# Maximal torus

In the mathematical theory of compact Lie groups a special role is played by torus subgroups (not to be confused with the mathematical torus), 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 **T**^{n}). 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. **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 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 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*.^{[1]} - 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.^{[2]}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

- 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 - Helgason, Sigurdur (1978),
*Differential geometry, Lie groups, and symmetric spaces*, Academic Press, ISBN 0821828487 - Hochschild, G. (1965),
*The structure of Lie groups*, Holden-Day