Normal Lie subgroup - Knowlpedia

Let $G$ be a Lie group. A Lie subgroup $N\subseteq G$ is normal if

gNg^{-1}=N\qquad\text{for every }g\in G.

Thus $N$ is invariant under the conjugation action of $G$ .

Infinitesimal characterization

Let $\mathfrak g=\operatorname{Lie}(G)$ and $\mathfrak n=\operatorname{Lie}(N)$ . If $N$ is normal, then $\mathfrak n$ is an ideal in $\mathfrak g$ . Conversely, if $G$ is connected and $\mathfrak n\subseteq\mathfrak g$ is an ideal, then the connected Lie subgroup integrating $\mathfrak n$ is normal in $G$ .

Quotients

If $N$ is closed and normal, then $G/N$ is a quotient Lie group whose Lie algebra is

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

Remarks

Normal Lie subgroups permit Lie-group quotients, while ideals play the corresponding infinitesimal role.