Relationship of Simple Lie Algebras To Groups
Secondly the Lie algebra only determines uniquely the simply connected (universal) cover G* of the component containing the identity of a Lie group G. It may well happen that G* isn't actually a simple group, for example having a non-trivial center. We have therefore to worry about the global topology, by computing the fundamental group of G (an abelian group: a Lie group is an H-space). This was done by Élie Cartan.
For an example, take the special orthogonal groups in even dimension. With the non-identity matrix −I in the center, these aren't actually simple groups; and having a twofold spin cover, they aren't simply-connected either. They lie 'between' G* and G, in the notation above.
Read more about this topic: Simple Lie Group
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