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 Tn).

A noncompact Lie group need not have any nontrivial tori (e.g. Rn).

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.

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(2n) 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

This is also a maximal torus in the group SO(2n+1) where the action fixes the remaining direction.

Thus both SO(2n) and SO(2n+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. 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] This theorem has the following consequences: 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

that is invariant under the adjoint action of connected compact Lie groups.

The root system, as a subset of the Lie algebra

[6] The root system is a key tool in understanding the classification and representation theory of G. 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.

We now list some consequences of these main results.

The representation theory of G is essentially determined by T and W. As an example, consider the case

maps each standard basis element

to a multiple of some other standard basis element

permutes the standard basis elements, up to multiplication by some constants.

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

is the normalized Haar measure on T.[10] Here Δ is given by the Weyl denominator formula and

An important special case of this result occurs when f is a class function, that is, a function invariant under conjugation.

Then the Weyl integral formula for class functions takes the following explicit form:[11] Here