Lie group action

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

Definition

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

Properties

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

• the stabilizers ${\displaystyle G_{x}\subseteq G}$ of the group action are closed, thus are Lie subgroups of ${\displaystyle G}$
• the orbits ${\displaystyle G\cdot x\subseteq M}$ 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 ${\displaystyle G}$, the following are Lie group actions:

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

• the adjoint action of ${\displaystyle G}$ on its Lie algebra ${\displaystyle {\mathfrak {g}}}$.

Other examples of Lie group actions include:

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

More precisely, fixing any ${\displaystyle x\in M}$, the orbit map ${\displaystyle \sigma _{x}:G\to M,g\mapsto g\cdot x}$ is differentiable and one can compute its differential at the identity ${\displaystyle e\in G}$. If ${\displaystyle X\in {\mathfrak {g}}}$, then its image under ${\displaystyle \mathrm {d} _{e}\sigma _{x}\colon {\mathfrak {g}}\to T_{x}M}$ is a tangent vector at ${\displaystyle x}$, and varying ${\displaystyle x}$ one obtains a vector field on ${\displaystyle M}$. The minus of this vector field, denoted by ${\displaystyle X^{\#}}$, is also called the fundamental vector field associated with ${\displaystyle X}$ (the minus sign ensures that ${\displaystyle {\mathfrak {g}}\to {\mathfrak {X}}(M),X\mapsto X^{\#}}$ 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.^[1]

Properties

An infinitesimal Lie algebra action ${\displaystyle {\mathfrak {g}}\to {\mathfrak {X}}(M)}$ is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of ${\displaystyle \mathrm {d} _{e}\sigma _{x}\colon {\mathfrak {g}}\to T_{x}M}$ is the Lie algebra ${\displaystyle {\mathfrak {g}}_{x}\subseteq {\mathfrak {g}}}$ of the stabilizer ${\displaystyle G_{x}\subseteq G}$.

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

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 ${\displaystyle G_{x}\subseteq G}$ are compact
• the orbits ${\displaystyle G\cdot x\subseteq M}$ are embedded submanifolds
• the orbit space ${\displaystyle M/G}$ is Hausdorff

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

Structure of the orbit space

Given a Lie group action of ${\displaystyle G}$ on ${\displaystyle M}$, the orbit space ${\displaystyle M/G}$ does not admit in general a manifold structure. However, if the action is free and proper, then ${\displaystyle M/G}$ has a unique smooth structure such that the projection ${\displaystyle M\to M/G}$ is a submersion (in fact, ${\displaystyle M\to M/G}$ is a principal ${\displaystyle G}$-bundle).^[2]

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

Equivariant cohomology

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

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

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

See also

• Hamiltonian group action
• Equivariant differential form
• Isotropy representation

Notes

1. Palais, Richard S. (1957). "A global formulation of the Lie theory of transformation groups". Memoirs of the American Mathematical Society (22): 0. doi:10.1090/memo/0022. ISSN 0065-9266.
2. Lee, John M. (2012). Introduction to smooth manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-9982-5. OCLC 808682771.

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