Normal Lie subgroup

A Lie subgroup invariant under conjugation; infinitesimally, it corresponds to an ideal.

Normal Lie subgroup

Let $G$ be a Lie group .

Definition

A Lie subgroup $N\subseteq G$ is normal if

gNg^{-1}=N \quad \text{for all } g\in G,

i.e. $N$ is invariant under the conjugation action of $G$ on itself.

Infinitesimal characterization

Let $\mathfrak g=\operatorname{Lie}(G)$ and $\mathfrak n=\operatorname{Lie}(N)$ (viewed inside $\mathfrak g$ using the subgroup Lie algebra lemma ). Then:

If $N$ is normal in $G$ , $\mathfrak n$ is an ideal in $\mathfrak g$ .
Conversely, if $G$ is connected and $\mathfrak n$ is an ideal, then the connected subgroup integrating $\mathfrak n$ (via the Lie correspondence ) is normal in $G$ .

Quotients

If $N$ is closed and normal, then the quotient set $G/N$ carries a natural structure of Lie group quotient , and its Lie algebra is the quotient Lie algebra

\operatorname{Lie}(G/N)\cong \mathfrak g/\mathfrak n.

Context

Normal Lie subgroups are the geometric mechanism for building new Lie groups from old ones by quotienting, while ideals play the parallel role on the Lie algebra side.