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 special orthogonal group SO(2n) is a given by the set of all simultaneous rotations in n pairwise orthogonal 2-planes. This is also a maximal torus in the group SO(2n+1) where the action fixes the remaining direction. Therefore, 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.
Read more about this topic: Maximal Torus
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