Editing Simple Lie group (section)

===Compact Lie groups===
{{Main|root system}}
Every simple complex Lie algebra has a unique real form whose corresponding centerless Lie group is [[Compact space|compact]]. It turns out that the simply connected Lie group in these cases is also compact. Compact Lie groups have a particularly tractable representation theory because of the [[Peter–Weyl theorem]]. Just like simple complex Lie algebras, centerless compact Lie groups are classified by Dynkin diagrams (first classified by [[Wilhelm Killing]] and [[Élie Cartan]]).

[[File:Finite_Dynkin_diagrams.svg|Dynkin diagrams|480px]]

For the infinite (A, B, C, D) series of Dynkin diagrams, a connected compact Lie group associated to each Dynkin diagram can be explicitly described as a matrix group, with the corresponding centerless compact Lie group described as the quotient by a subgroup of scalar matrices. For those of type A and C we can find explicit matrix representations of the corresponding simply connected Lie group as matrix groups.