Table of Lie groups

This article gives a table of some common Lie groups and their associated Lie algebras.

The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).

For more examples of Lie groups and other related topics see the list of simple Lie groups; the Bianchi classification of groups of up to three dimensions; and the list of Lie group topics.

Column legend

Table legend:

with Lie bracket the cross product; also isomorphic to su(2) and to so(3,R)

The dimensions given are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.

1

The dimensions given are dimensions over C. Note that every complex Lie algebra can also be viewed as a real Lie algebra of twice the dimension.

the commutator

the commutator

where J is the standard skew-symmetric matrix

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