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In [[mathematics]], a '''reductive group''' is an [[algebraic group]] ''G'' over an algebraically closed field such that the [[unipotent radical]] of ''G'' is trivial (i.e., the group of unipotent elements of the [[Radical of an algebraic group|radical]] of ''G''). Any [[semisimple algebraic group]] is reductive, as is any [[algebraic torus]] and any [[general linear group]]. More generally, over fields that are not necessarily algebraically closed, a reductive group is a smooth affine algebraic group such that the unipotent radical of ''G'' over the algebraic closure is trivial. The intervention of an algebraic closure in this definition is necessary to include the case of imperfect ground fields, such as local and global function fields over finite fields. Algebraic groups over (possibly imperfect) fields ''k'' such that the ''k''-unipotent radical is trivial are called [[pseudo-reductive group]]s. | |||
The name comes from the [[Semisimple algebraic group|complete reducibility]] of [[linear representation]]s of such a group, which is a property in fact holding for representations of the algebraic group over [[field (mathematics)|field]]s of [[characteristic (field)|characteristic]] zero. (This only applies to representations of the algebraic group: finite dimensional representations of the underlying discrete group need not be completely reducible even in characteristic 0.) [[Haboush's theorem]] shows that a certain rather weaker property called geometric reductivity holds for reductive groups in the positive characteristic case. | |||
If ''G'' ≤ '''GL'''<sub>''n''</sub> is a smooth closed <math>k</math>-subgroup that acts irreducibly on affine <math>n</math>-space over <math>k</math>, then ''G'' is reductive.<ref>See {{harvnb|Springer|1998}}, exercise 2.4.15</ref> It follows that '''GL'''<sub>''n''</sub> and '''SL'''<sub>''n''</sub> are reductive (the latter being even [[semisimple algebraic group|semisimple]]). | |||
==Lie group case== | |||
{{details|Reductive Lie algebra}} | |||
More generally, in the case of [[Lie group]]s, a '''reductive Lie group''' ''G'' can be defined in terms of its [[Lie algebra]], namely a reductive Lie group is one whose Lie algebra ''g'' is a [[reductive Lie algebra]]; concretely, a Lie algebra that is the sum of an abelian and a [[semisimple Lie algebra|semisimple]] Lie algebra. Sometimes the condition that [[identity component]] ''G''<sup>0</sup> of ''G'' is of [[finite index]] is added. | |||
A Lie algebra is reductive if and only if its [[Adjoint representation of a Lie group|adjoint representation]] is completely reducible, but this does not imply that all of its finite dimensional representations are completely reducible. The concept of reductive is not quite the same for Lie groups as it is for algebraic groups because a reductive Lie group can be the group of real points of a [[unipotent algebraic group]]. | |||
For example, the one-dimensional, abelian Lie algebra '''R''' is obviously reductive, and is the Lie algebra of both a reductive algebraic group '''G'''<sub>''m''</sub> (the [[multiplicative group]] of nonzero real numbers) and also a unipotent (non-reductive) algebraic group '''G'''<sub>''a''</sub> (the [[Abelian group|additive group]] of real numbers). These are not isomorphic as ''algebraic groups''; at the Lie algebra level we see the same structure, but this is not enough to make any stronger assertion (essentially because the [[exponential map#Lie theory|exponential map]] is not an algebraic function). | |||
== | ==See also== | ||
*[[Luna's slice theorem]] | |||
*[[Root datum]] | |||
*[[Pseudo-reductive group]] | |||
==Notes== | |||
<references/> | |||
==References== | |||
* {{Citation| last1=Borel| first1=Armand| author-link=Armand Borel| year=1991| title=Linear Algebraic Groups| local= New York| edition=2nd| volume = 126| series=[[Graduate Texts in Mathematics]] | publisher=Springer-Verlag| isbn = 978-0-387-97370-8 | url=http://www.springer.com/mathematics/algebra/book/978-0-387-97370-8}}. | |||
*A. Borel, [[J. Tits]], [http://www.numdam.org/item?id=PMIHES_1965__27__55_0 ''Groupes réductifs''] Publ. Math. IHES, 27 (1965) pp. 55–150; [http://www.numdam.org/item?id=PMIHES_1972__41__253_0 ''Compléments à l'article «Groupes réductifs».''] Publications Mathématiques de l'IHÉS, 41 (1972), p. 253–276 | |||
*[[François Bruhat|Bruhat, François]]; Tits, Jacques ''Groupes réductifs sur un corps local'' : [http://www.numdam.org/item?id=PMIHES_1972__41__5_0 I. Données radicielles valuées.] Publications Mathématiques de l'IHÉS, 41 (1972), p. 5–251 [http://www.numdam.org/item?id=PMIHES_1984__60__5_0 II. Schémas en groupes. Existence d'une donnée radicielle valuée.] Publications Mathématiques de l'IHÉS, 60 (1984), p. 5–184 | |||
*{{springer|id=R/r080440|title=Reductive group|author=[[Vladimir L. Popov|V.L. Popov]]}} | |||
*{{springer|id=l/l058500|title=Lie algebra, reductive|author=A.L. Onishchik}} | |||
*{{citation | |||
|first=Tonny A. | |||
|last=Springer | |||
|authorlink=T. A. Springer | |||
|chapterurl=http://www.ams.org/online_bks/pspum331/pspum331-ptI-1.pdf | |||
|chapter=Reductive groups | |||
|pages=3–27 | |||
|url=http://www.ams.org/online_bks/pspum331/ | |||
|title=Automorphic forms, representations, and L-functions | |||
|volume=1 | |||
|ISBN=0-8218-3347-2 | |||
|year=1979 | |||
}} | |||
* {{Citation | last1=Springer | first1=Tonny A. | authorlink=T. A. Springer | title=Linear algebraic groups | publisher=Birkhäuser Boston | location=Boston, MA | edition=2nd | series=Progress in Mathematics | isbn=978-0-8176-4021-7 | id={{MathSciNet | id = 1642713}} | year=1998 | volume=9}} | |||
[[Category:Representation theory of algebraic groups]] | |||
[[Category:Lie groups]] | |||
Revision as of 23:07, 2 November 2013
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial (i.e., the group of unipotent elements of the radical of G). Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group. More generally, over fields that are not necessarily algebraically closed, a reductive group is a smooth affine algebraic group such that the unipotent radical of G over the algebraic closure is trivial. The intervention of an algebraic closure in this definition is necessary to include the case of imperfect ground fields, such as local and global function fields over finite fields. Algebraic groups over (possibly imperfect) fields k such that the k-unipotent radical is trivial are called pseudo-reductive groups.
The name comes from the complete reducibility of linear representations of such a group, which is a property in fact holding for representations of the algebraic group over fields of characteristic zero. (This only applies to representations of the algebraic group: finite dimensional representations of the underlying discrete group need not be completely reducible even in characteristic 0.) Haboush's theorem shows that a certain rather weaker property called geometric reductivity holds for reductive groups in the positive characteristic case.
If G ≤ GLn is a smooth closed -subgroup that acts irreducibly on affine -space over , then G is reductive.[1] It follows that GLn and SLn are reductive (the latter being even semisimple).
Lie group case
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More generally, in the case of Lie groups, a reductive Lie group G can be defined in terms of its Lie algebra, namely a reductive Lie group is one whose Lie algebra g is a reductive Lie algebra; concretely, a Lie algebra that is the sum of an abelian and a semisimple Lie algebra. Sometimes the condition that identity component G0 of G is of finite index is added.
A Lie algebra is reductive if and only if its adjoint representation is completely reducible, but this does not imply that all of its finite dimensional representations are completely reducible. The concept of reductive is not quite the same for Lie groups as it is for algebraic groups because a reductive Lie group can be the group of real points of a unipotent algebraic group.
For example, the one-dimensional, abelian Lie algebra R is obviously reductive, and is the Lie algebra of both a reductive algebraic group Gm (the multiplicative group of nonzero real numbers) and also a unipotent (non-reductive) algebraic group Ga (the additive group of real numbers). These are not isomorphic as algebraic groups; at the Lie algebra level we see the same structure, but this is not enough to make any stronger assertion (essentially because the exponential map is not an algebraic function).
See also
Notes
- ↑ See Template:Harvnb, exercise 2.4.15
References
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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