A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to the standard torus Tn). 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. 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.
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 2-planes. 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.
- 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.
- 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. 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.
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.
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