Group Actions| Group Actions |
In General
> s.a. lie group / Homogeneous
Space; representations.
$ Def: An action of a
group G on a set X is a homomorphism θ:
G → M(X), where M(X)
is the group of all invertible maps X → X.
$ Right action: A map
X × G → X, (x, g)
\(\mapsto\) x · g, that satisfies,
∀x ∈ X and ∀g,
g' ∈ G, (i) x · e
= x; (ii) (x · g) · g'
= x · (gg').
* Remark: Left action
is analogous; The distinction between left and right is meaningful
only when G is non-Abelian.
* Orbits: Given a group
action G × X → X, the equivalence
classes X/~, where x ~ x' iff ∃ g
∈ G such that gx = x'.
* Stabilizer of a point:
Given some point x ∈ X, the subgroup of G
of all g such that xg = x (in right-action
notation).
* Examples: Of a Lie
group on itself by left-multiplication (free), or conjugation (not free);
Of the Poincaré group on Minkowski spacetime (not free); Of a
Lie group on a manifold (a realization, usually required to be
a smooth diffeomorphism); On a vector space (a representation).
* Realization of a group:
A mapping σ: G → Diff(X) from the
group to the diffeomorphisms of a manifold X, g →
σg, which is
a homomorphism, i.e., a smooth group action on a manifold; Special
cases: The realization is faithful if the mapping is injective.
@ General references: in Reid 70 [on Klein's program for geometry];
Rudolph & Weiss AM(00)m.DS [amenable groups, entropy and mixing].
@ Non-linear realizations: Isham et al AP(71) [of spacetime symmetries];
> s.a. types of yang-mills theories.
> Online resources:
see MathWorld page;
Wikipedia page.
Effective Action of a Group on a Manifold
$ Def: G acts effectively
on X if σg(x)
= x for all x ∈ X implies g = e.
* Idea: Only the identity leaves
all of X invariant, but some gs can have fixed points;
This condition is weaker than that defining free action.
Free Action of a Group on a Manifold
$ Def: The action
σg: X
→ X of a group G on a manifold X is free
iff only e has fixed points, i.e.,
g ∈ G & g ≠ e implies σg(x) ≠ x , for all x ∈ X .
* Example:
The left action of a group on itself.
* Relationships:
The condition is stronger than the one for an effective action.
Proper Discontinuous Action of a Group
$ Def: A group G acts
properly discontinuously on a topological space X if (1) For all
x ∈ X there is a neighborhood U of x
such that ∀g ∈ G, g ≠ e,
U ∩ gU = Ø (X/G ∈ Man);
(2, optional) if for some p, q ∈ X there
is no g ∈ G such that gp = q, then
there are neighborhoods U of p and V of q
such that ∀g ∈ G, gU ∩ V
= Ø (X/G is Hausdorff).
* Example: Any finite group acting
without fixed points on a Hausdorff space acts properly discontinuously.
Transitive Action of a Group on a Manifold
$ Def: A group action σ:
G × X → X is said to be transitive if any two x,
y ∈ X can be connected by a g ∈ G, i.e.,
for all x, y ∈ X, ∃ g ∈ G such
that σg(x) = y.
* Special case: If this g is
unique, the action is called simply transitive.
Group Actions on Manifolds with Other Structure
@ Preserving geodesics: Matveev JDG-m.DG/04 [Lichnerowicz-Obata conjecture].
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send feedback and suggestions to bombelli at olemiss.edu – modified 11 aug 2018