In differential geometry, a Lie group action is a group action adapted to the smooth setting: is a Lie group, is a smooth manifold, and the action map is differentiable.

Definition

edit

Let   be a (left) group action of a Lie group   on a smooth manifold  ; it is called a Lie group action (or smooth action) if the map   is differentiable. Equivalently, a Lie group action of   on   consists of a Lie group homomorphism  . A smooth manifold endowed with a Lie group action is also called a  -manifold.

Properties

edit

The fact that the action map   is smooth has a couple of immediate consequences:

Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action.

Examples

edit

For every Lie group  , the following are Lie group actions:

  • the trivial action of   on any manifold;
  • the action of   on itself by left multiplication, right multiplication or conjugation;
  • the action of any Lie subgroup   on   by left multiplication, right multiplication or conjugation;
  • the adjoint action of   on its Lie algebra  .

Other examples of Lie group actions include:

Infinitesimal Lie algebra action

edit

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   induces an infinitesimal Lie algebra action on  , i.e. a Lie algebra homomorphism  . Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism  , and interpreting the set of vector fields   as the Lie algebra of the (infinite-dimensional) Lie group  .

More precisely, fixing any  , the orbit map   is differentiable and one can compute its differential at the identity  . If  , then its image under   is a tangent vector at  , and varying   one obtains a vector field on  . The minus of this vector field, denoted by  , is also called the fundamental vector field associated with   (the minus sign ensures that   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

edit

An infinitesimal Lie algebra action   is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of   is the Lie algebra   of the stabilizer  .

On the other hand,   in general not surjective. For instance, let   be a principal  -bundle: the image of the infinitesimal action is actually equal to the vertical subbundle  .

Proper actions

edit

An important (and common) class of Lie group actions is that of proper ones. Indeed, such a topological condition implies that

  • the stabilizers   are compact
  • the orbits   are embedded submanifolds
  • the orbit space   is Hausdorff

In general, if a Lie group   is compact, any smooth  -action is automatically proper. An example of proper action by a not necessarily compact Lie group is given by the action a Lie subgroup   on  .

Structure of the orbit space

edit

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

The fact that   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",   becomes instead an orbifold (or quotient stack).

Equivariant cohomology

edit

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

 ,

where the right-hand side denotes the de Rham cohomology of the manifold  .

See also

edit

Notes

edit
  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

edit
  • 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