Lie transformation group

A smooth action of a connected Lie group on a smooth manifold , that is, a smooth mapping (of class ) such that

I) for all , ;

II) for all ( is the identity of the group ).

An action that also satisfies the condition

III) if for all , then , is said to be effective.

Examples of Lie transformation groups. Any smooth linear representation of a Lie group in a finite-dimensional vector space ; the action of a Lie group on itself by means of left or right translations, or , respectively ; the action of a Lie group on itself by means of inner automorphisms, ; and a one-parameter transformation group, that is, the smooth action of the group on a manifold .

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 one considers a local Lie group (cf. Lie group, local), that is, a neighbourhood of the identity in some Lie group, and instead of an open subset .

If is a Lie transformation group on , then by choosing a suitable neighbourhood in and an open subset 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 and if is sufficiently small, then globalization is possible (see [2]).

One sometimes considers Lie transformation groups of class , , or (analytic), that is, it is assumed that belongs to the corresponding class. If is continuous, then for it to belong to or it is sufficient that for any the transformation of should belong to this class (see [3]). In particular, the specification of a Lie transformation group on is equivalent to the specification of a continuous homomorphism into the group of diffeomorphisms of , endowed with the natural topology.

To any Lie transformation group corresponds a homomorphism of the Lie algebra of into the Lie algebra of smooth vector fields on , which sets up a correspondence between an element and the velocity field of the one-parameter transformation group

where , and is the exponential mapping (see [5]). If is effective, then is injective. For a connected group the homomorphism completely determines the Lie transformation group. Conversely, to any homomorphism corresponds a local Lie transformation group [6]. If all vector fields of are complete (that is, their integral curves are defined for all ), then there is a global Lie transformation group on for which . It is sufficient to require that as a Lie algebra is generated by complete vector fields; the completeness condition is automatically satisfied if is compact [4].

If is a Lie transformation group of a manifold , then the stationary subgroup for any point is a closed Lie subgroup of ; it is also called the stabilizer, or isotropy subgroup, of the point . The corresponding Lie subalgebra consists of all such that . The subalgebra depends continuously on in the natural topology on the set of all subalgebras of [7]. The orbit of the point is an immersed submanifold of diffeomorphic to . If is compact, then all orbits are compact imbedded submanifolds. Examples of non-imbedded orbits are given by the action of the group on the torus

given by the formula

where is irrational.

Two Lie transformation groups , , are said to be similar if there is a diffeomorphism such that , , . 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 and 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 is a locally compact group which acts continuously and effectively on a manifold by means of transformations, then is a Lie group and the action is .

For this theorem is due to S. Bochner and D. Montgomery, for to M. Kuranishi, see [a1], Chapt. V.

References

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