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 $ G $ be a topological space, $ e $ an element of it, $ \Theta $ and $ \Omega $ open subsets of $ G $ and $ G \times G $, respectively, where $ e \in \Theta $, and let $ i : \Theta \rightarrow G $ and $ m : \Omega \rightarrow G $ be continuous mappings. Then the system $ ( G , e , \Theta , \Omega , i , m ) $ is called a local topological group if the following conditions are satisfied:

1) $ ( e , g ) $ and $ ( g , e ) \in \Omega $ for any $ g \in G $ and $ m ( ( e , g ) ) = m ( ( g , e ) ) = g $;

2) if $ g , h , t \in G $ and $ ( g , h ) $, $ ( h , t ) $, $ ( ( g , h ) , t ) $, $ ( g , ( h , t ) ) \in \Omega $, then $ m ( ( m ( ( g , h ) ) , t ) ) = m ( ( g , m ( ( h , t ) ) ) ) $;

3) $ ( g , i ( g) ) $ and $ ( i ( g) , g ) \in \Omega $ for any $ g \in \Theta $ and $ m ( ( g , i ( g) ) ) = m ( ( i ( g), g)) = e $.

The local topological group $ ( G , e , \Theta , \Omega , i , m ) $ is usually denoted simply by $ G $; the element $ m ( ( g , h )) $ is denoted by $ gh $ and called the product of $ g $ and $ h $; the element $ i ( g) $ is denoted by $ g ^ {-} 1 $ and called the inverse of $ g $; the element $ e $ is called the identity of $ G $. If $ ( g , h ) \in \Omega $, one says that the product of $ g $ and $ h $ is defined; if $ g \in \Theta $, one says that an inverse element is defined for $ g $.

These operations on $ G $( which are not defined for all elements) induce the structure of a local topological group in an arbitrary neighbourhood of the identity $ e $ of $ G $. Let $ G _ {1} $ and $ G _ {2} $ be two local topological groups. A local homomorphism of $ G _ {1} $ into $ G _ {2} $ is a continuous mapping $ f $ of a neighbourhood $ U _ {1} $ of the identity $ e _ {1} $ of $ G _ {1} $ into a neighbourhood $ U _ {2} $ of the identity $ e _ {2} $ of $ G _ {2} $ such that $ f ( e _ {1} ) = e _ {2} $ and for any elements $ g , h \in U _ {1} $ whose product is defined in $ G _ {1} $ the product of the elements $ f ( g) $ and $ f ( h) $ is also defined in $ G _ {2} $ and $ f ( g h ) = f ( g) f ( h) $. Two local homomorphisms of $ G _ {1} $ into $ G _ {2} $ are said to be equivalent if they coincide in a neighbourhood of the identity of $ G _ {1} $. Suppose that the local homomorphism $ f $ is a homeomorphism of the neighbourhoods $ U _ {1} $ and $ U _ {2} $ and that the inverse mapping $ f ^ { - 1 } : U _ {2} \rightarrow U _ {1} $ is a local homomorphism of $ G _ {2} $ to $ G _ {1} $. Then $ f $ is called a local isomorphism of $ G _ {1} $ and $ G _ {2} $. 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 $ ( G , e , \Theta , \Omega , i , m ) $ be a local topological group and let $ H $ be a subset of $ G $ containing $ e $ such that in a neighbourhood $ U $ of $ e $ in $ G $ the set $ U \cap H $ is closed. Suppose also that for any $ g \in H \cap \Theta $ the element $ i ( g) $ belongs to $ H $ and that the set

$$ \Omega _ {H} = \ \{ {( g , h ) \in \Omega \cap ( H \times H ) } : { m ( ( g , h ) ) \in H } \} $$

is open in $ H \times H $( under the assumption that $ H $ is endowed with the topology induced from $ G $). Then the system

$$ ( H , e , \Theta \cap H , \Omega _ {H} , i \mid _ {H} ,\ m \mid _ {\Omega _ {H} } ) $$

is a local topological group, called a local subgroup of $ G $. 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)