The Group of Extensions of a Topological Local Group

Full text

World Applied Sciences Journal 17 (12): 1588-1591, 2012
ISSN 1818-4952
© IDOSI Publications, 2012
Corresponding Author: H. Sahleh, Faculty of Mathematical Sciences, University of Guilan, P.O.Box 1914, Rasht, Iran
1588
Extending Topological Local Groups to Topological Groups
H. Sahleh and A. Hosseini
Faculty of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht, Iran
Abstract: The aim of this paper is to extend a topological local group X to a group G.Then with the
translation topology G will be a topological group.
Key words: Local group • topological local group • translation topology • enlargeable groups • monodrome
INTRODUCTION
The studies for lie groups; a special case of local
lie groups, goes back to 1936's. For an expository
article see [1]. Elie Cartan showed that every local lie
group contains a neighborhood of identity which is
homeomorphic to a neighborhood of the identity of a lie
group [1, 8] and Pontryagin spotted that a local lie
group is basis for a lie group [8]. The question then
arose as to whether every local lie group is contained in
a lie group. Olver showed that if a local lie group has
the associative law then it embeds into a lie group [7].
Topological local groups (or local topological
groups) are local lie groups without manifold property,
which means that the group multiplication and
inversion operations only being defined for elements
sufficiently near the identity.
Sharma showed that any topology which makes the
center of a group G a topological group can be extended
in such a way that G will be a topological group
[10]. In this paper we show that under what conditions
a topological local group in center of G
can be
extended to a topological group G. The idea is
motivated by [2, 12].
In section 1, we give definitions and results which
will be needed in the sequel. In the section2, the
monodrome of a topological local group is defined. In
section 3, we extend a topological local group to a
topological group.
PRELIMINARY
In this section we give definitions and results
which will be needed in the sequel.
Definition 1.1: If X is a set,
(n)n
DX
⊂ is subset of the
cartesian product X
n of n copies of X and
(n)(n)
f:DX
→, (n)
1n12
f(x,...,x)=x...x
, then ƒ(n) will be
called an n-array local operation on X. Denote ƒ(2) (x,y)
by xy.
Definition 1.2: A triple
(2)(2)
(X,f,D)
is a local group if
X is a set and a subset (2)
DXX
⊂×
and (2)(2)
f:DX
→a
binary local operation such that:
If xy and yz exist then either both (xy)z and x(yz)
exist and (xy)z = x(yz) or both (xy)z and x(yz) do
not exist;
There exists an element e∈D such that ex and xe
exist for every x∈D and xe = ex; for every x∈D(1) there
exist an unique x
-1∈D(1) such that xx-1 and x-1x exist
and xx-1 = x
-1x; If xy exists then y-1x-1 exists and
(xy)-1 = y-1x-1.
Definition 1.3: Let X be a local group, we call X an
n-assocative if
1. Local operation ƒ(n) is defined for every k<n.
2. There exists (k)(l)
1kk1n
f(x,...,x)f(x,...,x)
+ for every
k,l<n such that k+l = n and
(k)(l)(n)
1kk1n1n
f(x,...,x)f(x,...,x)=f(x,...,x)
+
Definition 1.4: We call X the global associative if:
1. Conditions in Definition 1.3 for every n hold ;
2. For all local operation
(n)
1
f
,
(n)
2
f
and for every
n-tuple
(n)n
12n
(x,x,...,x)DX
∈⊂
(n)(n)
1n1n
12
f(x,...,x)=f(x,...,x)
Definition 1.5: A n-array local opration in a local group
X is called a word if it is an n-assocative.
Definition 1.6: A topological local group is a four-tuple
(2)(2)
(X,f,D,J)
, where
(2)(2)
(X,f,D)
is a local group and
J is a Hausdorff topology on X and D(2) an open subset
of X×X such that the maps X→X,
1
xx
−
a
and
(2)(2)
f:DX
→, (x, y)→xy are continuous.

World Appl. Sci. J., 17 (12): 1588-1591, 2012
1589
Remark 1: Let X be a local group and ℜ be a
collection of sublocal groups such that
1. With any pair 12
V,V
∈ℜ
, there exists a V3∈ℜ with
312
VVV
⊂∩.
2. For any a∈X and V∈ℜ there is a W∈ℜ, such that
aW, Wa-1 are defined and aWa-1⊂V.
3. With any V∈ℜ there is a W∈ℜ such that W2⊂V.
There is a unique topology in X in which ℜ is a
neighborhood base at e.
Then, X is a topological local group [13].
A continuous map of topological local groups
φ:X→X′, will be called a homomorphism of topological
local group, if x,y∈X and xy∈X then φ(x)φ(y) exists in
X′ and φ(xy) = φ(x)φ(y).
With these morphisms the topological local groups
form a category which contains the subcategory of
topological groups.
A homomorphism φ:X→X′ will be called strong if,
for every x,y∈X, the existence of φ(x)φ(y) implies the
existence of x,y∈X. A morphism will be called a
monomorphism (epimorphism) if, it is injective
(surjective).
A subset H of a local group will be called sublocal
group (symmetric subset) if it contains the identity and
also if x∈H then x-1∈H.
THE MONODROME TOPOLOGICAL GROUP
OF A TOPOLOGICAL LOCAL GROUP
In [11] enlargement of a local group X and
monodrome were introduced. Now we define them in
the topological context.
Let X be a topological local group. Suppose U1 and
U2 are open neighborhoods of e in X and 11
f:UX
′
→
and 22
f:UX
′
→ are both morphisms. We say that ƒ1
and ƒ2 are equivalent if there exists an open
neighborhood U3 of e in X such that
312
UUU
⊆∩ and
1U2U
33
f|=f|
.
Definition 2.1: We say that a topological local group X
is enlargeable if there exists a topological group G and
a morphism φ:X→G such that φ:X→φ(X) is a
homeomorphism related to the equivalent class.
Definition 2.2: Let X be a topological local group and
G a topological group and X⊂G. Then G is called an
X-monodrome if
1. X generates G
topologically (i.e: X is the smallest
closed sublocal group in G which generate G)
2. For a topological group H and every continuous
homomorphism ψ:X→H there exists a continuous
homomorphism ν:G→H such that the following
diagram commutes.
(2.1)
Lemma 2.3: (Uniqueness) Let G be a topological group
which is an X-monodrome with embedding φ. Suppose
H is a topological group, ψ:X→H a continuous
homomorphism, H is generated by ψ (X) and ν′:H→G
is a continuous homomorphism. If the following
diagram commutes
(2.2)
then ν′ is a homeomorphism and H is an X-monodrome
with embedding ψ.
Proof: Combining the diagrams (2.1) and (2.2), we
obtain
Since
(((x)))=(x)
′
ννψψ for every x∈X and H is
topologically generated by ψ(X), then
H
=Id
′νν
o
In
fact, ν′:H→G is surjective, because
(X)=(X)
′
νψφ
o
generates G. Hence, ν′ is a homeomorphism with the
inverse ν.
Definition 2.4: Let F be a free group on a local group
X. Then, u∈F is called an e-element if
12
n
n
12
u=xx...x
εε
ε
,where εi∈{1,-1} and 12n
x,x,...,xX
∈
such that (n) 12
n
n12
f(x,x,...,x)
εε ε is an n-associative and is
equaled to the identity e in X.
A topological local group X can be embedded in
the factor group F over a normal subgroup N which
contains the e-elements.
Proposition 2.5: Let X be a local group with the global
associative property and F the free group on X. Let
N⊂F be the set of all e-elements of F. Then N is a
normal subgroup of F and if
F
:F
N
φ→ denotes the
natural homomorphism, then the restriction of φ to X is

World Appl. Sci. J., 17 (12): 1588-1591, 2012
1590
injective and F/N is a monodrome for X with
embedding
F
:X
N
φ→.
Proof: Let X be a local group with the global
associative property. By Malcev [6], X can be extended
to a group G. Now by [12, Lemma 3.2], it can be
embedded in F/N.
In the next part, we will extend the topology on the
factor group F/N which changes it to a topological
group.
THE EXTENDING TOPOLOGY
Sharma in [10] has shown that any topology which
makes the center of a group G a topological group can
be extended in such a way that G will be a topological
group. Let G be a group and X a proper sublocal group
of G. If X is a topological local group, a natural
question to ask is when can the topology on X be
extended to a topology on G that makes G a topological
group?
When X is a topological group and a subgroup of a
group G, the question was answered by Clark [2].
In this section we only assume that G is a group
and X a Huasdorff topological sublocal group of G with
the topology τ.
Let U = {Uα} be a family of sublocal groups
of X and also a base for the topology of X at the
identity element e∈X (see Remark1). The collection
L
U={gU|UUandgG}
αα
∈∈ is called the left
translation base and τL the topology induced by UL, the
left translation topology. Similarly we define τR the
right translation topology.
Theorem 3.1: [2] The group G with a translation
topology is a topological group if and only if τL = τR.
Definition 3.2: [8] A topological sublocal group of
topological local group X is a set X′⊆X containing e for
which there exists a symmetric open neighborhood V of
the identity e such that
1. X′⊆V and X′ is closed in V;
2. If x∈ X′ and x-1∈V, then x-1∈ X′;
3. If (x,y)∈D(2) and xy∈V, then xy∈X′.
We call X′ a topological sublocal group of X with
the associated neighborhood V.
Definition 3.3: A topological normal sublocal group
of X is a topological sublocal group X′ of X with
the associated neighborhood V and if y∈V and
x∈X′ are such that yxy -1 is defined and yxy-1∈V,
then yxy-1∈X′.
Theorem 3.4: Let X be an open normal topological
sublocal group in a group G. Then G is a topological
group with the translation topology if and only
if g
f:XX
→,
1
xgxg
−
a
is a homeomorphism of
topological local groups for g∈G, gxg-1∈X.
Proof: It is known that if G is a topological group then
ƒg is a homeomorphism.
Conversely, suppose that g
f:XX
→ is a
homeomorphism for g∈G. Let a,b∈G and W
1 be an
open neighborhood of e∈X. Then abW1 is a basic open
set of ab∈G. So W2 = W1∩X is an open subset of X.
Let
1
322
W=WW
−
∪ be a symmetric open set in X and
abW3 an open set of ab in G. By Remark1, there exists
an open neighborhood K of e such that 2
3
KW
⊂. By
[3], for every b∈X there is an open set U1⊂X, e∈U1
and an open set 1
WX
′
⊂
,
1
bW
′
∈ such that
11
bUW
′
⊂,
1
11
bWU
−′⊂. For 1`
bX
−
∈
, there is an open set U2⊂X,
e⊂U2 and an open set 2
WX
′⊂, 1
2
bW
−
′
∈ such that
1
22
UbW
−
′
⊂,
22
WbU
′⊂. Let
412
U=UU
∩. Now
consider
412
W=WW
′′
∩, then
44
bUW
⊂, 1
44
UbW
−⊂.
Let
543435
U=UWWW=W
∩⊂∩ . Then
55
bUW
⊂ and likewise 1
55
UbW
−⊂. Now we consider
5
U=KU
∩. Then
53
bUWW
⊂⊂, 1
3
UbW
−⊂ and
2
3
UW
⊂.
We have bxb-1∈X for x∈U, since X is a normal
sublocal group. Then bUb-1 exists. Let T = bUb-1. Since
ƒg is a homeomorphism then T is an open neighborhood
of e in G. Thus by Theorem3.1,
12
3
(aT)(bU)=abbTbU=abUabW
−⊂
Therefore, multiplication is continuous. A similar
argument shows that the inverse operation is also
continuous.
The following corollary is an immediate
consequence of Proposition 2.5 and Theorem 3.4.
Corollary 3.5: Every Hausdorff topological local group
X with the global associative property embeds in a
group G. If the map
1
g
f:xgxg
−
a is a homeomorphism
of topological local groups, then G is a topological
group.

World Appl. Sci. J., 17 (12): 1588-1591, 2012
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CONCLUSION
The most common version of Hilbert’s fifth
problem asks whether every locally Euclidean
topological group is a Lie group.
Goldbring in [3] solved this problem for local Lie
groups, by methods from nonstandard analysis.
Now we define Topological local groups without
locally Euclidean property. In topological local groups,
if the products x⋅y; y⋅z; x⋅(y⋅z) and (x⋅y)⋅z are all
defined, then x⋅(y⋅z) = (x⋅y)⋅z. This condition is
called”local associativity” [7]. A much stronger
condition for a local group is ”global associativity” in
which, given any finite sequence of elements from the
topological local group and two different ways of
introducing parentheses in the sequence, if both
products thus formed exist, then these two products are
in fact equal.
Let X be a topological local group. We can answer
the following question: When can the topology on X be
extended to a topology on G that makes it a topological
group. A topological local group can be the basis that
generates transformations with a natural interpretation
for physical applications [4, 9]. Kloss. B.M, in [5]
defined a regular completely additive measure µ on the
class of all Borel subsets of bicompact topological
group G with µ(G) = 1. Now we can define this regular
completely additive measure on a topological local
group G and find the same results.
REFERENCES
1. Cartan, E., 1936. La topologie de groupes de Lie.
Exposes de Geometrie. No.8. Hermann, Paris.
2. Clark, B., M. Dooley and V. Schneider, 1985.
Extending topologies from subgroups to groups.
Topolgy Proceedings, 10: 251-257.
3. Goldbring, I., 2010. Hilbert's fifth problem for
local groups. Annales. Math., 172 (2): 1269-1314.
4. Johnson, J.E., 1985. Markov-type Lie groups in
GL(n, R). J. Math. Phys., 26: 252-258.
5. Kloss, B.M., 1959. Probability Distributions on
Bicompact Topological Groups. Theory Probab.
Appl., 4: 237-270.
6. Malcev, A.I., 1941. Sur les groupes topologiques
locaux et comlets. Acad. Sci. URSS (N.S), 32:
606-608.
7. Olver, P.J., 1996. Non-Associative local Lie
groups. Joural of Lie Theory, 6: 23-51.
8. Pontryagin, S., 1966. Topological groups, 2nd Edn.
Gordon and Breach, Science Publisher.
9. Qinxiu Sun, 2012. Generalization of H-
pseudoalgebraic structures. J. Math. Phys., 53:
012105.
10. Sharma, P.L., 1981. Hausdorff topologies on
groups I. Math. Japonica, 26: 555-556.
11. Swierczkowski, S., 1967. Cohomology of Local
Group Extensions. Trans. Amer. Math. Soc.,
128 (2): 291-320.
12. Swierczkowski, S., 1965. Embedding Theorems for
Local Analytic Groups. Acta. Math., 114: 207-235.
13. Van Est, W.T., 1962. Local and global groups.
Indag. Math., 24: 391-425.