Editing Lie group (section)

== Classification ==
Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis.  What must be understood is the nature of 'small' transformations, for example, rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra ([[Sophus Lie|Lie]] himself called them "infinitesimal groups"). It can be defined because Lie groups are smooth manifolds, so have [[tangent space]]s at each point.

The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a [[Direct sum of modules|direct sum]] of an [[abelian Lie algebra]] and some number of [[simple Lie group|simple]] ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the [[simple Lie group|simple Lie algebras]] of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" A<sub>''n''</sub>, B<sub>''n''</sub>, C<sub>''n''</sub> and D<sub>''n''</sub>, which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E<sub>8</sub> is the largest of these.

Lie groups are classified according to their algebraic properties ([[simple group|simple]], [[semisimple group|semisimple]], [[solvable group|solvable]], [[nilpotent group|nilpotent]], [[abelian group|abelian]]), their [[connectedness]] ([[connected space|connected]] or [[simply connected space|simply connected]]) and their [[compact space|compactness]].

A first key result is the [[Levi decomposition]], which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup. 
* Connected [[compact Lie group]]s are all known: they are finite central quotients of a product of copies of the circle group ''S''<sup>1</sup> and simple compact Lie groups (which correspond to connected [[Dynkin diagram]]s).
* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Solvable groups are too messy to classify except in a few small dimensions.
* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1s on the diagonal of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Like solvable groups, nilpotent  groups are too messy to classify except in a few small dimensions.
* [[Simple Lie group]]s are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example, [[SL2(R)|SL(2, '''R''')]] is simple according to the second definition but not according to the first. They have all been [[list of simple Lie groups|classified]] (for either definition).
* [[Semisimple group|Semisimple]] Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras.{{sfn|ps=|Helgason|1978|p=131}} They are central extensions of products of simple Lie groups.

The [[identity component]] of any Lie group is an open [[normal subgroup]], and the [[quotient group]] is a [[discrete group]]. The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group ''G'' can be decomposed into discrete, simple, and abelian groups in a canonical way as follows.  Write 
: ''G''<sub>con</sub> for the connected component of the identity
: ''G''<sub>sol</sub> for the largest connected normal solvable subgroup
: ''G''<sub>nil</sub> for the largest connected normal nilpotent subgroup
so that we have a sequence of normal subgroups
: {{math|1 ⊆ ''G''<sub>nil</sub> ⊆ ''G''<sub>sol</sub> ⊆ ''G''<sub>con</sub> ⊆ ''G''}}.
Then
: ''G''/''G''<sub>con</sub> is discrete
: ''G''<sub>con</sub>/''G''<sub>sol</sub> is a [[group extension|central extension]] of a product of [[list of simple Lie groups|simple connected Lie groups]].
: ''G''<sub>sol</sub>/''G''<sub>nil</sub> is abelian. A connected [[abelian Lie group]] is isomorphic to a product of copies of  '''R''' and the [[circle group]] ''S''<sup>1</sup>.
: ''G''<sub>nil</sub>/1 is nilpotent, and therefore its ascending central series has all quotients abelian.

This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.
* The [[diffeomorphism|diffeomorphism group]] of a Lie group acts transitively on the Lie group
* Every Lie group is [[parallelizable]], and hence an [[orientable manifold]] (there is a [[fibre bundle|bundle isomorphism]] between its [[tangent bundle]] and the product of itself with the [[tangent space]] at the identity)