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
.
Lie group, local. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_local&oldid=47631