- 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 classificationof groups of dimension at most 3, and the List of Lie group topics.
Real Lie groups and their algebras
* CM: Is this group "G" compact? (Yes or No)
* : Gives the
group of componentsof "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).
* : Gives the
fundamental groupof "G" whenever "G" is connected. The group is simply connectedif and only if the fundamental group is trivial (denoted by 0).
* UC: If "G" is not simply connected, gives the
universal coverof "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.
* 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.
Wikimedia Foundation. 2010.
Look at other dictionaries:
List of simple Lie groups — In mathematics, the simple Lie groups were classified by Élie Cartan.The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. See also the table of Lie groups for a smaller list of… … Wikipedia
List of Lie groups topics — This is a list of Lie group topics, by Wikipedia page. Contents 1 Examples 2 Lie algebras 3 Foundational results 4 Semisimple theory … Wikipedia
Lie group — Lie groups … Wikipedia
Table of Clebsch-Gordan coefficients — This is a table of Clebsch Gordan coefficients used for adding angular momentum values in quantum mechanics. The overall sign of the coefficients for each set of constant j 1, j 2, j is arbitrary to some degree and has been fixed according to the … Wikipedia
List of Lie group topics — This is a list of Lie group topics, by Wikipedia page. Examples See Table of Lie groups for a list *General linear group, special linear group **SL2(R) **SL2(C) *Unitary group, special unitary group **SU(2) **SU(3) *Orthogonal group, special… … Wikipedia
Examples of groups — Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here. Contents 1 Permutations of a set of three elements 2 The group of translations of the plane 3 The symme … Wikipedia
Representation theory of finite groups — In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. See the article on group representations for an introduction. This article discusses the representation theory of… … Wikipedia
Cayley table — A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group s elements in a square table reminiscent of an addition or… … Wikipedia
History of the periodic table — The history of the periodic table reflects over a century of growth in the understanding of chemical properties, and culminates with the publication of the first actual periodic table by Dmitri Mendeleev in 1869. While Mendeleev built upon… … Wikipedia
File Allocation Table — For other uses, see Fat (disambiguation). FAT Developer Microsoft Full Name File Allocation Table FAT12 (12‑bit version) FAT16/FAT16B (16‑bit versions) FAT32 (32‑bit version with 28 bits used) Introduced … Wikipedia