In differential geometry, a Lie group action is a group action adapted to the smooth setting:
is a smooth manifold, and the action map is differentiable.
Equivalently, a Lie group action of
consists of a Lie group homomorphism
A smooth manifold endowed with a Lie group action is also called a
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.
, the following are Lie group actions: Other examples of Lie group actions include: Following the spirit of the Lie group-Lie algebra correspondence, Lie group actions can also be studied from the infinitesimal point of view.
induces an infinitesimal Lie algebra action on
Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism
, and interpreting the set of vector fields
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
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] An infinitesimal Lie algebra action
is injective if and only if the corresponding global Lie group action is free.
-bundle: the image of the infinitesimal action is actually equal to the vertical subbundle
An important (and common) class of Lie group actions is that of proper ones.
Indeed, such a topological condition implies that In general, if a Lie group
An example of proper action by a not necessarily compact Lie group is given by the action a Lie subgroup
does not admit in general a manifold structure.
has a unique smooth structure such that the projection
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.
becomes instead an orbifold (or quotient stack).
An application of this principle is the Borel construction from algebraic topology.
denote the universal bundle, which we can assume to be a manifold since
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