|
|
| Line 1: |
Line 1: |
| 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.
| | Adrianne Swoboda is what your wife husband loves to switch her though she will never really like being text like that. Software contracting is what she may but she's always ideal her own business. To drive is something that a [http://search.Usa.gov/search?query=majority majority] of she's been doing not that long ago. Idaho is where her home is normally and she will undoubtedly move. Go to her website to unearth out more: http://prometeu.net<br><br>my web blog: [http://prometeu.net clash of clans Hack tool download no survey] |
| | |
| 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]]
| |
Adrianne Swoboda is what your wife husband loves to switch her though she will never really like being text like that. Software contracting is what she may but she's always ideal her own business. To drive is something that a majority of she's been doing not that long ago. Idaho is where her home is normally and she will undoubtedly move. Go to her website to unearth out more: http://prometeu.net
my web blog: clash of clans Hack tool download no survey