nLab simple Lie group

Simple Lie groups

Simple Lie groups

Definition

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

Via Lie's theorems this is equivalent to the Lie algebra $\mathfrak{g}$ of $G$ 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 $Conn Lie Grp$). 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.

graphics grabbed from Schwichtenberg

Examples

• The special unitary group $SU(N+1)$ is simple for all $N \geq 1$.

• The special orthogonal group $SO(n)$ is simple at least for $n \geq 8$ (corresponding to SO(8)), hence so are the corresponding spin groups.

References

• Notes on simple Lie algebras and Lie groups (pdf)