/tenants who are fascinated by renting and investing in Singapore properties. I'm dedicated to offer quality service to all my purchasers. Along with my local & abroad database, I'm all the time keen to tell & update my buyers on the newest and 'finest-buy' properties in Singapore. When property is made obtainable to the customer for occupation; or For extra information on GST therapy for transactions related to property builders, please refer to GST Information for Property Developer (263KB) A true story re-told about John the Property Agent. (Not his actual name and gender) Understand Property Investing and Sub-prime Asia Pacific Property Investment Information

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