# Simple Lie Group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself. A direct sum of simple Lie algebras is called a semisimple Lie algebra.

An equivalent definition of a simple Lie group follows from the Lie correspondence: a connected Lie group is simple if its Lie algebra is simple. An important technical point is that a simple Lie group may contain discrete normal subgroups, hence being a simple Lie group is different from being simple as an abstract group.

Simple Lie groups include many classical Lie groups, which provide a group-theoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen programme. It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics.

While the notion of a simple Lie group is satisfying from the axiomatic perspective, in applications of Lie theory, such as the theory of Riemannian symmetric spaces, somewhat more general notions of semisimple and reductive Lie groups proved to be even more useful. In particular, every connected compact Lie group is reductive, and the study of representations of general reductive groups is a major branch of representation theory.

### Other articles related to "simple lie group, lie group, simple, group, simple lie groups, lie, groups":

List Of Simple Lie Groups - Simple Lie Groups
... Unfortunately there is no generally accepted definition of a simple Lie group, and in particular it is not necessarily defined as a Lie group that is simple as an abstract group ... Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether R is a simple Lie group ... The most common definition implies that simple Lie groups must be connected, and non-abelian, but are allowed to have a non-trivial center ...
Wess–Zumino–Witten Model - Action - Topological Obstructions
... They are maps from flat 3-space into the Lie group G ... However, the homotopy group π3(G) = Z for any compact, connected simple Lie group G ... that coupling constant k must be an integer when G is a connected, compact, simple Lie group ...
Simple Lie Group - Simply Laced Groups
... A simply laced group is a Lie group whose Dynkin diagram only contain simple links, and therefore all the nonzero roots of the corresponding Lie algebra have the same length ... The A, D and E series groups are all simply laced, but no group of type B, C, F, or G is simply laced ...
Lie - Religious Perspectives - In Zoroastrianism
... which is real, and Druj, which is "the Lie" ... Later on the Lie became personified as Angra Mainyu, a figure similar to the Christian Devil, who was portrayed as the eternal opponent of Ahura Mazda (God) ...
Classification of Riemannian Symmetric Spaces - Classification Scheme
... G is a (real) simple Lie group B ... G is either the product of a compact simple Lie group with itself (compact type), or a complexification of such a Lie group (non-compact type) ... described by the classification of simple Lie groups ...

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