Effective action

A Lie group action with trivial kernel; equivalently, the only element acting as the identity on the space is $e$.

Effective action

Let $G$ be a Lie group acting smoothly on a manifold $M$ , i.e. a smooth action of a Lie group .

Definition

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 its kernel is trivial, i.e. $\ker(G\curvearrowright M)=\{e\}$ .

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 $G/\ker$ on $M$ is effective and has the same orbits (compare orbits ) and stabilizers up to the quotient (compare stabilizer ).

Motivation

Effectiveness is the correct “no redundancy” condition: if an action is not effective, one can replace $G$ by the smaller group $G/\ker$ without changing the geometry of the orbit decomposition or isotropy behavior.