Quotient Lie group

Let $G$ be a Lie group (see Lie group ) and let $N\trianglelefteq G$ be a closed normal Lie subgroup (see normal Lie subgroup and closed subgroup theorem ). The quotient set $G/N$ carries a natural smooth manifold structure such that:

• the quotient map $q:G\to G/N$ is a smooth submersion,
• the group operation induced from $G$ is smooth,
• and $q$ is a Lie group homomorphism (see Lie group homomorphism ).

The closedness of $N$ is essential: if $N$ is not closed, then $G/N$ may fail to be Hausdorff and need not admit a Lie group structure.

Infinitesimally, if $\mathfrak g=\mathrm{Lie}(G)$ and $\mathfrak n=\mathrm{Lie}(N)$ (see Lie algebra of a Lie group ), then $\mathfrak n$ is an ideal in $\mathfrak g$ and the induced map on Lie algebras identifies

as a quotient Lie algebra .

This construction is the global counterpart of “modding out by an ideal,” and it is fundamental in building new Lie groups from old ones (compare covering Lie groups and universal covering groups ).