Unions of Parafree Lie Algebras

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 someworks 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.


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][3][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.
be a properly ascending series of parafree Lie algebras of the same finite rank .Then, (i)  is residually nilpotent; (ii)  is residually finite; (iii)  has the same lower central sequence as some free Lie algebra; (iv)  is not free.

Preliminaries
Let  be a field and  be a Lie algebra over .By   (), we denote the th term of the lower central series of .Let   ∈   and   be the equivalence class of   and let  be the set of equivalence classes.A short calculation shows that  is a Lie algebra.
It is not difficult to see that  has the same mapping property as does the direct limit.So we can obtain that the direct limit exists and is equal to .Hence, direct limit of the Lie algebras   may be viewed as the union of these algebras.That is, lim For a background of the theory of systems of objects and their limits, we refer to [7].
Let  be a nonempty set and let () be the free Lie algebra freely generated by  over .Let  be a Lie algebra and let  be any subset of .By ⟨⟩, we denote the subalgebra of  generated by the set .Throughout this work, all Lie algebras will be considered over a fixed field .
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 if and only if ,  have the same cardinality.
It is easy to see that the following theorem is an immediate consequence of Theorem 9 of the Section 4.2.4 of [8].
Theorem 9. Let  = ()/  (()) and let  be a subset of  which is linearly independent modulo the derived algebra  2 ().Then, Using Theorem 8 and Lemma 2.10.1 of [9], we obtain the following well-known result.

Parafree Unions
be a properly ascending series of parafree Lie algebras of the same finite rank .Then, (i)  is residually nilpotent; (ii)  is residually finite; (iii)  has the same lower central sequence as some free Lie algebra; (iv)  is not free.
Proof.(i) For  ≥ 1, let   be a paragenerating set for   .Then,   freely generates modulo  2 (  ), a free abelian Lie algebra.Since all the algebras   have the same finite rank  and for  ≥ ,   ⊆   , then   freely generates modulo  2 (  ), a free abelian Lie algebra.Therefore, by Lemma 10   freely generates modulo   (  ), a free nilpotent Lie algebra for every  = 2, 3, . ... Clearly, the set   is linearly independent modulo  2 (  ).By Lemma 10, the subalgebra ⟨  ⟩ of   is isomorphic to   /  (  ).Hence, we obtain Thus, By (7) and the second isomorphism theorem, it follows that Since the algebras   are parafree Lie algebras of rank , then there exist free Lie algebras   of rank  such that Using ( 8) and ( 11), we obtain It is clear that   (  ) ⊆   ∩   (  ).By hopficity of   /  (  ), it follows that Now, let   :   →   be the inclusion map for all ,  such that  ≤ .Then, the system of Lie algebras   and the homomorphisms   provide us with a direct system.Let  be the direct limit of this system. may be viewed as the union of its subalgebras   : By definition of the direct limit,  is a Lie algebra and   () = ⋃ ∞ =1   (  ).
(ii) Since   is parafree, there exists a free Lie algebra   of rank  such that where  ≥ 2,  ≥ 1.It is well known that free nilpotent Lie algebras of finite rank are finite-dimensional algebras.Thus, by (17), we have that   /  (  ) is a finite-dimensional algebra.It follows immediately that   is residually finite.Now, we are going to prove the equality where  ≥ 1. Clearly   +   () ⊆ .Now, suppose that  ∈ .
(iii) From the proof of (ii) (using (17) and ( 22)), we have where   is a free Lie algebra of rank ,  ≥ 2. Thus,  has the same lower central sequence with the free Lie algebra   .So if we combine this result with (i), we observe that  is parafree of rank .
(iv) Since  is the direct limit of the parafree Lie algebras   , it is not finitely generated.Hence,  is not free since it is not finitely generated, but / 2 () is.
The following corollary is an immediate consequence of Theorem 11.
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 integer  ≥ 1, let   be a free Lie algebra of rank two.Then, the union ⋃ ∞ =1   of   is a parafree Lie algebra of rank two.Moreover, ⋃ ∞ =1   is not free.
Proof.For  ≥ 1, consider the free Lie algebra   generated by the free generating set   = {  , V  }, where  ≥ 1.Let  +1 be the map defined by More generally, if  is a property or a class of Lie algebras, then  is called residually  if, given any nontrivial element  ∈ , there exists an ideal  of  such that  ∉  with / ∈ .We say that two Lie algebras  and  have the same lower central sequence if /  () ≅ /  () for every  ≥ 1.A system ({  } ∈ , {  } ≤ ) of Lie algebras   is called a directed system, if (i)  is a directed set.(ii) If , ,  ∈ ,  ≤  ≤ , then   :   →   is a Lie algebra homomorphism such that   =     Definition 6.The direct limit of a directed system ({  } ∈ , {  } ≤ ) of Lie algebras is a pair (, {  } ∈ ), where  is a Lie algebra and each   :   →  is a Lie algebra homomorphism such that One defines a relation ∼ on  by   ∼   ,   ∈   ,   ∈   , if there exists  ∈ , ,  ≤  such that   (  ) =   (  ).