---
tag: AM7YJP8AR6
kind: concept
name: Simple Lie group
msc: ["22E46", "22E15", "22E10"]
specializes: [XJS07AJ50Y]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_Lie_group
same_as:
  wikidata: Q1003162
  wikipedia: Simple Lie group
---

## Summary

A simple Lie group is a connected [[Lie group|7JBRGCVJRB]] whose Lie algebra is a simple Lie algebra.
Such a Lie algebra is not abelian and has no ideals other than $0$ and itself.
Simple Lie groups are the building blocks of the [[semisimple Lie groups|XJS07AJ50Y]], and through them of a large part of Lie theory.
Up to coverings and central quotients, they are classified by the simple Lie algebras.
These form the classical families $A_n, B_n, C_n, D_n$ and five exceptional types.

## Explanations

### L1 human

A simple Lie group is a continuous group of symmetries that cannot be broken into smaller continuous pieces.
Think of the rotations of space: there is no smaller family of rotations, closed under conjugation and depending continuously on parameters, other than the trivial one.
Simplicity is checked on the infinitesimal level, on the Lie algebra, so a simple Lie group may still have a finite centre.
For example, $SU(2)$ is a simple Lie group even though its centre $\set{I, -I}$ is a normal subgroup.
The rotation group $SO(3)$, the groups $SL(n, \R)$ for $n \ge 2$ and the exceptional group $E_8$ are other examples.