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Lie group action

## 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 ${\displaystyle \sigma :G\times M\to M,(g,x)\mapsto g\cdot x}$ 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:

## 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.[1]

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).[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).

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 ${\displaystyle M_{G}}$.