Download Topological loops and their multiplication groups

Topological loops and their
multiplication groups
Ágota Figula
A set L with a binary operation (x, y) 7→ x ∗ y
is called a loop if there exists an element e ∈ L
such that
x=e∗x=x∗e
holds for all x ∈ L and the equations
a ∗ y = b and x ∗ a = b
have precisely one solution which we denote by
y = a\b and x = b/a.
The left and right multiplications
λa = y 7→ a ∗ y : L → L,
ρa : y 7→ y ∗ a : L → L,
a ∈ L, are permutations of the loop L.
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The group M ult(L) = hλa, ρa; a ∈ Li is called
the multiplication group of L. The stabilizer
of the identity element e ∈ L in M ult(L) is
denoted by Inn(L) and we say that Inn(L) is
the inner mapping group of L.
The kernel of a homomorphism α : (L, ·) →
(L0, ∗) of a loop L into a loop L0 is a normal
subloop N of L.
The centre Z(L) of a loop L consists of all elements z which satisfy the equations
zx · y = z · xy, x · yz = xy · z,
xz · y = x · zy, zx = xz
for all x, y ∈ L.
If we put Z0 = e, Z1 = Z(L) and
Zi/Zi−1 = Z(L/Zi−1), then we obtain a series
of normal subloops of L. If Zn−1 is a proper
subloop of L but Zn = L, then L is centrally
nilpotent of class n.
The multiplication groups of centrally nilpotent loops are solvable. If the multiplication
group of a loop L is nilpotent, then the loop
L is centrally nilpotent.
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Important correspondences between normal
subloops of L and certain normal subgroups of
M ult(L) are the following:
Let L be a loop with multiplication group M ult(L)
and identity element e.
(i) Let α be a homomorphism of the loop L
onto the loop α(L) with kernel N . Then N is
a normal subloop of L and α induces a homomorphism of the group M ult(L) onto the group
M ult(α(L)).
Denote by M (N ) the set {m ∈ M ult(L); xN =
m(x)N for all x ∈ L}. Then M (N ) is a normal subgroup of M ult(L) containing the multiplication group M ult(N ) of the loop N , the
multiplication group of the factor loop L/N is
isomorphic to M ult(L)/M (N ) and the inner
mapping group of the factor loop L/N is isomorphic to Inn(L)/(M (N ) ∩ Inn(L)).
(ii) For every normal subgroup N of M ult(L)
the orbit N (e) is a normal subloop of L. Moreover, N ≤ M (N (e)).
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Theorem 1. A group K is isomorphic to
the multiplication group of a loop if and only
if there exists a subgroup S containing no
non-trivial normal subgroup of K and two
transversals A and B to S in the group K
satisfying a−1b−1ab ∈ S for every a ∈ A and
b ∈ B and K = hA, Bi.
The theory of topological loops L is the theory
of the continuous binary operations (x, y) 7→
x ∗ y, (x, y) 7→ x/y, (x, y) 7→ x\y on the
topological manifold L.
Connected topological loops having a Lie group
G as the group generated by their left multiplications can be treated as sharply transitive
sections σ : G/H → G, where H is the stabilizer of the identity element e ∈ L in the group
G.
Proposition 2. There exists no proper 1dimensional topological loop L having a Lie
group as the multiplication group M ult(L) of
L.
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We call a simply connected Lie group of dimension n an elementary filiform Lie group Fn if
its Lie algebra has a basis {e1, · · · , en}, n ≥ 3,
such that [e1, ei] = ei+1 for 2 ≤ i ≤ n − 1 and
all other products are zero.
Theorem 3. Let L be a 2-dimensional proper
connected topological loop. The group M ult(L)
topologically generated by all multiplications
of L is a Lie group if and only if M ult(L)
is an elementary filiform Lie group Fn with
n ≥ 4 and the group G topologically generated by the left multiplications of L is an
elementary filiform Lie group Fm, where 3 ≤
m ≤ n. Moreover, the inner mapping group
Inn(L) corresponds to the abelian subalgebra
he2, e3, · · · , en−1i.
A 2-dimensional simply connected loop LF is
called an elementary filiform loop if its multiplication group is an elementary filiform group
Fn, n ≥ 4.
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Every solvable Lie group has a one- or twodimensional connected normal subgroup K. If
the multiplication group M ult(L) of a threedimensional proper topological loop L is solvable, then the orbit K(e) is a connected normal
subloop of L. Since the inner mapping group
Inn(L) does not contain any non-trivial normal
subgroup of M ult(L) the subloop K(e) has dimension 1 or 2.
Proposition 4. Let L be a 3-dimensional
proper connected simply connected topological loop such that its multiplication group
M ult(L) is a solvable Lie group. We assume
that L has a one-dimensional connected normal subloop N .
(a) If the factor loop L/N is isomorphic to
R2, then N ∼
= R is a central subgroup of
L. The multiplication group M ult(L) is a
semidirect product of the abelian group M ∼
=
Rm, for some m ≥ 2, by a group Q ∼
= R2
such that M = Z × Inn(L), where Z ∼
= N is
contained in the centre of M ult(L).
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(b) The loop L is an extension of the Lie
group R by the non-abelian two-dimensional
Lie group L2 and the multiplication group
M ult(L) has a normal subgroup S such that
the factor group M ult(L)/S is isomorphic to
L2 × L 2 .
(c) The loop L is an extension of the Lie
group R by a two-dimensional elementary filiform loop LF and the multiplication group
M ult(L) contains a normal subgroup S such
that the factor group M ult(L)/S is isomorphic to an elementary filiform Lie group Fn,
n ≥ 4.
To determine which at most 5-dimensional solvable Lie groups K can occur as the multiplication group of 3-dimensional topological loops we
proceed in the following way.
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1) We consider the 1-dimensional normal subgroups N of K and investigate the structure of
the factor groups K/N . Every 3-dimensional
proper simply connected loop L having an at
most 5-dimensional solvable Lie group K as the
multiplication group of L is a central extension
of the group R by the group R2.
2) We determine all abelian normal subgroup
M of codimension 2 of K. As the factor group
K/M is abelian, M contains the commutator
subgroup of K.
3) We determine the 2-dimensional subgroups
S of M which does not contain any non-trivial
normal subgroup of K.
4) We select those subgroups S of K which
satisfy the property the normalizer NK (S) =
Z(K) × S, where Z(K) is the centre of the
group K. These groups S can occur as the inner mapping group of L.
5) For the remaining pairs (K, S) we have to
find transversals A and B to S in K such that
a−1b−1ab ∈ S for every a ∈ A and b ∈ B and
K = hA, Bi.
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For 4-dimensional solvable Lie groups the assertion in the first step follows immediately since
there does not exist any proper factor group of
K which is isomorphic either to L2 × L2 or
to Fn, n ≥ 4. Each 5-dimensional solvable
Lie group has a normal subgroup N such that
the factor group K/N is neither isomorphic to
L2 × L2 nor isomorphic to Fn, n ≥ 4. Hence
the factor loop L/N (e) must be isomorphic to
R2 and we get case (a) in Proposition 4.
The second step excludes for instance solvable
Lie groups having non-abelian commutator subgroup. As dim K ≤ 5 and dim M = dim K −2
solvable Lie groups having 4-dimensional commutator subgroup are also excluded.
The fourth step excludes for instance the 5dimensional solvable Lie groups having abelian
nilradical of dimension 4.
After these steps among the 39 classes of 5dimensional indecoposable solvable Lie algebras
it remains to investigate only 4 classes.
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Using the fifth step we can prove that the 4dimensional solvable Lie groups as well as the 5dimensional indecomposable Lie groups are not
multiplication groups of 3-dimensional topological loops.
Theorem 5. Each 3-dimensional connected
topological loop L such that its multiplication
group is an at most 5-dimensional solvable
Lie group is centrally nilpotent of class 2.
The multiplication group M ult(L) of simply
connected topological loop L is isomorphic to
one of the following Lie groups:
(a) The direct product of the Lie group R
and the 4-dimensional Lie group which can
be represented as the group of matrices


 w

w
e −xe y z






 0 ew 0 y 
 ; x, y, z, w ∈ R .
g(x, y, z, w) = 

0
0 1 0






0
0 0 1
(b) The direct product of the 3-dimensional
non-abelian nilpotent Lie group F3 and the
group L2.
(c) The Lie group L2 × L2 × R.
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(d) The direct product of the 3-dimensional
solvable Lie groups which have only discrete
centre and the Lie group R2.
(e) The nilpotent Lie group F4 × R.
If L has 2-dimensional centre, then we have
proved that the groups R × Fn, n ≥ 4, where
Fn is the n-dimensional elementary filiform Lie
group, are multiplication groups of L.
Also the direct products Fn ×Z Fm of the elementary filiform Lie groups Fn, Fm with amalgamated center Z such that (n, m) ∈ N\{(3, 3)}
are multiplication groups of 3-dimensional topological loops L. These loops L have 1-dimensional
centre Z(L) such that L/Z(L) is the abelian
group R2.
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A transitive action of a Lie group G on a manifold M is called minimal, if it is locally effective and if G does not contain subgroups acting
transitively on M . The minimal actions of nonsolvable Lie groups on 3-dimensional manifolds
are classified.
Theorem 6. There exists no 3-dimensional
proper connected topological loop L such that
its multiplication group M ult(L) acts minimally on L and M ult(L) is a non-solvable
Lie group.
A connected simply connected topological loop
L with dimension 3 such that M ult(L) is a Lie
group is homeomorphic either to the 3-sphere
S 3 or to the affine space R3.
Proposition 7. Let L be a 3-dimensional
quasi-simple connected simply connected topological loop such that the multiplication group
M ult(L) of L is a Lie group.
(a) If L is homeomorphic to S 3, then the
group M ult(L) is either quasi-simple or isomorphic to the semidirect product Spin3(R)o
SO3(R).
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(b) If L is homeomorphic to R3 and the group
M ult(L) is not quasi-simple, then one of the
following holds:
(i) If M ult(L) is semi-simple, then it is iso^
morphic to the semidirect product P SL
2 (R)o
P SL2(R).
(ii) If M ult(L) is not semi-simple, then it
is the semidirect product R3 o S, where S is
isomorphic either to Spin3(R) or to SL3(R)
respectively to P SL2(R) and acts irreducibly
on R3.
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