Connected Lie group

A connected Lie group is a Lie group $G$ whose underlying topological space (hence smooth manifold) is connected.

A basic structural fact is that every Lie group has a distinguished connected, normal Lie subgroup: the identity component

Then:

• $G^\circ$ is an open-and-closed embedded Lie subgroup and a normal Lie subgroup of $G$.
• The connected components of $G$ are precisely the left (or right) cosets of $G^\circ$.
• The quotient $G/G^\circ$ is a discrete group (often called the component group).

The Lie algebra $\mathfrak g = \mathrm{Lie}(G)$ controls the local connected structure through the exponential map : for any neighborhood $U\subset \mathfrak g$ of $0$, the subgroup generated by $\exp(U)$ equals $G^\circ$. In particular, $G$ is connected iff it is generated by exponentials of elements of $\mathfrak g$.

Context. Many classification and representation-theoretic statements are stated for connected (or connected compact) groups because the discrete component group $G/G^\circ$ contributes separate, essentially “finite/discrete” data on top of the Lie-algebraic information.