Lie group action

In differential geometry, a Lie group action is a group action adapted to the smooth setting: $G$ is a Lie group, $M$ is a smooth manifold, and the action map is differentiable.

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Definition

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

Properties

The fact that the action map $\sigma$ is smooth has a couple of immediate consequences:

* the stabilizers $G_x \subseteq G$ of the group action are closed, thus are Lie subgroups of ''$G$''
* the orbits $G \cdot x \subseteq M$ 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 $G$, the following are Lie group actions:
*the trivial action of $G$ on any manifold;
*the action of $G$ on itself by left multiplication, right multiplication or conjugation;
* the action of any Lie subgroup $H \subseteq G$ on $G$ by left multiplication, right multiplication or conjugation;

*the adjoint action of $G$ on its Lie algebra $\mathfrak$.

Other examples of Lie group actions include:
* the action of $\mathbb$ on $M$ given by the flow of any complete vector field;
* the actions of the general linear group $\operatorname(n,\mathbb)$ and of its Lie subgroups $G\subseteq\operatorname(n,\mathbb)$ on $\mathbb^n$ 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 $\sigma: G \times M \to M$ induces an infinitesimal Lie algebra action on $M$, i.e. a Lie algebra homomorphism $\mathfrak \to \mathfrak(M)$. Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism $G \to \mathrm(M)$, and interpreting the set of vector fields $\mathfrak(M)$ as the Lie algebra of the (infinite-dimensional) Lie group $\mathrm(M)$.

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

Properties

An infinitesimal Lie algebra action $\mathfrak \to \mathfrak(M)$ is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of $\mathrm_e\sigma_x\colon \mathfrak\to T_xM$ is the Lie algebra $\mathfrak_x \subseteq \mathfrak$ of the stabilizer $G_x \subseteq G$.

On the other hand, $\mathfrak \to \mathfrak(M)$ in general not surjective. For instance, let $\pi: P \to M$ be a principal $G$-bundle: the image of the infinitesimal action is actually equal to the vertical subbundle $T^\pi P \subset TP$.

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 $G_x \subseteq G$ are compact
* the orbits $G \cdot x \subseteq M$ are embedded submanifolds
* the orbit space $M/G$ is Hausdorff

In general, if a Lie group $G$ is compact, any smooth $G$-action is automatically proper. An example of proper action by a not necessarily compact Lie group is given by the action a Lie subgroup $H \subseteq G$ on $G$.

Structure of the orbit space

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

The fact that $M/G$ 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", $M/G$ becomes instead an orbifold (or quotient stack).

Equivariant cohomology

An application of this principle is the Borel construction from algebraic topology. Assuming that $G$ is compact, let $EG$ denote the universal bundle, which we can assume to be a manifold since $G$ is compact, and let $G$ act on $EG \times M$ diagonally. The action is free since it is so on the first factor and is proper since $G$ is compact; thus, one can form the quotient manifold $M_G = (EG \times M)/G$ and define the equivariant cohomology of ''M'' as
:$H^*_G(M) = H^*_(M_G)$,
where the right-hand side denotes the de Rham cohomology of the manifold $M_G$.

See also

* Hamiltonian group action
* Equivariant differential form
* Isotropy representation

Notes

References

*Michele Audin, ''Torus actions on symplectic manifolds'', Birkhauser, 2004
*John Lee, ''Introduction to smooth manifolds'', chapter 9,
*Frank Warner, ''Foundations of differentiable manifolds and Lie groups'', chapter 3, {{ISBN, 978-0-387-90894-6

Group actions
Lie groups