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 and first properties

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.

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 $GL(n,\mathbb)$ and of its Lie subgroups $G \subseteq GL(n,\mathbb)$ 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 $d_e \sigma_x: \mathfrak \to T_x M$ 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.

Moreover, 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 $d_e \sigma_x: \mathfrak \to T_x M$ 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).

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

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 (mathematics)
Lie groups