# Maximal torus

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,

*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, the maximal torus consists of all block-diagonal matrices with 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 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 [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 denote the Lie algebra of *T*, let denote the Lie algebra of , and let denote the complexification of . Then we say that an element is a **root** for *G* relative to *T* if and there exists a nonzero such that

for all . Here is a fixed inner product on that is invariant under the adjoint action of connected compact Lie groups.

The root system, as a subset of the Lie algebra of *T*, has all the usual properties of a root system, except that the roots may not span .[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,

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 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 with being the diagonal subgroup of . Then belongs to if and only if maps each standard basis element to a multiple of some other standard basis element , that is, if and only if permutes the standard basis elements, up to multiplication by some constants. The Weyl group in this case is then the permutation group on 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:

where is the normalized volume measure on the quotient manifold and is the normalized Haar measure on *T*.[10] Here Δ is given by the Weyl denominator formula and 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

Consider as an example the case , with being the diagonal subgroup. Then the Weyl integral formula for class functions takes the following explicit form:[11]

Here , the normalized Haar measure on is , and denotes the diagonal matrix with diagonal entries and .

## See also

## 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