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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 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. 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 ''n'' pairwise orthogonal 2-planes. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Maximal torus」の詳細全文を読む スポンサード リンク
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