Editing Simple Lie group (section)

{{Short description|Connected non-abelian Lie group lacking nontrivial connected normal subgroups}}
{{About|the Killing-Cartan classification|a smaller list of groups that commonly occur in [[theoretical physics]]|Table of Lie groups|groups of dimension at most 3|Bianchi classification}}
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{{Lie groups|Simple}}
In mathematics, a '''simple Lie group''' is a [[connected space|connected]] [[nonabelian group|non-abelian]] [[Lie group]] ''G'' which does not have nontrivial connected [[normal subgroup]]s. The list of simple Lie groups can be used to read off the list of [[simple Lie algebra]]s and [[Riemannian symmetric space]]s.

Together with the commutative Lie group of the real numbers, <math>\mathbb{R}</math>, and that of the unit-magnitude complex numbers, [[Circle group|U(1)]] (the unit circle), simple Lie groups give the atomic "building blocks" that make up all (finite-dimensional) connected Lie groups via the operation of [[group extension]]. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "[[special linear group]]" SL(''n'', <math>\mathbb{R}</math>) of ''n'' by ''n'' matrices with determinant equal to 1 is simple for all odd ''n''&nbsp;> 1, when it is isomorphic to the [[projective special linear group]].

The first classification of simple Lie groups was by [[Wilhelm Killing]], and this work was later perfected by [[Élie Cartan]]. The final classification is often referred to as Killing-Cartan classification.