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.

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

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 $\operatorname {GL} (n,\mathbb {R} )$ and of its Lie subgroups $G\subseteq \operatorname {GL} (n,\mathbb {R} )$ 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 $\mathrm {d} _{e}\sigma _{x}\colon {\mathfrak {g}}\to T_{x}M$ 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.

Properties

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 $\mathrm {d} _{e}\sigma _{x}\colon {\mathfrak {g}}\to T_{x}M$ 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).

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

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

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, ISBN 978-1-4419-9981-8
• Frank Warner, Foundations of differentiable manifolds and Lie groups, chapter 3, ISBN 978-0-387-90894-6