Effective action - Knowlpedia

Let $G$ be a Lie group acting smoothly on a manifold $M$ . The kernel of the action is

\ker(G\curvearrowright M) := \{g\in G : g\cdot x = x \text{ for all }x\in M\}.

The action is effective if $\ker(G\curvearrowright M)=\{e\}$ .

Basic structure

The kernel is a closed normal Lie subgroup of $G$ .
Every action factors through an effective action of the quotient group: the induced action of $G/\ker(G\curvearrowright M)$ on $M$ is effective and has the same orbits.

Motivation

An ineffective action contains group elements that are invisible on $M$ ; passing to the quotient group by the kernel removes this redundancy.