# Unions of Parafree Lie Algebras

Authors: Naime Ekici; Zehra Velioğlu
Journal: Algebra
 (2014)

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## Abstract

We consider unions of parafree Lie algebras and we prove that such unions are again parafree under some conditions.


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## Body

## 1. Introduction

The study of Lie algebras which share many properties with a free Lie algebra has begun with Baur [1]. This extraordinary class of Lie algebras is called the class of parafree Lie algebras. These Lie algebras arose from Baumslag’s works about parafree groups. In [2–4], Baumslag has introduced parafree groups and he obtained some interesting results about these groups. In his doctoral dissertation [1], Baur has defined parafree Lie algebras as in the group case and he proved the existence of a nonfree parafree Lie algebra [5]. Baumslag’s and Baur’s works have given a start for studies in the theory of parafree Lie algebras. Although there are some works about parafree Lie algebras in the literature, many questions about them have remained unanswered. This fact and paucity of studies about parafree Lie algebras motivated us for this work.The objective of this work is to investigate the ascending unions of parafree Lie algebras. Our main theorem is as follows.Theorem 1.
Let
L
1
⊂
L
2
⊂
⋯
⊂
L
=
⋃
n
=
1
∞
L
n be a properly ascending series of parafree Lie algebras of the same finite rank r. Then, (i)
L is residually nilpotent;
(ii)
L is residually finite;
(iii)
L has the same lower central sequence as some free Lie algebra;
(iv)
L is not free.


## 2. Preliminaries

LetK be a field and L be a Lie algebra over K. By γ
n
(
L
), we denote the nth term of the lower central series of L. If ⋂
n
=
1
∞
γ
n
(
L
)
=
{
0
}, then L is called residually nilpotent; equivalently, given any 0
≠
u
∈
L, there exists an ideal J of L such that u
∉
J with L
/
J being nilpotent. More generally, if P is a property or a class of Lie algebras, then L is called residually P if, given any nontrivial element u
∈
L, there exists an ideal J of L such that u
∉
J with L
/
J
∈
P.We say that two Lie algebrasL and G have the same lower central sequence if L
/
γ
n
(
L
)
≅
G
/
γ
n
(
G
) for every n
≥
1.Definition 2.
LetL be a Lie algebra. L is called a parafree Lie algebra over a set X if(i)
L is residually nilpotent;
(ii)
L has the same lower central sequence as a free Lie algebra generated by the set X.The cardinality ofX is called the rank of L.Definition 3.
LetL be a parafree Lie algebra and let Y be a subset of L. Y is called a paragenerating set if it freely generates L modulo γ
2
(
L
).We will use some functorial properties of Lie algebras.Definition 4.
A directed setI is a partially ordered set such that, for each pair i
,
j
∈
I, there exists a k
∈
I for which i
≤
k and j
≤
k.Definition 5.
A system(
{
L
i
}
i
∈
I
,
{
φ
i
j
}
i
≤
j
) of Lie algebras L
i is called a directed system, if(i)
I is a directed set.
(ii)
Ifi
,
j
,
k
∈
I, i
≤
j
≤
k, then φ
i
j
:
L
i
→
L
j is a Lie algebra homomorphism such that
(1)
φ
i
k
=
φ
j
k
φ
i
j
and for eachi
∈
I
(2)
φ
i
i
=
i
d
L
i
.Definition 6.
The direct limit of a directed system(
{
L
i
}
i
∈
I
,
{
φ
i
j
}
i
≤
j
) of Lie algebras is a pair (
D
,
{
ϕ
α
}
α
∈
I
), where D is a Lie algebra and each ϕ
α
:
L
α
→
D is a Lie algebra homomorphism such that(i)
ϕ
β
φ
α
β
=
ϕ
α, for α
≤
β
,
α
,
β
∈
I;
(ii)
if there is a Lie algebraM together with maps π
i
:
L
i
→
M such that π
j
φ
i
j
=
π
i, for each i
≤
j, then there exists a unique Lie algebra homomorphism π
:
D
→
M such that π
ϕ
i
=
π
i.
The direct limit, if it exists, is unique up to isomorphism. We denote it bylim
⁡
⟶
⁡
L
i.Let(
{
L
i
}
i
∈
I
,
{
φ
i
j
}
i
≤
j
) be a directed system of Lie algebras and P
=
⋃
i
=
1
∞
L
i. Now, we will construct the direct limit lim
⁡
⟶
⁡
L
i [6].Definition 7.
One defines a relation~ on P by u
i
~
u
j
,
u
i
∈
L
i
,
u
j
∈
L
j, if there exists k
∈
I
,
i
,
j
≤
k such that φ
i
k
(
u
i
)
=
φ
j
k
(
u
j
).
Letu
i
∈
L
i and u
i
¯ be the equivalence class of u
i and let L be the set of equivalence classes. A short calculation shows that L is a Lie algebra.
It is not difficult to see thatL has the same mapping property as does the direct limit. So we can obtain that the direct limit exists and is equal to L. Hence, direct limit of the Lie algebras L
i may be viewed as the union of these algebras. That is,
(3)
lim
⁡
⟶
⁡
L
i
=
L
=
⋃
i
=
1
∞
L
i
.
For a background of the theory of systems of objects and their limits, we refer to [7].LetX be a nonempty set and let F
(
X
) be the free Lie algebra freely generated by X over K. Let G be a Lie algebra and let Y be any subset of G. By 〈
Y
〉, we denote the subalgebra of G generated by the set Y. Throughout this work, all Lie algebras will be considered over a fixed field K.Any relation among free generators of a free Lie algebra of a variety is an identical relation in this variety. The following proposition is an obvious corollary of Proposition 4.1.2.2 of [8].Proposition 8.
LetX
,
Y be two nonempty sets. Then,
(4)
F
(
X
)
γ
n
(
F
(
X
)
)
≅
F
(
Y
)
γ
n
(
F
(
Y
)
)
,
if and only if X
,
Y have the same cardinality.It is easy to see that the following theorem is an immediate consequence of Theorem 9 of the Section4.2
.
4 of [8].Theorem 9.
LetG
=
F
(
X
)
/
γ
n
(
F
(
X
)
) and let Y be a subset of G which is linearly independent modulo the derived algebra γ
2
(
G
). Then,
(5)
〈
Y
〉
≅
F
(
Y
)
γ
n
(
F
(
Y
)
)
.Using Theorem 8 and Lemma2.10.1 of [9], we obtain the following well-known result.Lemma 10.
Suppose that a setY freely generates F
=
F
(
X
) modulo γ
2
(
F
). Then, Y freely generates F modulo γ
n
(
F
) for n
=
2,3
,
…


## 3. Parafree Unions

Theorem 11.
LetL
1
⊂
L
2
⊂
⋯
⊂
L
=
⋃
n
=
1
∞
L
n be a properly ascending series of parafree Lie algebras of the same finite rank r. Then,(i)
L is residually nilpotent;
(ii)
L is residually finite;
(iii)
L has the same lower central sequence as some free Lie algebra;
(iv)
L is not free.Proof.
(i) Fori
≥
1, let X
i be a paragenerating set for L
i. Then, X
i freely generates modulo γ
2
(
L
i
), a free abelian Lie algebra. Since all the algebras L
i have the same finite rank r and for j
≥
i
,
L
i
⊆
L
j, then X
i freely generates modulo γ
2
(
L
j
), a free abelian Lie algebra. Therefore, by Lemma 10
X
i freely generates modulo γ
n
(
L
j
), a free nilpotent Lie algebra for every n
=
2,3
,
…. Clearly, the set X
i is linearly independent modulo γ
2
(
L
j
). By Lemma 10, the subalgebra 〈
X
i
〉 of L
j is isomorphic to L
j
/
γ
n
(
L
j
). Hence, we obtain
(6)
L
j
=
〈
X
i
〉
+
γ
n
(
L
j
)
.
Thus,
(7)
L
j
=
L
i
+
γ
n
(
L
j
)
,
where
i
≤
j
.
By (7) and the second isomorphism theorem, it follows that
(8)
L
j
γ
n
(
L
j
)
=
(
L
i
+
γ
n
(
L
j
)
)
γ
n
(
L
j
)
≅
L
i
L
i
∩
γ
n
(
L
j
)
.
Since the algebras L
j are parafree Lie algebras of rank r, then there exist free Lie algebras F
j of rank r such that
(9)
L
j
γ
n
(
L
j
)
≅
F
j
γ
n
(
F
j
)
,
j
≥
1
.
By Proposition 8, we have
(10)
F
j
γ
n
(
F
j
)
≅
F
i
γ
n
(
F
i
)
,
i
,
j
≥
1
.
Hence,
(11)
L
j
γ
n
(
L
j
)
≅
L
i
γ
n
(
L
i
)
.
Using (8) and (11), we obtain
(12)
L
i
γ
n
(
L
i
)
≅
L
i
L
i
∩
γ
n
(
L
j
)
.
It is clear that γ
n
(
L
i
)
⊆
L
i
∩
γ
n
(
L
j
). By hopficity of L
i
/
γ
n
(
L
i
), it follows that
(13)
γ
n
(
L
i
)
=
L
i
∩
γ
n
(
L
j
)
.
Now, let φ
i
j
:
L
i
→
L
j be the inclusion map for all i
,
j such that i
≤
j. Then, the system of Lie algebras L
i and the homomorphisms φ
i
j provide us with a direct system. Let P be the direct limit of this system. P may be viewed as the union of its subalgebras L
i:
(14)
P
=
⋃
i
=
1
∞
L
i
=
L
.
By definition of the direct limit, L is a Lie algebra and γ
n
(
L
)
=
⋃
i
=
1
∞
γ
n
(
L
i
).
We now computeL
i
∩
γ
n
(
L
). Consider
(15)
L
i
∩
γ
n
(
L
)
=
L
i
∩
(
⋃
k
=
1
∞
γ
n
(
L
k
)
)
=
(
L
i
∩
γ
n
(
L
1
)
)
∪
⋯
∪
(
L
i
∩
γ
n
(
L
i
)
)
∪
(
L
i
∩
γ
n
(
L
i
+
1
)
)
∪
⋯
=
γ
n
(
L
1
)
∪
⋯
∪
γ
n
(
L
i
)
∪
γ
n
(
L
i
)
∪
⋯
=
γ
n
(
L
i
)
.
Hence, we get
(16)
γ
n
(
L
i
)
=
L
i
∩
γ
n
(
L
)
.
Now, suppose that 0
≠
l
∈
L. Then, l
∈
L
i for some i. Therefore, l
∉
γ
n
(
L
i
) for some n since L
i is residually nilpotent. By (16), this implies l
∉
γ
n
(
L
). Hence, L is residually nilpotent.
(ii) SinceL
i is parafree, there exists a free Lie algebra F
i of rank r such that
(17)
L
i
γ
n
(
L
i
)
≅
F
i
γ
n
(
F
i
)
,
where n
≥
2
,
i
≥
1. It is well known that free nilpotent Lie algebras of finite rank are finite-dimensional algebras. Thus, by (17), we have that L
i
/
γ
n
(
L
i
) is a finite-dimensional algebra. It follows immediately that L
i is residually finite.
Now, we are going to prove the equality(18)
L
=
L
i
+
γ
n
(
L
)
,
where i
≥
1. Clearly L
i
+
γ
n
(
L
)
⊆
L. Now, suppose that l
∈
L. Then, l
∈
L
j for some j. By virtue of (7), l can be written as
(19)
l
=
l
1
+
l
2
,
where l
1
∈
L
i
,
l
2
∈
γ
n
(
L
j
). Since, for each j
,
n
≥
1
γ
n
(
L
j
)
⊆
γ
n
(
L
), it follows that l
2
∈
γ
n
(
L
) and, hence, L
⊆
L
i
+
γ
n
(
L
). Therefore,
(20)
L
=
L
i
+
γ
n
(
L
)
.
Using (20) and the second isomorphism theorem, we obtain
(21)
L
γ
n
(
L
)
≅
L
i
+
γ
n
(
L
)
γ
n
(
L
)
≅
L
i
L
i
∩
γ
n
(
L
)
.
Hence, by (16),
(22)
L
γ
n
(
L
)
≅
L
i
γ
n
(
L
i
)
,
i
≥
1
,
n
≥
2
.
Finite dimensionality of L
i
/
γ
n
(
L
i
) implies that L
/
γ
n
(
L
) is a finite-dimensional algebra. Now, let 0
≠
l
∈
L. Then, l
∈
L
i for some i. Therefore, l
∉
γ
n
(
L
i
) for some n. By (16), l
∉
γ
n
(
L
). Hence, L is residually finite.
(iii) From the proof of (ii) (using (17) and (22)), we have
(23)
L
γ
n
(
L
)
≅
L
i
γ
n
(
L
i
)
≅
F
i
γ
n
(
F
i
)
,
where F
i is a free Lie algebra of rank r, n
≥
2. Thus, L has the same lower central sequence with the free Lie algebra F
i. So if we combine this result with (i), we observe that L is parafree of rank r.
(iv) SinceL is the direct limit of the parafree Lie algebras L
i, it is not finitely generated. Hence, L is not free since it is not finitely generated, but L
/
γ
2
(
L
) is.The following corollary is an immediate consequence of Theorem11.Corollary 12.
Union of a properly ascending series of parafree Lie algebras of the same finite rank is a parafree Lie algebra.Corollary 13.
For any integeri
≥
1, let F
i be a free Lie algebra of rank two. Then, the union ⋃
i
=
1
∞
F
i of F
i is a parafree Lie algebra of rank two. Moreover, ⋃
i
=
1
∞
F
i is not free.Proof.
Fori
≥
1, consider the free Lie algebra F
i generated by the free generating set X
i
=
{
u
i
,
v
i
}, where i
≥
1. Let ϕ
i
i
+
1 be the map defined by
(24)
ϕ
i
i
+
1
:
u
i
⟶
[
u
i
+
1
,
v
i
+
1
]
,
v
i
⟶
[
[
u
i
+
1
,
v
i
+
1
]
,
v
i
+
1
]
,
where [
,
] is the Lie product. The subalgebra generated by the elements [
u
i
+
1
,
v
i
+
1
] and [
[
u
i
+
1
,
v
i
+
1
]
,
v
i
+
1
] of F
i
+
1 is a free Lie algebra of rank 2. Therefore, the map ϕ
i
i
+
1 can be extended to an embedding
(25)
ϕ
¯
i
i
+
1
:
F
i
⟶
F
i
+
1
.
Clearly, the subalgebra ϕ
¯
i
i
+
1
(
F
i
) is generated by
(26)
{
[
u
i
+
1
,
v
i
+
1
]
,
[
[
u
i
+
1
,
v
i
+
1
]
,
v
i
+
1
]
}
and it is a proper subalgebra of F
i
+
1. Furthermore, the algebras F
i and the monomorphisms ϕ
i
i
+
1 form a directed system. Let P be the direct limit of this system. Then, P may be viewed as the properly ascending union of the algebras F
i. That is,
(27)
P
=
⋃
i
=
1
∞
F
i
.
Hence, by Theorem 11, P is parafree. From the proof of Theorem 11 (ii), we have
(28)
P
γ
n
(
P
)
≅
F
i
γ
n
(
F
i
)
,
n
≥
2
.
This shows that the rank of P is two. Now, recall the construction of the direct limit and the relation ~. It is easy to prove that for all w
∈
L
i
w
~
ϕ
i
i
+
1
(
w
). Therefore, P
=
[
P
,
P
]. Hence, P is not free.Corollary 14.
There exists a parafree Lie algebra of rank two such that every finitely generated quotient algebra ofP can be generated by two elements.Proof.
Let the algebrasF
i be as in Corollary 13. Then, P
=
⋃
i
=
1
∞
F
i is a parafree Lie algebra of rank two. Let J be any ideal of P such that P
/
J is finitely generated. Let a
+
J
∈
P
/
J. Then, a
∈
F
i for some i. Thus a
+
J
∈
F
i
+
J
/
J for some i. So a
+
J
∈
⋃
i
=
1
∞
(
F
i
+
J
/
J
). This shows that P
/
J
⊆
⋃
i
=
1
∞
(
F
i
+
J
/
J
). Therefore,
(29)
P
J
=
⋃
i
=
1
∞
(
F
i
+
J
J
)
.
Assume that P
/
J is finitely generated. Since every finite subset of P
/
J is included in a suitable subalgebra (
F
i
+
J
)
/
J, then
(30)
P
J
=
(
F
i
+
J
)
J
for some i. Hence, P
/
J is generated by two elements.



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*Source: 385397-2014-08-13.xml*