# Maximal torus

__: Maximal compact connected Abelian Lie subgroup.__

**Short description**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^{[1]} the standard torus **T**^{n}). A **maximal torus** is one which is maximal among such subgroups. That is, *T* is a maximal torus if for any 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. **R**^{n}).

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,

- [math]\displaystyle{ 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 *n* − 1.

A maximal torus in the special orthogonal group SO(2*n*) is given by the set of all simultaneous rotations 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]\displaystyle{ 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 **H**.

## Properties

Let *G* be a compact, connected Lie group and let [math]\displaystyle{ \mathfrak g }[/math] be the Lie algebra of *G*. The first main result is the torus theorem, which may be formulated as follows:^{[2]}

**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.^{[3]} - 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.^{[4]} - The maximal tori in
*G*are exactly the Lie subgroups corresponding to the maximal abelian subalgebras of [math]\displaystyle{ \mathfrak g }[/math]^{[5]}(cf. Cartan subalgebra) - Every element of
*G*lies in some maximal torus; thus, the exponential map for*G*is surjective. - If
*G*has dimension*n*and rank*r*then*n*−*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 weights for the adjoint action of *T* on the complexified Lie algebra of *G*. To be more explicit, let [math]\displaystyle{ \mathfrak t }[/math] denote the Lie algebra of *T*, let [math]\displaystyle{ \mathfrak g }[/math] denote the Lie algebra of [math]\displaystyle{ G }[/math], and let [math]\displaystyle{ \mathfrak g_{\mathbb C}:=\mathfrak g\oplus i\mathfrak g }[/math] denote the complexification of [math]\displaystyle{ \mathfrak g }[/math]. Then we say that an element [math]\displaystyle{ \alpha\in\mathfrak t }[/math] is a **root** for *G* relative to *T* if [math]\displaystyle{ \alpha\neq 0 }[/math] and there exists a nonzero [math]\displaystyle{ X\in\mathfrak g_{\mathbb C} }[/math] such that

- [math]\displaystyle{ \mathrm{Ad}_{e^H}(X)=e^{i\langle\alpha,H\rangle}X }[/math]

for all [math]\displaystyle{ H\in\mathfrak t }[/math]. Here [math]\displaystyle{ \langle\cdot,\cdot\rangle }[/math] is a fixed inner product on [math]\displaystyle{ \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]\displaystyle{ \mathfrak t }[/math] of *T*, has all the usual properties of a root system, except that the roots may not span [math]\displaystyle{ \mathfrak t }[/math].^{[6]} The root system is a key tool in understanding the classification and 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]\displaystyle{ W(T,G) := N_G(T)/C_G(T). }[/math]

Fix a maximal torus [math]\displaystyle{ 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*.^{[7]} - The Weyl group is generated by reflections about the roots of the associated Lie algebra.
^{[8]}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.^{[9]}In fact, the space of conjugacy classes in*G*is homeomorphic to the orbit space*T*/*W*. - The Weyl group acts by (outer) automorphisms 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 representation theory of *G* is essentially determined by *T* and *W*.

As an example, consider the case [math]\displaystyle{ G=SU(n) }[/math] with [math]\displaystyle{ T }[/math] being the diagonal subgroup of [math]\displaystyle{ G }[/math]. Then [math]\displaystyle{ x\in G }[/math] belongs to [math]\displaystyle{ N(T) }[/math] if and only if [math]\displaystyle{ x }[/math] maps each standard basis element [math]\displaystyle{ e_i }[/math] to a multiple of some other standard basis element [math]\displaystyle{ e_j }[/math], that is, if and only if [math]\displaystyle{ 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]\displaystyle{ 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{ \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]\displaystyle{ d[y] }[/math] is the normalized volume measure on the quotient manifold [math]\displaystyle{ G/T }[/math] and [math]\displaystyle{ dt }[/math] is the normalized Haar measure on *T*.^{[10]} Here Δ is given by the Weyl denominator formula and [math]\displaystyle{ |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{ \displaystyle{\int_G f(g)\, dg = |W|^{-1} \int_T f(t) |\Delta(t)|^2\, dt.} }[/math]

Consider as an example the case [math]\displaystyle{ G=SU(2) }[/math], with [math]\displaystyle{ T }[/math] being the diagonal subgroup. Then the Weyl integral formula for class functions takes the following explicit form:^{[11]}

- [math]\displaystyle{ \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]\displaystyle{ |W|=2 }[/math], the normalized Haar measure on [math]\displaystyle{ T }[/math] is [math]\displaystyle{ \frac{d\theta}{2\pi} }[/math], and [math]\displaystyle{ \mathrm{diag}\left(e^{i\theta}, e^{-i\theta}\right) }[/math] denotes the diagonal matrix with diagonal entries [math]\displaystyle{ e^{i\theta} }[/math] and [math]\displaystyle{ e^{-i\theta} }[/math].

## See also

- Compact group
- Cartan subgroup
- Cartan subalgebra
- Toral Lie algebra
- Bruhat decomposition
- Weyl character formula
- Representation theory of a connected compact Lie group

## References

- ↑ Hall 2015 Theorem 11.2
- ↑ Hall 2015 Lemma 11.12
- ↑ Hall 2015 Theorem 11.9
- ↑ Hall 2015 Theorem 11.36 and Exercise 11.5
- ↑ Hall 2015 Proposition 11.7
- ↑ Hall 2015 Section 11.7
- ↑ Hall 2015 Theorem 11.36
- ↑ Hall 2015 Theorem 11.36
- ↑ Hall 2015 Theorem 11.39
- ↑ Hall 2015 Theorem 11.30 and Proposition 12.24
- ↑ Hall 2015 Example 11.33

- Adams, J. F. (1969),
*Lectures on Lie Groups*, University of Chicago Press, ISBN 0226005305 - Bourbaki, N. (1982),
*Groupes et Algèbres de Lie (Chapitre 9)*, Éléments de Mathématique, Masson, ISBN 354034392X - Dieudonné, J. (1977),
*Compact Lie groups and semisimple Lie groups, Chapter XXI*, Treatise on analysis,**5**, Academic Press, ISBN 012215505X - Duistermaat, J.J.; Kolk, A. (2000),
*Lie groups*, Universitext, Springer, ISBN 3540152938 - Hall, Brian C. (2015),
*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*, Graduate Texts in Mathematics,**222**(2nd ed.), Springer, ISBN 978-3319134666 - Helgason, Sigurdur (1978),
*Differential geometry, Lie groups, and symmetric spaces*, Academic Press, ISBN 0821828487 - Hochschild, G. (1965),
*The structure of Lie groups*, Holden-Day

Original source: https://en.wikipedia.org/wiki/Maximal torus.
Read more |