Editing Simple Lie group (section)

==Full classification==
Simple Lie groups are fully classified. The classification is usually stated in several steps, namely:
* [[Semisimple Lie algebra#Classification|Classification of simple complex Lie algebras]] The classification of simple Lie algebras over the complex numbers by [[Dynkin diagrams]].
* [[Satake diagram|Classification of simple real Lie algebras]] Each simple complex Lie algebra has several [[Real form (Lie theory)|real forms]], classified by additional decorations of its Dynkin diagram called [[Satake diagram]]s, after [[Ichirô Satake]].
* '''Classification of centerless simple Lie groups''' For every (real or complex) simple Lie algebra <math>\mathfrak{g}</math>, there is a unique "centerless" simple Lie group <math>G</math> whose Lie algebra is <math>\mathfrak{g}</math> and which has trivial [[Center (group theory)|center]].
* [[List of simple Lie groups|Classification of simple Lie groups]]
One can show that the [[fundamental group]] of any Lie group is a discrete [[Abelian group|commutative group]]. Given a (nontrivial) subgroup <math>K\subset \pi_1(G)</math> of the fundamental group of some Lie group <math>G</math>, one can use the theory of [[covering space]]s to construct a new group <math>\tilde{G}^K</math> with <math>K</math> in its center. Now any (real or complex) Lie group can be obtained by applying this construction to centerless Lie groups. Note that real Lie groups obtained this way might not be real forms of any complex group. A very important example of such a real group is the [[metaplectic group]], which appears in infinite-dimensional representation theory and physics. When one takes for <math>K\subset \pi_1(G)</math> the full fundamental group, the resulting Lie group <math>\tilde{G}^{K = \pi_1(G)}</math> is the universal cover of the centerless Lie group <math>G</math>, and is simply connected. In particular, every (real or complex) Lie algebra also corresponds to a unique connected and [[Simply connected space|simply connected]] Lie group <math>\tilde{G}</math> with that Lie algebra, called the "simply connected Lie group" associated to <math>\mathfrak{g}.</math>

===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.