Editing Maximal torus (section)

{{Short description|Maximal compact connected Abelian Lie subgroup.}}
In the [[mathematics|mathematical]] theory of [[compact Lie group]]s 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 space|compact]], [[connected space|connected]], [[abelian group|abelian]] [[Lie subgroup]] of ''G'' (and therefore isomorphic to<ref>{{harvnb|Hall|2015}} Theorem 11.2</ref> the standard torus '''T'''<sup>''n''</sup>). A '''maximal torus''' is one which is maximal among such subgroups. That is, ''T'' is a maximal torus if for any torus ''T''&prime; containing ''T'' we have ''T'' = ''T''&prime;. Every torus is contained in a maximal torus simply by [[Dimension (mathematics and physics)|dimensional]] considerations. A noncompact Lie group need not have any nontrivial tori (e.g. '''R'''<sup>''n''</sup>).

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 (group theory)|conjugate]]. For [[semisimple Lie group|semisimple]] groups the rank is equal to the number of nodes in the associated [[Dynkin diagram]].