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In the mathematical theory of compact Lie groups a special role is played by torus subgroups (not to be confused with the mathematical torus), 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 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).

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• Maximal torus
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• In the mathematical theory of compact Lie groups a special role is played by torus subgroups (not to be confused with the mathematical torus), 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 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).
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has abstract
• In the mathematical theory of compact Lie groups a special role is played by torus subgroups (not to be confused with the mathematical torus), 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 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.
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