Lie group action

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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 [math]\displaystyle{ \sigma: G \times M \to M, (g, x) \mapsto g \cdot x }[/math] 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 [math]\displaystyle{ \sigma }[/math] is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism [math]\displaystyle{ G \to \mathrm{Diff}(M) }[/math]. A smooth manifold endowed with a Lie group action is also called a G-manifold.

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

  • the stabilizers [math]\displaystyle{ G_x \subseteq G }[/math] of the group action are closed, thus are Lie subgroups of G
  • the orbits [math]\displaystyle{ G \cdot x \subseteq M }[/math] 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 [math]\displaystyle{ H \subseteq G }[/math] on G by left multiplication, right multiplication or conjugation
  • the adjoint action of G on its Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math].

Other examples of Lie group actions include:

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


More precisely, fixing any [math]\displaystyle{ x \in M }[/math], the orbit map [math]\displaystyle{ \sigma_x : G \to M, g \mapsto g \cdot x }[/math] is differentiable and one can compute its differential at the identity [math]\displaystyle{ e \in G }[/math]. If [math]\displaystyle{ X \in \mathfrak{g} }[/math], then its image under [math]\displaystyle{ d_e \sigma_x: \mathfrak{g} \to T_x M }[/math] is a tangent vector at x, and varying x one obtains a vector field on M. The minus of this vector field, denoted by [math]\displaystyle{ X^\# }[/math], is also called the fundamental vector field associated with X (the minus sign ensures that [math]\displaystyle{ \mathfrak{g} \to \mathfrak{X}(M), X \mapsto X^\# }[/math] 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 [math]\displaystyle{ \mathfrak{g} \to \mathfrak{X}(M) }[/math] is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of [math]\displaystyle{ d_e \sigma_x: \mathfrak{g} \to T_x M }[/math] is the Lie algebra [math]\displaystyle{ \mathfrak{g}_x \subseteq \mathfrak{g} }[/math] of the stabilizer [math]\displaystyle{ G_x \subseteq G }[/math]. On the other hand, [math]\displaystyle{ \mathfrak{g} \to \mathfrak{X}(M) }[/math] in general not surjective. For instance, let [math]\displaystyle{ \pi: P \to M }[/math] be a principal G-bundle: the image of the infinitesimal action is actually equal to the vertical subbundle [math]\displaystyle{ T^\pi P \subset TP }[/math].

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 [math]\displaystyle{ G_x \subseteq G }[/math] are compact
  • the orbits [math]\displaystyle{ G \cdot x \subseteq M }[/math] are embedded submanifolds
  • the orbit space [math]\displaystyle{ M/G }[/math] 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 [math]\displaystyle{ H \subseteq G }[/math] on G.

Structure of the orbit space

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

The fact that [math]\displaystyle{ M/G }[/math] 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", [math]\displaystyle{ M/G }[/math] 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 [math]\displaystyle{ EG }[/math] denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on [math]\displaystyle{ EG \times M }[/math] 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 [math]\displaystyle{ M_G = (EG \times M)/G }[/math] and define the equivariant cohomology of M as

[math]\displaystyle{ H^*_G(M) = H^*_{\text{dr}}(M_G) }[/math],

where the right-hand side denotes the de Rham cohomology of the manifold [math]\displaystyle{ M_G }[/math].

See also

References

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