Local topological group
A topological group in which the group operations are defined only for elements sufficiently close to the identity. The introduction of local topological groups was inspired by the study of the local structure of topological groups (that is, their structure in an arbitrary small neighbourhood of the identity, see [1]). The precise definition of a local topological group is as follows.
Let be a topological space,
an element of it,
and
open subsets of
and
, respectively, where
, and let
and
be continuous mappings. Then the system
is called a local topological group if the following conditions are satisfied:
1) and
for any
and
;
2) if and
,
,
,
, then
;
3) and
for any
and
.
The local topological group is usually denoted simply by
; the element
is denoted by
and called the product of
and
; the element
is denoted by
and called the inverse of
; the element
is called the identity of
. If
, one says that the product of
and
is defined; if
, one says that an inverse element is defined for
.
These operations on (which are not defined for all elements) induce the structure of a local topological group in an arbitrary neighbourhood of the identity
of
. Let
and
be two local topological groups. A local homomorphism of
into
is a continuous mapping
of a neighbourhood
of the identity
of
into a neighbourhood
of the identity
of
such that
and for any elements
whose product is defined in
the product of the elements
and
is also defined in
and
. Two local homomorphisms of
into
are said to be equivalent if they coincide in a neighbourhood of the identity of
. Suppose that the local homomorphism
is a homeomorphism of the neighbourhoods
and
and that the inverse mapping
is a local homomorphism of
to
. Then
is called a local isomorphism of
and
. Two local topological groups between which there is a local isomorphism are said to be locally isomorphic. For example, any local topological group is locally isomorphic to an arbitrary neighbourhood of the identity of it.
As an example of a local topological group one can take any topological group (and hence any neighbourhood of the identity of it). In the theory of local topological groups the main question is to what extent this example has a general character; that is, whether any local topological group is locally isomorphic to some topological group. In the general case the answer is negative (see [4]), but in the important special case of finite-dimensional local Lie groups (cf. Lie group, local) it is affirmative.
Just as in the theory of topological groups, in the theory of local topological groups one can define the concepts of (local) subgroups, normal subgroups, cosets, and quotient groups. For example, let be a local topological group and let
be a subset of
containing
such that in a neighbourhood
of
in
the set
is closed. Suppose also that for any
the element
belongs to
and that the set
![]() |
is open in (under the assumption that
is endowed with the topology induced from
). Then the system
![]() |
is a local topological group, called a local subgroup of . For the definitions of a normal subgroup, cosets with respect to a subgroup and a quotient group, see [1].
References
[1] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) |
[2] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) |
[3] | J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) |
[4] | S. Lie, F. Engel, "Theorie der Transformationsgruppen" , 1–3 , Leipzig (1930) |
Local topological group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_topological_group&oldid=47684