Table of Lie groups


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 dimension at most 3, and the List of Lie group topics.

Real Lie groups and their algebras

Column legend
* CM: Is this group "G" compact? (Yes or No)
* pi_0: Gives the group of components of "G". The order of the component group gives the number of connected components. The group is connected if and only if the component group is trivial (denoted by 0).
* pi_1: Gives the fundamental group of "G" whenever "G" is connected. The group is simply connected if and only if the fundamental group is trivial (denoted by 0).
* UC: If "G" is not simply connected, gives the universal cover of "G".

Complex Lie algebras

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.

References

*
* Vincent Arsigny, Pierre Fillard, Xavier Pennec and Nicholas Ayache, "Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices" (2006) SIAM Journal on Matrix Analysis and Applications. Note: In press. Preprint available at ftp://ftp-sop.inria.fr/epidaure/Publications/Arsigny/arsigny_siam_tensors.pdf.


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