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Summary

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.

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 $G\to \mathrm {Diff} (M)$ . 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 ${\mathfrak {g}}$ .

Other examples of Lie group actions include:

• the action of $\mathbb {R}$ on M given by the flow of any complete vector field
• the actions of the general linear group $GL(n,\mathbb {R} )$ and of its Lie subgroups $G\subseteq GL(n,\mathbb {R} )$ on $\mathbb {R} ^{n}$ 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 ${\mathfrak {g}}\to {\mathfrak {X}}(M)$ . Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism $G\to \mathrm {Diff} (M)$ , and interpreting the set of vector fields ${\mathfrak {X}}(M)$ as the Lie algebra of the (infinite-dimensional) Lie group $\mathrm {Diff} (M)$ .

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 $X\in {\mathfrak {g}}$ , then its image under $d_{e}\sigma _{x}:{\mathfrak {g}}\to T_{x}M$ 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 ${\mathfrak {g}}\to {\mathfrak {X}}(M),X\mapsto X^{\#}$ 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 ${\mathfrak {g}}\to {\mathfrak {X}}(M)$ is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of $d_{e}\sigma _{x}:{\mathfrak {g}}\to T_{x}M$ is the Lie algebra ${\mathfrak {g}}_{x}\subseteq {\mathfrak {g}}$ of the stabilizer $G_{x}\subseteq G$ . On the other hand, ${\mathfrak {g}}\to {\mathfrak {X}}(M)$ 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

$H_{G}^{*}(M)=H_{\text{dr}}^{*}(M_{G})$ ,

where the right-hand side denotes the de Rham cohomology of the manifold $M_{G}$ .