In the mathematicaltheory ofcompactLie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie groupG is a compact, connected, abelianLie subgroup of G. 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. 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 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 has rank n. A maximal torus in the special unitary group SU ⊂ U is just the intersection of T and SU which is a torus of dimension n − 1. A maximal torus in the special orthogonal group SO is given by the set of all simultaneous rotations in any fixed choice of n pairwise orthogonal planes. 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 where the action fixes the remaining direction. Thus both SO and SO have rank n. For example, in the rotation group SO the maximal tori are given by rotations about a fixed axis. The symplectic group Sp 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: This theorem has the following consequences:
All maximal tori in G are conjugate.
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.
The maximal tori in G are exactly the Lie subgroups corresponding to the maximal abelian subalgebras of
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. The root system is a key tool in understanding the classification and representation theory of G.
Given a torus T, the Weyl group of Gwith respect toT 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. 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.
The Weyl group is generated by reflections about the roots of the associated Lie algebra. 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.
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.
Suppose f is a continuous function on G. Then the integral over G of f with respect to the normalized Haar measuredg may be computed as follows: where is the normalized volume measure on the quotient manifold and is the normalized Haar measure on T. 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: Here, the normalized Haar measure on is, and denotes the diagonal matrix with diagonal entries and.
