Editing Simple Lie group (section)

==Definition==
Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that is [[simple group|simple]] as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether <math>\mathbb{R}</math> is a simple Lie group.

The most common definition is that a Lie group is simple if it is connected, non-abelian, and every closed ''connected'' normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a non-trivial center, but <math>\mathbb{R}</math> is not simple.

In this article the connected simple Lie groups with trivial center are listed.  Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a [[covering space#universal covers|universal cover]] whose center is the [[fundamental group]] of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center.

===Alternatives===
An equivalent definition of a simple Lie group follows from the [[Lie correspondence]]: A connected Lie group is simple if its [[Lie algebra]] is [[simple Lie algebra|simple]]. An important technical point is that a simple Lie group may contain ''discrete'' normal subgroups. For this reason, the definition of a simple Lie group is not equivalent to the definition of a Lie group that is [[simple group|simple as an abstract group]].

Simple Lie groups include many [[classical Lie group]]s, which provide a group-theoretic underpinning for [[spherical geometry]], [[projective geometry]] and related geometries in the sense of [[Felix Klein]]'s [[Erlangen program]]. It emerged in the course of [[list of simple Lie groups|classification]] of simple Lie groups that there exist also several [[exceptional object|exceptional]] possibilities not corresponding to any familiar geometry. These ''exceptional groups'' account for many special examples and configurations in other branches of mathematics, as well as contemporary [[theoretical physics]].
<!-- ==Simple complex Lie groups== 
All (locally compact, connected) Lie groups are smooth [[manifolds]]. Mathematicians often study [[complex Lie group]]s, which are Lie groups with a [[Complex manifold|complex structure]] on the underlying manifold, which is required to be compatible with the group operations. A [[complex Lie group]] is called simple if it is connected as a topological space and its Lie algebra is simple as a [[complex Lie algebra]]. Note that the underlying Lie group may not be simple, although it will still be semisimple (see below). -->

As a counterexample, the [[general linear group]] is neither simple, nor [[semisimple Lie group|semisimple]]. This is because multiples of the identity form a nontrivial normal subgroup, thus evading the definition.  Equivalently, the corresponding [[Lie algebra]] has a degenerate [[Killing form]], because multiples of the identity map to the zero element of the algebra. Thus, the corresponding Lie algebra is also neither simple nor semisimple. Another counter-example are the [[special orthogonal group]]s in even dimension. These have the matrix <math>-I</math> in the [[center (group theory)|center]], and this element is path-connected to the identity element, and so these groups evade the definition. Both of these are [[reductive group]]s.