We now present an example of a group with an uncountable number of elements that is not a Lie group under a certain topology. The group given by
All of the preceding examples fall under the heading of the classical groups.
Then a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows:
The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, the topology of a Lie group together with the group law determines the geometry of the group.
Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras.
Methods for determining whether a Lie group is simply connected or not are discussed in the article on fundamental groups of Lie groups.
The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.
(In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.)
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.
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.