- Table of Lie groups
This article gives a table of some common

Lie group s and their associatedLie algebra s.The following are noted: the topological properties of the group (

dimension , connectedness, compactness, the nature of thefundamental group , and whether or not they aresimply 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 , theBianchi classification of groups of dimension at most 3, and theList of Lie group topics .**Real Lie groups and their algebras**Column legend

***CM**: Is this group "G" compact? (Yes or No)

***$pi\_0$**: Gives thegroup 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 thefundamental group of "G" whenever "G" is connected. The group issimply connected if and only if the fundamental group is trivial (denoted by 0).

***UC**: If "G" is not simply connected, gives theuniversal 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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