∞-Lie theory

# Simple Lie groups

## Definition

A simple Lie group is a connected Lie group with exactly two connected normal subgroups that is not abelian.

## Remarks

This is not the same thing as a simple object in LieGrp (or even in $Conn Lie Grp$). 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.