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In the [[mathematics|mathematical]] theory of [[compact Lie group]]s a special role is played by [[torus]] subgroups, in particular by the '''maximal torus''' subgroups. | |||
A '''torus''' in a compact [[Lie group]] ''G'' is a [[compact space|compact]], [[connected space|connected]], [[abelian group|abelian]] [[Lie subgroup]] of ''G'' (and therefore isomorphic to the standard torus '''T'''<sup>''n''</sup>). A '''maximal torus''' is one which is maximal among such subgroups. That is, ''T'' is a maximal torus if for any other torus ''T''′ containing ''T'' we have ''T'' = ''T''′. Every torus is contained in a maximal torus simply by [[Dimension (mathematics and physics)|dimensional]] considerations. A noncompact Lie group need not have any nontrivial tori (e.g. '''R'''<sup>''n''</sup>). | |||
The dimension of a maximal torus in ''G'' is called the '''rank''' of ''G''. The rank is [[well-defined]] since all maximal tori turn out to be [[conjugate (group theory)|conjugate]]. For [[semisimple Lie group|semisimple]] groups the rank is equal to the number of nodes in the associated [[Dynkin diagram]]. | |||
==Examples== | |||
The [[unitary group]] U(''n'') has as a maximal torus the subgroup of all [[diagonal matrices]]. That is, | |||
:<math>T = \left\{\mathrm{diag}(e^{i\theta_1},e^{i\theta_2},\dots,e^{i\theta_n}) : \forall j, \theta_j \in \mathbb R\right\}.</math> | |||
''T'' is clearly isomorphic to the product of ''n'' circles, so the unitary group U(''n'') has rank ''n''. A maximal torus in the [[special unitary group]] SU(''n'') ⊂ U(''n'') is just the intersection of ''T'' and SU(''n'') which is a torus of dimension ''n'' − 1. | |||
A maximal torus in the [[special orthogonal group]] SO(2''n'') is given by the set of all simultaneous [[rotation]]s in ''n'' pairwise orthogonal 2-planes. This is also a maximal torus in the group SO(2''n''+1) where the action fixes the remaining direction. Thus both SO(2''n'') and SO(2''n''+1) have rank ''n''. For example, in the [[rotation group SO(3)]] the maximal tori are given by rotations about a fixed axis. | |||
The [[symplectic group]] Sp(''n'') has rank ''n''. A maximal torus is given by the set of all diagonal matrices whose entries all lie in a fixed complex subalgebra of '''H'''. | |||
==Properties== | |||
Let ''G'' be a compact, connected Lie group and let <math>\mathfrak g</math> be the [[Lie algebra]] of ''G''. | |||
* A maximal torus in ''G'' is a maximal abelian subgroup, but the converse need not hold. | |||
* The maximal tori in ''G'' are exactly the Lie subgroups corresponding to the maximal abelian, diagonally acting subalgebras of <math>\mathfrak g</math> (cf. [[Cartan subalgebra]]) | |||
* Given a maximal torus ''T'' in ''G'', every element ''g'' ∈ ''G'' is [[Conjugacy_class|conjugate]] to an element in ''T''. | |||
* Since the conjugate of a maximal torus is a maximal torus, every element of ''G'' lies in some maximal torus. | |||
* All maximal tori in ''G'' are [[conjugate (group theory)|conjugate]]. Therefore, the maximal tori form a single [[conjugacy class]] among the subgroups of ''G''. | |||
* It follows that the dimensions of all maximal tori are the same. This dimension is the rank of ''G''. | |||
* If ''G'' has dimension ''n'' and rank ''r'' then ''n'' − ''r'' is even. | |||
== Weyl group == | |||
Given a torus ''T'' (not necessarily maximal), the [[Weyl group]] of ''G'' with respect to ''T'' can be defined as the [[normalizer]] of ''T'' modulo the [[centralizer]] of ''T''. That is, <math>W(T,G) := N_G(T)/C_G(T).</math> Fix a maximal torus <math>T = T_0</math> in ''G;'' then the corresponding Weyl group is called the Weyl group of ''G'' (it depends up to isomorphism on the choice of ''T''). The [[representation theory]] of ''G'' is essentially determined by ''T'' and ''W''. | |||
* The Weyl group acts by ([[outer automorphism|outer]]) [[automorphism]]s on ''T'' (and its Lie algebra). | |||
* The centralizer of ''T'' in ''G'' is equal to ''T'', so the Weyl group is equal to ''N''(''T'')/''T''. | |||
* The [[identity component]] of the normalizer of ''T'' is also equal to ''T''. The Weyl group is therefore equal to the [[component group]] of ''N''(''T''). | |||
* The normalizer of ''T'' is [[closed set|closed]], so the Weyl group is finite | |||
* Two elements in ''T'' are conjugate if and only if they are conjugate by an element of ''W''. That is, the conjugacy classes of ''G'' intersect ''T'' in a Weyl [[orbit (group theory)|orbit]]. | |||
* The space of conjugacy classes in ''G'' is homeomorphic to the [[orbit space]] ''T''/''W'' and, if ''f'' is a continuous function on ''G'' invariant under conjugation, the '''Weyl integration formula''' holds: | |||
::<math>\displaystyle{\int_G f(g)\, dg = |W|^{-1} \int_T f(t) |\Delta(t)|^2\, dt,}</math> | |||
:where Δ is given by the [[Weyl denominator formula]]. | |||
==See also== | |||
*[[Toral Lie algebra]] | |||
*[[Bruhat decomposition]] | |||
*[[Weyl character formula]] | |||
==References== | |||
*{{citation|last=Adams|first= J. F.|title= Lectures on Lie Groups|publisher=University of Chicago Press|year= 1969| | |||
id=ISBN 0226005305}} | |||
*{{citation|first=N.|last=Bourbaki|series=Éléments de Mathématique|title=Groupes et Algèbres de Lie (Chapitre 9)|publisher=Masson|year=1982|isbn=354034392X}} | |||
*{{citation|last=Dieudonné|first= J.|series =Treatise on analysis|title=Compact Lie groups and semisimple Lie groups, Chapter XXI|volume=5|publisher=Academic Press|year= 1977|id= ISBN 012215505X}} | |||
*{{citation|title=Lie groups|publisher=Springer|year= 2000|id=ISBN 3540152938|first=J.J.|last=Duistermaat|first2=A.|last2=Kolk|series=Universitext}} | |||
*{{citation|first=Sigurdur|last= Helgason|title=Differential geometry, Lie groups, and symmetric spaces|year=1978|publisher=Academic Press|id= ISBN 0821828487}} | |||
*{{citation|last=Hochschild|first=G.|title=The structure of Lie groups|year=1965|publisher=Holden-Day}} | |||
{{DEFAULTSORT:Maximal Torus}} | |||
[[Category:Lie groups]] | |||
[[Category:Representation theory of Lie groups]] | |||
Revision as of 15:40, 13 October 2013
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any other torus T′ containing T we have T = T′. Every torus is contained in a maximal torus simply by dimensional considerations. A noncompact Lie group need not have any nontrivial tori (e.g. Rn).
The dimension of a maximal torus in G is called the rank of G. The rank is well-defined since all maximal tori turn out to be conjugate. For semisimple groups the rank is equal to the number of nodes in the associated Dynkin diagram.
Examples
The unitary group U(n) has as a maximal torus the subgroup of all diagonal matrices. That is,
T is clearly isomorphic to the product of n circles, so the unitary group U(n) has rank n. A maximal torus in the special unitary group SU(n) ⊂ U(n) is just the intersection of T and SU(n) which is a torus of dimension n − 1.
A maximal torus in the special orthogonal group SO(2n) is given by the set of all simultaneous rotations in n pairwise orthogonal 2-planes. This is also a maximal torus in the group SO(2n+1) where the action fixes the remaining direction. Thus both SO(2n) and SO(2n+1) have rank n. For example, in the rotation group SO(3) the maximal tori are given by rotations about a fixed axis.
The symplectic group Sp(n) has rank n. A maximal torus is given by the set of all diagonal matrices whose entries all lie in a fixed complex subalgebra of H.
Properties
Let G be a compact, connected Lie group and let be the Lie algebra of G.
- A maximal torus in G is a maximal abelian subgroup, but the converse need not hold.
- The maximal tori in G are exactly the Lie subgroups corresponding to the maximal abelian, diagonally acting subalgebras of (cf. Cartan subalgebra)
- Given a maximal torus T in G, every element g ∈ G is conjugate to an element in T.
- Since the conjugate of a maximal torus is a maximal torus, every element of G lies in some maximal torus.
- All maximal tori in G are conjugate. Therefore, the maximal tori form a single conjugacy class among the subgroups of G.
- It follows that the dimensions of all maximal tori are the same. This dimension is the rank of G.
- If G has dimension n and rank r then n − r is even.
Weyl group
Given a torus T (not necessarily maximal), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is, Fix a maximal torus in G; then the corresponding Weyl group is called the Weyl group of G (it depends up to isomorphism on the choice of T). The representation theory of G is essentially determined by T and W.
- The Weyl group acts by (outer) automorphisms on T (and its Lie algebra).
- The centralizer of T in G is equal to T, so the Weyl group is equal to N(T)/T.
- The identity component of the normalizer of T is also equal to T. The Weyl group is therefore equal to the component group of N(T).
- The normalizer of T is closed, so the Weyl group is finite
- Two elements in T are conjugate if and only if they are conjugate by an element of W. That is, the conjugacy classes of G intersect T in a Weyl orbit.
- The space of conjugacy classes in G is homeomorphic to the orbit space T/W and, if f is a continuous function on G invariant under conjugation, the Weyl integration formula holds:
- where Δ is given by the Weyl denominator formula.
See also
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 - 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 - 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 - 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