Lie group action explained

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.

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Definition

Let

\sigma: G x M \to M, (g, x) \mapsto g ⋅ 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 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:G _{x} \subseteq G of the group action are closed, thus are Lie subgroups of G G ⋅ 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 ak{g} .

Other examples of Lie group actions include:

• the action of

R on M given by the flow of any complete vector field;\operatorname{GL}(n, R ) and of its Lie subgroups G\subseteq\operatorname{GL}(n, R ) on 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 x M \to M induces an infinitesimal Lie algebra action on M , i.e. a Lie algebra homomorphism ak{g} \to ak{X}(M) . Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism G \to Diff (M) , and interpreting the set of vector fields ak{X}(M) as the Lie algebra of the (infinite-dimensional) Lie group Diff (M) .

More precisely, fixing any

x \in M , the orbit map \sigma _{x} : G \to M, g \mapsto g ⋅ x is differentiable and one can compute its differential at the identity e \in G . If X \in ak{g} , then its image under d _{e\sigma} _{x\colon ak{g}\toT} _{xM} 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 ak{g} \to ak{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

ak{g} \to ak{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\colon ak{g}\toT} _{xM} is the Lie algebra ak{g} _{x} \subseteq ak{g} of the stabilizer G _{x} \subseteq G .

On the other hand,

ak{g} \to ak{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 ⋅ 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).^[2]

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 x 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 x M)/G and define the equivariant cohomology of M asH ^{*} _{G(M)} = H ^{*} _{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,
• Frank Warner, Foundations of differentiable manifolds and Lie groups, chapter 3,

Notes and References

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