Maximal torus

In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups. In a compact Lie group G there is to be found a maximal torus T; that is, a closed subgroup that is a torus, and of the largest possible dimension. That dimension is called the rank of G. The rank occurs as the number of nodes in the Dynkin diagram of a semisimple group.

For example, the Lie group SO(3) of rotations in three dimensions has as maximal torus T a circle group (a 1-torus, that is). It can be taken to be the group of rotations about the x-axis, parametrised by angle. According to general theory, all the maximal tori form a single conjugacy class of subgroups. The related group SU(2) also has rank 1, with a rotation group as maximal torus. The conjugacy of maximal tori implies that all the maximal tori SO(3) are the rotations about some fixed axis - so that we have surveyed them all. In general SO(2n) and SO(2n+1) have rank n. In those cases one can easily find explicit parameter angles for the maximal torus: that is, commuting one-parameter families of rotations exhibiting the torus as a product of circle groups.

The Weyl group W of G is the normalizer of T in G modulo the centralizer; or in other words the group of transformations of T into itself carried out by conjugation in G. The representation theory of G, when it is a connected group at least, is essentially determined by T and W.