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In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, a Lie group action is a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
adapted to the smooth setting: G is a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
, M is a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, and the action map is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
. __TOC__


Definition and first properties

Let \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 \sigma is differentiable. Equivalently, a Lie group action of G on M consists of a
Lie group homomorphism In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
G \to \mathrm(M). A smooth manifold endowed with a Lie group action is also called a ''G''-manifold. The fact that the action map \sigma is smooth has a couple of immediate consequences: * the stabilizers G_x \subseteq G of the group action are closed, thus are Lie subgroups of ''G'' * the
orbits In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
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 In topology, a continuous group action on a topological space ''X'' is a group action of a topological group ''G'' that is continuous: i.e., :G \times X \to X, \quad (g, x) \mapsto g \cdot x is a continuous map. Together with the group action, ''X'' ...
.


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 Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the chang ...
* the action of any Lie subgroup H \subseteq G on G by left multiplication, right multiplication or conjugation *the
adjoint action In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
of G on its Lie algebra \mathfrak. Other examples of Lie group actions include: * the action of \mathbb on M given by the flow of any complete vector field * the actions of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL(n,\mathbb) and of its Lie subgroups G \subseteq GL(n,\mathbb) on \mathbb^n by matrix multiplication *more generally, any
Lie group representation In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vec ...
on a vector space *any
Hamiltonian group action In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the ac ...
on a
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
*the transitive action underlying any
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
*more generally, the group action underlying any
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...


Infinitesimal Lie algebra action

Following the spirit of the
Lie group-Lie algebra correspondence A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be inter ...
, Lie group actions can also be studied from the infinitesimal point of view. Indeed, any Lie group action \sigma: G \times M \to M induces an infinitesimal Lie algebra action on M, i.e. a Lie algebra homomorphism \mathfrak \to \mathfrak(M). Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism G \to \mathrm(M), and interpreting the set of vector fields \mathfrak(M) as the Lie algebra of the (infinite-dimensional) Lie group \mathrm(M). More precisely, fixing any x \in M, the orbit map \sigma_x : G \to M, g \mapsto g \cdot x is differentiable and one can compute its differential at the identity e \in G. If X \in \mathfrak, then its image under d_e \sigma_x: \mathfrak \to T_x M is a
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
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 In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applicatio ...
associated with ''X'' (the minus sign ensures that \mathfrak \to \mathfrak(M), X \mapsto X^\# is a Lie algebra homomorphism). Conversely, by
Lie–Palais theorem In differential geometry, the Lie–Palais theorem states that an Group action (mathematics), action of a finite-dimensional Lie algebra on a Differentiable manifold, smooth compact manifold can be lifted to an action of a finite-dimensional Lie gr ...
, any abstract infinitesimal action of a (finite-dimensional) Lie algebra on a compact manifold can be integrated to a Lie group action. Moreover, an infinitesimal Lie algebra action \mathfrak \to \mathfrak(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: \mathfrak \to T_x M is the Lie algebra \mathfrak_x \subseteq \mathfrak of the stabilizer G_x \subseteq G. On the other hand, \mathfrak \to \mathfrak(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 Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
ones. Indeed, such a topological condition implies that * the stabilizers G_x \subseteq G are
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
* the orbits G \cdot 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 In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
M/G does not admit in general a manifold structure. However, if the action is
free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...
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). 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 In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
(or
quotient stack In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. T ...
). An application of this principle is the
Borel construction In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an or ...
from
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. Assuming that ''G'' is compact, let EG denote the
universal bundle In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group , is a specific bundle over a classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topo ...
, which we can assume to be a manifold since ''G'' is compact, and let ''G'' act on 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 M_G = (EG \times M)/G and define the
equivariant cohomology In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordi ...
of ''M'' as :H^*_G(M) = H^*_(M_G), where the right-hand side denotes the
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
of the manifold M_G.


See also

*
Hamiltonian group action In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the ac ...
*
Equivariant differential form In differential geometry, an equivariant differential form on a manifold ''M'' acted upon by a Lie group ''G'' is a polynomial map :\alpha: \mathfrak \to \Omega^*(M) from the Lie algebra \mathfrak = \operatorname(G) to the space of differential fo ...
*
isotropy representation In differential geometry, the isotropy representation is a natural linear representation of a Lie group, that is acting on a manifold, on the tangent space to a fixed point. Construction Given a Lie group action (G, \sigma) on a manifold ''M'', ...


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, {{ISBN, 978-0-387-90894-6 Group actions (mathematics) Lie groups