Connected Lie group - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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

G^\circ := \text{the connected component of } e \in G.

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).

Remarks

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.