# 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 and first properties

Let $\displaystyle{ \sigma: G \times M \to M, (g, x) \mapsto g \cdot x }$ 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 $\displaystyle{ \sigma }$ is differentiable. Equivalently, a Lie group action of G on 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 G-manifold.

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 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 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 $\displaystyle{ H \subseteq G }$ on G by left multiplication, right multiplication or conjugation
• the adjoint action of G on its Lie algebra $\displaystyle{ \mathfrak{g} }$.

Other examples of Lie group actions include:

• the action of $\displaystyle{ \mathbb{R} }$ on M given by the flow of any complete vector field
• the actions of the general linear group $\displaystyle{ GL(n,\mathbb{R}) }$ and of its Lie subgroups $\displaystyle{ G \subseteq GL(n,\mathbb{R}) }$ 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 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 $\displaystyle{ d_e \sigma_x: \mathfrak{g} \to T_x M }$ is a tangent vector at x, and varying x one obtains a vector field on M. The minus of this vector field, denoted by $\displaystyle{ X^\# }$, is also called the fundamental vector field associated with 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.

Moreover, 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 $\displaystyle{ d_e \sigma_x: \mathfrak{g} \to T_x M }$ 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 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 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 $\displaystyle{ H \subseteq G }$ on G.

## Structure of the orbit space

Given a Lie group action of G on 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 G-bundle).

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).

An application of this principle is the Borel construction from algebraic topology. Assuming that G is compact, let $\displaystyle{ EG }$ denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on $\displaystyle{ EG \times M }$ 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 $\displaystyle{ M_G = (EG \times M)/G }$ 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 $\displaystyle{ M_G }$.