Editing Maximal torus

{{Short description|Maximal compact connected Abelian Lie subgroup.}}
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<ref>{{harvnb|Hall|2015}} Theorem 11.2</ref> 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 torus ''T''&prime; containing ''T'' we have ''T'' = ''T''&prime;. 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\{\operatorname{diag}\left(e^{i\theta_1},e^{i\theta_2},\dots,e^{i\theta_n}\right) : \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&nbsp;''n''&nbsp;&minus;&nbsp;1.

A maximal torus in the [[special orthogonal group]] SO(2''n'') is given by the set of all simultaneous [[rotation]]s in any fixed choice of ''n'' pairwise orthogonal planes (i.e., two dimensional vector spaces). Concretely, one maximal torus consists of all block-diagonal matrices with <math>2\times 2</math> diagonal blocks, where each diagonal block is a rotation matrix.
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 [[Quaternion|'''H''']].

==Properties==
Let ''G'' be a compact, connected Lie group and let <math>\mathfrak g</math> be the [[Lie algebra]] of ''G''. The first main result is the torus theorem, which may be formulated as follows:<ref>{{harvnb|Hall|2015}} Lemma 11.12</ref>
:'''Torus theorem''':  If ''T'' is one fixed maximal torus in ''G'', then every element of ''G'' is conjugate to an element of ''T''. 
This theorem has the following consequences:
* All maximal tori in ''G'' are conjugate.<ref>{{harvnb|Hall|2015}} Theorem 11.9</ref> 
* All maximal tori have the same dimension, known as the ''rank'' of ''G''.
* A maximal torus in ''G'' is a maximal abelian subgroup, but the converse need not hold.<ref>{{harvnb|Hall|2015}} Theorem 11.36 and Exercise 11.5</ref>
* The maximal tori in ''G'' are exactly the Lie subgroups corresponding to the maximal abelian subalgebras of <math>\mathfrak g</math><ref>{{harvnb|Hall|2015}} Proposition 11.7</ref> (cf. [[Cartan subalgebra]])
* Every element of ''G'' lies in some maximal torus; thus, the [[Exponential_map_(Lie_theory)|exponential map]] for ''G'' is surjective.
* If ''G'' has dimension ''n'' and rank ''r'' then ''n'' &minus; ''r'' is even.

==Root system==
If ''T'' is a maximal torus in a compact Lie group ''G'', one can define a [[root system]] as follows. The roots are the [[Compact_group#Representation_theory_of_a_connected_compact_Lie_group|weights]] for the adjoint action of ''T'' on the complexified Lie algebra of ''G''. To be more explicit, let <math>\mathfrak t</math> denote the Lie algebra of ''T'', let <math>\mathfrak g</math> denote the Lie algebra of <math>G</math>, and let <math>\mathfrak g_{\mathbb C}:=\mathfrak g\oplus i\mathfrak g</math> denote the complexification of <math>\mathfrak g</math>. Then we say that an element <math>\alpha\in\mathfrak t</math> is a '''root''' for ''G'' relative to ''T'' if <math>\alpha\neq 0</math> and there exists a nonzero <math>X\in\mathfrak g_{\mathbb C}</math> such that
:<math>\mathrm{Ad}_{e^H}(X)=e^{i\langle\alpha,H\rangle}X</math>
for all <math>H\in\mathfrak t</math>. Here <math>\langle\cdot,\cdot\rangle</math> is a fixed inner product on <math>\mathfrak g</math> that is invariant under the adjoint action of connected compact Lie groups.

The root system, as a subset of the Lie algebra <math>\mathfrak t</math> of ''T'', has all the usual properties of a root system, except that the roots may not span <math>\mathfrak t</math>.<ref>{{harvnb|Hall|2015}} Section 11.7</ref> The root system is a key tool in understanding the [[Compact_group#Classification|classification]] and [[Compact_group#Representation_theory_of_a_connected_compact_Lie_group|representation theory]] of ''G''.

== 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 first two major results about the Weyl group are as follows.
* The centralizer of ''T'' in ''G'' is equal to ''T'', so the Weyl group is equal to ''N''(''T'')/''T''.<ref>{{harvnb|Hall|2015}} Theorem 11.36</ref>
* The Weyl group is generated by reflections about the roots of the associated Lie algebra.<ref>{{harvnb|Hall|2015}} Theorem 11.36</ref> Thus, the Weyl group of ''T'' is isomorphic to the [[Weyl group]] of the [[root system]] of the Lie algebra of ''G''.

We now list some consequences of these main results.
* Two elements in ''T'' are conjugate if and only if they are conjugate by an element of ''W''. That is, each conjugacy class of ''G'' intersects ''T'' in exactly one Weyl [[orbit (group theory)|orbit]].<ref>{{harvnb|Hall|2015}} Theorem 11.39</ref> In fact, the space of conjugacy classes in ''G'' is homeomorphic to the [[orbit space]] ''T''/''W''.
* The Weyl group acts by ([[outer automorphism|outer]]) [[automorphism]]s on ''T'' (and its Lie algebra).
* 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 Weyl group is finite.
The [[Compact_group#Representation_theory_of_a_connected_compact_Lie_group|representation theory]] of ''G'' is essentially determined by ''T'' and ''W''.

As an example, consider the case <math>G=SU(n)</math> with <math>T</math> being the diagonal subgroup of <math>G</math>. Then <math>x\in G</math> belongs to <math>N(T)</math> if and only if <math>x</math> maps each standard basis element <math>e_i</math> to a multiple of some other standard basis element <math>e_j</math>, that is, if and only if <math>x</math> permutes the standard basis elements, up to multiplication by some constants. The Weyl group in this case is then the permutation group on <math>n</math> elements.

==Weyl integral formula==
Suppose ''f'' is a continuous function on ''G''. Then the integral over ''G'' of ''f'' with respect to the normalized Haar measure ''dg'' may be computed as follows:
: <math>\displaystyle{\int_G f(g)\, dg = |W|^{-1} \int_T |\Delta(t)|^2\int_{G/T}f\left(yty^{-1}\right)\,d[y]\, dt,}</math>
where <math>d[y]</math> is the normalized volume measure on the quotient manifold <math>G/T</math> and <math>dt</math> is the normalized Haar measure on ''T''.<ref>{{harvnb|Hall|2015}} Theorem 11.30 and Proposition 12.24</ref> Here  Δ is given by the [[Weyl denominator formula]] and <math>|W|</math> is the order of the Weyl group. An important special case of this result occurs when ''f'' is a [[class function]], that is, a function invariant under conjugation. In that case, we have
: <math>\displaystyle{\int_G f(g)\, dg = |W|^{-1} \int_T f(t) |\Delta(t)|^2\, dt.}</math>
Consider as an example the case <math>G=SU(2)</math>, with <math>T</math> being the diagonal subgroup. Then the Weyl integral formula for class functions takes the following explicit form:<ref>{{harvnb|Hall|2015}} Example 11.33</ref>

: <math>\displaystyle{\int_{SU(2)} f(g)\, dg = \frac{1}{2} \int_0^{2\pi} f\left(\mathrm{diag}\left(e^{i\theta}, e^{-i\theta}\right)\right)\, 4\,\mathrm{sin}^2(\theta)  \, \frac{d\theta}{2\pi}.}</math>
Here <math>|W|=2</math>, the normalized Haar measure on <math>T</math> is <math>\frac{d\theta}{2\pi}</math>, and <math>\mathrm{diag}\left(e^{i\theta}, e^{-i\theta}\right)</math> denotes the diagonal matrix with diagonal entries <math>e^{i\theta}</math> and <math>e^{-i\theta}</math>.

==See also==
* [[Compact group]]
* [[Cartan subgroup]]
* [[Cartan subalgebra]]
* [[Toral Lie algebra]]
* [[Bruhat decomposition]]
* [[Weyl character formula]]
* [[Compact_group#Representation_theory_of_a_connected_compact_Lie_group|Representation theory of a connected compact Lie group]]

==References==
{{Reflist}}
* {{citation|last=Adams|first= J. F.|title= Lectures on Lie Groups|publisher=University of Chicago Press|year= 1969|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|isbn= 012215505X}}
* {{citation|title=Lie groups|publisher=Springer|year= 2000|isbn=3540152938|first=J.J.|last=Duistermaat|first2=A.|last2=Kolk|series=Universitext}}
* {{Citation| last=Hall|first=Brian C.|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
* {{citation|first=Sigurdur|last= Helgason|title=Differential geometry, Lie groups, and symmetric spaces|year=1978|publisher=Academic Press|isbn= 0821828487}}
* {{citation|last=Hochschild|first=G.|title=The structure of Lie groups|year=1965|publisher=Holden-Day}}

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[[Category:Lie groups]]
[[Category:Representation theory of Lie groups]]