Lie transformation group

A smooth action of a connected Lie group $ G $ on a smooth manifold $ M $, that is, a smooth mapping (of class $ C ^ \infty $) $ A : G \times M \rightarrow M $ such that

I) $ A ( g ^ \prime g ^ {\prime\prime} , m ) = A ( g ^ \prime , A ( g ^ {\prime\prime} , m ) ) $ for all $ g ^ \prime , g ^ {\prime\prime} \in G $, $ m \in G $;

II) $ A ( e , m ) = m $ for all $ m \in M $( $ e $ is the identity of the group $ G $).

An action $ A $ that also satisfies the condition

III) if $ A ( g , m ) = m $ for all $ m \in M $, then $ g = e $, is said to be effective.

Examples of Lie transformation groups. Any smooth linear representation of a Lie group $ G $ in a finite-dimensional vector space $ M $; the action of a Lie group $ G $ on itself by means of left or right translations, $ A ( g , m ) = gm $ or $ A ( g , m ) = m g ^ {-} 1 $, respectively $ ( g , m \in G ) $; the action of a Lie group $ G $ on itself by means of inner automorphisms, $ A ( g , m ) = gmg ^ {-} 1 $ $ ( g , m \in G ) $; and a one-parameter transformation group, that is, the smooth action of the group $ \mathbf R $ on a manifold $ M $.

Together with global Lie transformation groups defined above one also considers local Lie transformation groups, which are the main topic of the classical theory of Lie groups [1]. Instead of $ G $ one considers a local Lie group (cf. Lie group, local), that is, a neighbourhood $ U $ of the identity in some Lie group, and instead of $ M $ an open subset $ W \subset \mathbf R ^ {n} $.

If $ G $ is a Lie transformation group on $ M $, then by choosing a suitable neighbourhood $ U \ni e $ in $ G $ and an open subset $ W \subset M $ one obtains a local Lie transformation group. The reverse step, from a local Lie transformation group to a global one (globalization) is not always possible. However, if $ \mathop{\rm dim} M \leq 4 $ and if $ W $ is sufficiently small, then globalization is possible (see [2]).

One sometimes considers Lie transformation groups of class $ C ^ {k} $, $ 1 \leq k \leq \infty $, or $ C ^ {a} $( analytic), that is, it is assumed that $ A $ belongs to the corresponding class. If $ A $ is continuous, then for it to belong to $ C ^ {k} $ or $ C ^ {a} $ it is sufficient that for any $ g \in G $ the transformation $ A _ {g} : m \rightarrow A ( g , m) $ of $ M $ should belong to this class (see [3]). In particular, the specification of a Lie transformation group $ G $ on $ M $ is equivalent to the specification of a continuous homomorphism $ G \rightarrow \mathop{\rm Diff} M $ into the group $ \mathop{\rm Diff} M $ of diffeomorphisms of $ M $, endowed with the natural topology.

To any Lie transformation group corresponds a homomorphism $ A _ {*} : \mathfrak g \rightarrow \Phi ( M) $ of the Lie algebra $ \mathfrak g $ of $ G $ into the Lie algebra $ \Phi ( M) $ of smooth vector fields on $ M $, which sets up a correspondence between an element $ X \in \mathfrak g $ and the velocity field of the one-parameter transformation group

$$ ( t , m) \rightarrow A ( \mathop{\rm exp} tX , m ) , $$

where $ t \in \mathbf R $, $ m \in M $ and $ \mathop{\rm exp} : \mathfrak g \rightarrow G $ is the exponential mapping (see [5]). If $ G $ is effective, then $ A _ {*} $ is injective. For a connected group $ G $ the homomorphism $ A _ {*} $ completely determines the Lie transformation group. Conversely, to any homomorphism $ \beta : \mathfrak g \rightarrow \Phi ( M) $ corresponds a local Lie transformation group [6]. If all vector fields of $ \beta ( \mathfrak g ) $ are complete (that is, their integral curves $ x ( t) $ are defined for all $ t $), then there is a global Lie transformation group $ G $ on $ M $ for which $ \mathop{\rm Im} A _ {*} = \mathfrak g $. It is sufficient to require that as a Lie algebra $ \beta ( \mathfrak g ) $ is generated by complete vector fields; the completeness condition is automatically satisfied if $ M $ is compact [4].

If $ G $ is a Lie transformation group of a manifold $ M $, then the stationary subgroup $ G _ {m} = \{ {g \in G } : {A ( g , m ) = m } \} $ for any point $ m \in M $ is a closed Lie subgroup of $ G $; it is also called the stabilizer, or isotropy subgroup, of the point $ m $. The corresponding Lie subalgebra $ \mathfrak g _ {m} \subset \mathfrak g $ consists of all $ X \in \mathfrak g $ such that $ A _ {*} ( X) _ {m} = 0 $. The subalgebra $ \mathfrak g _ {m} $ depends continuously on $ m $ in the natural topology on the set of all subalgebras of $ \mathfrak g $[7]. The orbit $ G ( m) = \{ {A ( g , m ) } : {g \in G } \} $ of the point $ m $ is an immersed submanifold of $ M $ diffeomorphic to $ G / G _ {m} $. If $ G $ is compact, then all orbits are compact imbedded submanifolds. Examples of non-imbedded orbits are given by the action of the group $ \mathbf R $ on the torus

$$ T ^ {2} = \{ {( z _ {1} , z _ {2} ) } : {z _ {i} \in \mathbf C , | z _ {i} | = 1 , i = 1, 2 } \} $$

given by the formula

$$ A ( t , ( z _ {1} , z _ {2} ) ) = ( e ^ {it} z _ {1} , e ^ {i \alpha t } z _ {2} ) , $$

where $ \alpha \in \mathbf R $ is irrational.

Two Lie transformation groups $ A _ {i} : G \times M _ {i} \rightarrow M _ {i} $, $ i = 1 , 2 $, are said to be similar if there is a diffeomorphism $ f : M _ {1} \rightarrow M _ {2} $ such that $ A _ {1} ( g , m ) = A _ {2} ( g , f ( m) ) $, $ g \in G $, $ m \in M _ {1} $. An important problem in the theory of transformation groups is the problem of classifying Lie transformation groups up to similarity. At present (1989) it has been solved only in certain special cases. S. Lie [1] gave a classification of local Lie transformation groups in domains of $ \mathbf R ^ {1} $ and $ \mathbf R ^ {2} $ up to local similarity. A partial classification has been carried out for Lie transformation groups on three-dimensional manifolds. Compact Lie transformation groups have also been well studied. For transitive Lie transformation groups see Homogeneous space.

References

[1]	S. Lie, "Theorie der Transformationsgruppen" Math. Ann. , 16 (1880) pp. 441–528
[2]	G. Mostow, "The extensibility of local Lie groups of transformations and groups on surfaces" Ann. of Math. (2) , 52 (1950) pp. 606–636
[3]	S. Bochner, D. Montgomery, "Groups of differentiable and real or complex analytic transformations" Ann. of Math. (2) , 46 (1945) pp. 685–694
[4]	R. Palais, "A global formulation of the Lie theory of transformation groups" Mem. Amer. Math. Soc. , 22 (1957) pp. 1–123
[5]	R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Birkhäuser (1972)
[6]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[7]	R. Richardson, "On the variation of isotropy subalgebras" , Proc. Conf. Transformation Groups, New Orleans, 1967 , Springer (1968) pp. 429–440
[8]	N.G. Chebotarev, "The theory of Lie groups" , Moscow-Leningrad (1940) (In Russian)

Comments

If $ G $ is a locally compact group which acts continuously and effectively on a $ C ^ {k} $ manifold by means of $ C ^ {k} $ transformations, then $ G $ is a Lie group and the action $ G \times M \rightarrow M $ is $ C ^ {k} $.

For $ k \geq 2 $ this theorem is due to S. Bochner and D. Montgomery, for $ k = 1 $ to M. Kuranishi, see [a1], Chapt. V.

References

[a1]	D. Montgomery, L. Zippin, "Topological transformation groups" , Interscience (1964)