Lie group, local

local analytic group

An analytic manifold over a field that is complete with respect to some non-trivial absolute value, which is endowed with a distinguished element (the identity), an open subset and a pair of analytic mappings of the manifold into and of the neighbourhood into itself, for which:

1) in some neighbourhood of one has ;

2) in some neighbourhood of one has ;

3) for some neighbourhood of one has and , where are arbitrary elements of .

Local Lie groups first made their appearance in the work of S. Lie and his school (see [1]) as local Lie transformation groups (cf. Lie transformation group).

Let and be two local Lie groups with identities and , respectively. A local homomorphism of into (denoted by ) is an analytic mapping of some neighbourhood in for which and for and in some neighbourhood of . The naturally defined composition of local homomorphisms is also a local homomorphism. Local homomorphisms that coincide in some neighbourhood of are said to be equivalent. If there are local homomorphism and such that the compositions and are equivalent to the identity mappings, then the local Lie groups and are said to be equivalent.

Examples. Let be an analytic group with identity and an open neighbourhood of in . Then the analytic structure on induces an analytic structure on , and the operations of multiplication and taking the inverse of an element in convert into a local Lie group (in particular, itself can be regarded as a local Lie group). All local Lie groups obtainable in this way from a fixed analytic group are equivalent to one another.

One of the fundamental questions in the theory of Lie groups is the question of how general a character the example given above has, that is, whether every local Lie group is (up to equivalence) a neighbourhood of some analytic group. The answer to this question is affirmative (see [2], [3], [4]; in the case of local Banach Lie groups the answer is negative, see [4]).

The most important tool for studying local Lie groups is the correspondence between the local Lie group and its Lie algebra. Namely, let be a local Lie group over a field and let be the identity of it. The choice of a chart of the analytic manifold at the point makes it possible to identify some neighbourhood of in with some neighbourhood of the origin in the -dimensional coordinate space , so that becomes a local Lie group. Let be a neighbourhood of the origin in the local Lie group such that for any a product is defined. Then, in coordinate form, multiplication in in the neighbourhood is specified by analytic functions

where , , are, respectively, the coordinates of the points and . In a sufficiently small neighbourhood of the origin the function is represented as the sum of a convergent power series (also denoted by henceforth), and the presence in of an identity and the associative law is expressed by the following properties of these series, regarded as formal power series in variables:

a) and for all ;

b) for all .

Properties a) and b) imply that the system of formal power series is a formal group. In particular, the homogeneous component of degree 2 of each of the series is a bilinear form on , that is, it has the form

which makes it possible to define a multiplication on according to the rule:

With respect to this multiplication is a Lie algebra. The structure of a Lie algebra carries over to the tangent space to at by means of the chart , defined above, by the isomorphism . The formal groups and defined by different charts are isomorphic, and the structure of a Lie algebra on does not depend on the choice of the chart . The Lie algebra is called the Lie algebra of a local Lie group. For any local homomorphism of a local Lie group its differential at the identity is a homomorphism of Lie algebras, which implies that the correspondence between a local Lie group and its Lie algebra is functorial. In particular, equivalent local Lie groups have isomorphic Lie algebras.

If the field has characteristic 0, then the construction given above, which goes back to Lie [1], makes it possible to reduce the study of properties of local Lie groups to the study of the corresponding properties of their Lie algebras. In this case the Lie algebra determines the local Lie group uniquely up to equivalence. Namely, the chart can be chosen so that the product in the local Lie group is expressed as a convergent series (the so-called Campbell–Hausdorff series) of elements of obtained from and by means of the commutation operation and multiplication by elements of (see Campbell–Hausdorff formula). Conversely, for an arbitrary finite-dimensional Lie algebra over the Campbell–Hausdorff series converges in some neighbourhood of the origin in and determines in this neighbourhood the structure of a local Lie group with Lie algebra . Thus, for any given Lie algebra there is a unique (up to equivalence) local Lie group with as its Lie algebra. Moreover, every homomorphism of Lie algebras is induced by a unique homomorphism of the corresponding local Lie groups. In other words, the correspondence between a local Lie group and its Lie algebra defines an equivalence of the category of local Lie groups and the category of finite-dimensional Lie algebras over . Moreover, the correspondence between a local Lie group and the corresponding formal group defines an equivalence of the category of local Lie groups and the category of formal groups over .

The Lie algebra can also be defined for any local Banach Lie group; the main result about the equivalence of the categories of local Lie groups and Lie algebras can be generalized to this case (see [2]).

References

[1]	S. Lie, F. Engel, "Theorie der Transformationsgruppen" , 1–3 , Leipzig (1888–1893)
[2]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[3]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[4]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[5]	N.G. Chebotarev, "The theory of Lie groups" , Moscow-Leningrad (1940) (In Russian)

Comments

The equivalences of categories between local Lie groups, formal groups and Lie algebras over a field only hold for fields of characteristic zero. In particular, for a field of characteristic there are at least countably many non-isomorphic -dimensional formal groups over , while there is of course only one -dimensional Lie algebra over .