Lie group action

In differential geometry, a Lie group action is a group action adapted to the smooth setting: [math]\displaystyle{ G }[/math] is a Lie group, [math]\displaystyle{ M }[/math] is a smooth manifold, and the action map is differentiable.

Definition and first properties

Let [math]\displaystyle{ \sigma: G \times M \to M, (g, x) \mapsto g \cdot x }[/math] be a (left) group action of a Lie group [math]\displaystyle{ G }[/math] on a smooth manifold [math]\displaystyle{ M }[/math]; it is called a Lie group action (or smooth action) if the map [math]\displaystyle{ \sigma }[/math] is differentiable. Equivalently, a Lie group action of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ M }[/math] consists of a Lie group homomorphism [math]\displaystyle{ G \to \mathrm{Diff}(M) }[/math]. A smooth manifold endowed with a Lie group action is also called a [math]\displaystyle{ G }[/math]-manifold.

The fact that the action map [math]\displaystyle{ \sigma }[/math] is smooth has a couple of immediate consequences:

• the stabilizers [math]\displaystyle{ G_x \subseteq G }[/math] of the group action are closed, thus are Lie subgroups of [math]\displaystyle{ G }[/math]
• the orbits [math]\displaystyle{ G \cdot x \subseteq M }[/math] of the group action are immersed submanifolds.

Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action.

Examples

For every Lie group [math]\displaystyle{ G }[/math], the following are Lie group actions:

• the trivial action of [math]\displaystyle{ G }[/math] on any manifold
• the action of [math]\displaystyle{ G }[/math] on itself by left multiplication, right multiplication or conjugation
• the action of any Lie subgroup [math]\displaystyle{ H \subseteq G }[/math] on [math]\displaystyle{ G }[/math] by left multiplication, right multiplication or conjugation

• the adjoint action of [math]\displaystyle{ G }[/math] on its Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math].

Other examples of Lie group actions include:

• the action of [math]\displaystyle{ \mathbb{R} }[/math] on M given by the flow of any complete vector field
• the actions of the general linear group [math]\displaystyle{ \operatorname{GL}(n,\mathbb{R}) }[/math] and of its Lie subgroups [math]\displaystyle{ G\subseteq\operatorname{GL}(n,\mathbb{R}) }[/math] on [math]\displaystyle{ \mathbb{R}^n }[/math] by matrix multiplication
• more generally, any Lie group representation on a vector space
• any Hamiltonian group action on a symplectic manifold
• the transitive action underlying any homogeneous space
• more generally, the group action underlying any principal bundle

Infinitesimal Lie algebra action

Following the spirit of the Lie group-Lie algebra correspondence, Lie group actions can also be studied from the infinitesimal point of view. Indeed, any Lie group action [math]\displaystyle{ \sigma: G \times M \to M }[/math] induces an infinitesimal Lie algebra action on [math]\displaystyle{ M }[/math], i.e. a Lie algebra homomorphism [math]\displaystyle{ \mathfrak{g} \to \mathfrak{X}(M) }[/math]. Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism [math]\displaystyle{ G \to \mathrm{Diff}(M) }[/math], and interpreting the set of vector fields [math]\displaystyle{ \mathfrak{X}(M) }[/math] as the Lie algebra of the (infinite-dimensional) Lie group [math]\displaystyle{ \mathrm{Diff}(M) }[/math].

More precisely, fixing any [math]\displaystyle{ x \in M }[/math], the orbit map [math]\displaystyle{ \sigma_x : G \to M, g \mapsto g \cdot x }[/math] is differentiable and one can compute its differential at the identity [math]\displaystyle{ e \in G }[/math]. If [math]\displaystyle{ X \in \mathfrak{g} }[/math], then its image under [math]\displaystyle{ \mathrm{d}_e\sigma_x\colon \mathfrak{g}\to T_xM }[/math] is a tangent vector at [math]\displaystyle{ x }[/math], and varying [math]\displaystyle{ x }[/math] one obtains a vector field on [math]\displaystyle{ M }[/math]. The minus of this vector field, denoted by [math]\displaystyle{ X^\# }[/math], is also called the fundamental vector field associated with [math]\displaystyle{ X }[/math] (the minus sign ensures that [math]\displaystyle{ \mathfrak{g} \to \mathfrak{X}(M), X \mapsto X^\# }[/math] is a Lie algebra homomorphism).

Conversely, by Lie–Palais theorem, any abstract infinitesimal action of a (finite-dimensional) Lie algebra on a compact manifold can be integrated to a Lie group action.

Moreover, an infinitesimal Lie algebra action [math]\displaystyle{ \mathfrak{g} \to \mathfrak{X}(M) }[/math] is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of [math]\displaystyle{ \mathrm{d}_e\sigma_x\colon \mathfrak{g}\to T_xM }[/math] is the Lie algebra [math]\displaystyle{ \mathfrak{g}_x \subseteq \mathfrak{g} }[/math] of the stabilizer [math]\displaystyle{ G_x \subseteq G }[/math]. On the other hand, [math]\displaystyle{ \mathfrak{g} \to \mathfrak{X}(M) }[/math] in general not surjective. For instance, let [math]\displaystyle{ \pi: P \to M }[/math] be a principal [math]\displaystyle{ G }[/math]-bundle: the image of the infinitesimal action is actually equal to the vertical subbundle [math]\displaystyle{ T^\pi P \subset TP }[/math].

Proper actions

An important (and common) class of Lie group actions is that of proper ones. Indeed, such a topological condition implies that

• the stabilizers [math]\displaystyle{ G_x \subseteq G }[/math] are compact
• the orbits [math]\displaystyle{ G \cdot x \subseteq M }[/math] are embedded submanifolds
• the orbit space [math]\displaystyle{ M/G }[/math] is Hausdorff

In general, if a Lie group [math]\displaystyle{ G }[/math] is compact, any smooth [math]\displaystyle{ G }[/math]-action is automatically proper. An example of proper action by a not necessarily compact Lie group is given by the action a Lie subgroup [math]\displaystyle{ H \subseteq G }[/math] on [math]\displaystyle{ G }[/math].

Structure of the orbit space

Given a Lie group action of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ M }[/math], the orbit space [math]\displaystyle{ M/G }[/math] does not admit in general a manifold structure. However, if the action is free and proper, then [math]\displaystyle{ M/G }[/math] has a unique smooth structure such that the projection [math]\displaystyle{ M \to M/G }[/math] is a submersion (in fact, [math]\displaystyle{ M \to M/G }[/math] is a principal [math]\displaystyle{ G }[/math]-bundle).

The fact that [math]\displaystyle{ M/G }[/math] is Hausdorff depends only on the properness of the action (as discussed above); the rest of the claim requires freeness and is a consequence of the slice theorem. If the "free action" condition (i.e. "having zero stabilizers") is relaxed to "having finite stabilizers", [math]\displaystyle{ M/G }[/math] becomes instead an orbifold (or quotient stack).

An application of this principle is the Borel construction from algebraic topology. Assuming that [math]\displaystyle{ G }[/math] is compact, let [math]\displaystyle{ EG }[/math] denote the universal bundle, which we can assume to be a manifold since [math]\displaystyle{ G }[/math] is compact, and let [math]\displaystyle{ G }[/math] act on [math]\displaystyle{ EG \times M }[/math] diagonally. The action is free since it is so on the first factor and is proper since [math]\displaystyle{ G }[/math] is compact; thus, one can form the quotient manifold [math]\displaystyle{ M_G = (EG \times M)/G }[/math] and define the equivariant cohomology of M as

[math]\displaystyle{ H^*_G(M) = H^*_{\text{dr}}(M_G) }[/math],

where the right-hand side denotes the de Rham cohomology of the manifold [math]\displaystyle{ M_G }[/math].

See also

• Hamiltonian group action
• Equivariant differential form
• isotropy representation

References

• Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004
• John Lee, Introduction to smooth manifolds, chapter 9, ISBN:978-1-4419-9981-8
• Frank Warner, Foundations of differentiable manifolds and Lie groups, chapter 3, ISBN:978-0-387-90894-6