Editing Compact group (section)

===Maximal tori and root systems===
{{See also|Maximal torus|Root system}}
A key idea in the study of a connected compact Lie group ''K'' is the concept of a ''maximal torus'', that is a subgroup ''T'' of ''K'' that is isomorphic to a product of several copies of <math>S^1</math> and that is not contained in any larger subgroup of this type. A basic example is the case <math>K = \operatorname{SU}(n)</math>, in which case we may take <math>T</math> to be the group of diagonal elements in <math>K</math>. A basic result is the ''torus theorem'' which states that every element of <math>K</math> belongs to a maximal torus and that all maximal tori are conjugate.

The maximal torus in a compact group plays a role analogous to that of the [[Semisimple Lie algebra#Cartan subalgebras and root systems|Cartan subalgebra]] in a complex semisimple Lie algebra. In particular, once a maximal torus <math>T\subset K</math> has been chosen, one can define a [[root system]] and a [[Weyl group]] similar to what one has for [[semisimple Lie algebra]]s.<ref>{{harvnb|Hall|2015}} Chapter 11</ref> These structures then play an essential role both in the classification of connected compact groups (described above) and in the representation theory of a fixed such group (described below).

The root systems associated to the simple compact groups appearing in the classification of simply connected compact groups are as follows:<ref>{{harvnb|Hall|2015}} Section 7.7</ref>
*The special unitary groups <math>\operatorname{SU}(n)</math> correspond to the root system <math>A_{n-1}</math>
*The odd spin groups <math>\operatorname{Spin}(2n+1)</math> correspond to the root system <math>B_{n}</math>
*The compact symplectic groups <math>\operatorname{Sp}(n)</math> correspond to the root system <math>C_{n}</math>
*The even spin groups <math>\operatorname{Spin}(2n)</math> correspond to the root system <math>D_{n}</math>
*The exceptional compact Lie groups correspond to the five exceptional root systems G<sub>2</sub>, F<sub>4</sub>, E<sub>6</sub>, E<sub>7</sub>, or E<sub>8</sub>