nLab
effective group action
Contents
this entry is about the concept in group theory ; for the concept in quantum field theory see at effective action functional ; for disambiguation see effective action

Context
Group Theory
group theory

Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
Contents
Idea
A group action is effective if no group element other than the neutral element acts trivially on all elements of the space, hence if no other element acts the way the neutral element does.

Definition
A group action of a group (group object ) $G$ on a set (object ) $X$ is effective if $\underset{x \in X}{\forall} g x = x$ implies that $g = e$ is the neutral element .

Beware the similarity to and difference with free action : a free action is effective, but an effective action need not be free.

Last revised on November 4, 2021 at 05:56:42.
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