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 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 formula_2 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 formula_3 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:
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 formula_5 denote the Lie algebra of "T", let formula_3 denote the Lie algebra of formula_7, and let formula_8 denote the complexification of formula_3. Then we say that an element formula_10 is a root for "G" relative to "T" if formula_11 and there exists a nonzero formula_12 such that
for all formula_14. Here formula_15 is a fixed inner product on formula_3 that is invariant under the adjoint action of connected compact Lie groups.
The root system, as a subset of the Lie algebra formula_5 of "T", has all the usual properties of a root system, except that the roots may not span formula_5. 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 formula_20 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 formula_21 with formula_22 being the diagonal subgroup of formula_7. Then formula_24 belongs to formula_25 if and only if formula_26 maps each standard basis element formula_27 to a multiple of some other standard basis element formula_28, that is, if and only if formula_26 permutes the standard basis elements, up to multiplication by some constants. The Weyl group in this case is then the permutation group on formula_30 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 formula_32 is the normalized volume measure on the quotient manifold formula_33 and formula_34 is the normalized Haar measure on "T". Here Δ is given by the Weyl denominator formula and formula_35 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 formula_37, with formula_22 being the diagonal subgroup. Then the Weyl integral formula for class functions takes the following explicit form:
Here formula_40, the normalized Haar measure on formula_22 is formula_42, and formula_43 denotes the diagonal matrix with diagonal entries formula_44 and formula_45.