nLab
simple Lie group

Simple Lie groups

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Simple Lie groups

Definition

A simple Lie group is a non-abelian connected Lie group GG with no nontrivial connected normal subgroups.

Via Lie's theorems this is equivalent to the Lie algebra 𝔤\mathfrak{g} of GG being a simple Lie algebra (over the real numbers).

Remark

This is not the same thing as a simple object in LieGrp (or even in ConnLieGrpConn Lie Grp). However, the Lie algebra associated to a simple Lie group is always a simple Lie algebra (although that definition also requires a non-abelian clause).

Classification

The classification of simple Lie groups consists of four infinite series – the classical Lie groups – and five separate cases – the exceptional Lie groups.

See Wikipedia's list of simple Lie groups.

See also at ADE classification

Examples

References

  • Notes on simple Lie algebras and Lie groups (pdf, SimpleLieGroupNotes.pdf?)

See also

Last revised on October 3, 2018 at 04:00:27. See the history of this page for a list of all contributions to it.