Unions of Parafree Lie Algebras
We consider unions of parafree Lie algebras and we prove that such unions are again parafree under some conditions.
The study of Lie algebras which share many properties with a free Lie algebra has begun with Baur . 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 , Baur has defined parafree Lie algebras as in the group case and he proved the existence of a nonfree parafree Lie algebra . 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 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.
Let be a field and be a Lie algebra over . By , we denote the th term of the lower central series of . If , then is called residually nilpotent; equivalently, given any , there exists an ideal of such that with being nilpotent. 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 .
Definition 2. Let be a Lie algebra. is called a parafree Lie algebra over a set if(i) is residually nilpotent;(ii) has the same lower central sequence as a free Lie algebra generated by the set .
The cardinality of is called the rank of .
Definition 3. Let be a parafree Lie algebra and let be a subset of . is called a paragenerating set if it freely generates modulo .
We will use some functorial properties of Lie algebras.
Definition 4. A directed set is a partially ordered set such that, for each pair , there exists a for which and .
Definition 5. 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 and for each
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(i), for ;(ii)if there is a Lie algebra together with maps such that , for each , then there exists a unique Lie algebra homomorphism such that .
The direct limit, if it exists, is unique up to isomorphism. We denote it by .
Let be a directed system of Lie algebras and . Now, we will construct the direct limit .
Definition 7. One defines a relation on by , if there exists such that .
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,
For a background of the theory of systems of objects and their limits, we refer to .
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 18.104.22.168 of .
Proposition 8. Let be two nonempty sets. Then, 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 of .
Theorem 9. Let and let be a subset of which is linearly independent modulo the derived algebra . Then,
Using Theorem 8 and Lemma of , we obtain the following well-known result.
Lemma 10. Suppose that a set freely generates modulo . Then, freely generates modulo for
3. Parafree Unions
Theorem 11. Let 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 , let be a paragenerating set for . Then, freely generates modulo , a free abelian Lie algebra. Since all the algebras have the same finite rank and for , then freely generates modulo , a free abelian Lie algebra. Therefore, by Lemma 10 freely generates modulo , a free nilpotent Lie algebra for every . Clearly, the set is linearly independent modulo . By Lemma 10, the subalgebra of is isomorphic to . Hence, we obtain
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
By Proposition 8, we have
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 .
We now compute . Consider Hence, we get Now, suppose that . Then, for some . Therefore, for some since is residually nilpotent. By (16), this implies . Hence, is residually nilpotent.
(ii) Since is parafree, there exists a free Lie algebra of rank such that where . 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 . Clearly . Now, suppose that . Then, for some . By virtue of (7), can be written as where . Since, for each , it follows that and, hence, . Therefore, Using (20) and the second isomorphism theorem, we obtain Hence, by (16), Finite dimensionality of implies that is a finite-dimensional algebra. Now, let . Then, for some . Therefore, for some . By (16), . Hence, is residually finite.
(iii) From the proof of (ii) (using (17) and (22)), we have where is a free Lie algebra of rank , . 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 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 , let be a free Lie algebra of rank two. Then, the union of is a parafree Lie algebra of rank two. Moreover, is not free.
Proof. For , consider the free Lie algebra generated by the free generating set , where . Let be the map defined by where is the Lie product. The subalgebra generated by the elements and of is a free Lie algebra of rank 2. Therefore, the map can be extended to an embedding Clearly, the subalgebra is generated by and it is a proper subalgebra of . Furthermore, the algebras and the monomorphisms form a directed system. Let be the direct limit of this system. Then, may be viewed as the properly ascending union of the algebras . That is, Hence, by Theorem 11, is parafree. From the proof of Theorem 11 (ii), we have This shows that the rank of is two. Now, recall the construction of the direct limit and the relation . It is easy to prove that for all . Therefore, . Hence, is not free.
Corollary 14. There exists a parafree Lie algebra of rank two such that every finitely generated quotient algebra of can be generated by two elements.
Proof. Let the algebras be as in Corollary 13. Then, is a parafree Lie algebra of rank two. Let be any ideal of such that is finitely generated. Let . Then, for some . Thus for some . So . This shows that . Therefore, Assume that is finitely generated. Since every finite subset of is included in a suitable subalgebra , then for some . Hence, is generated by two elements.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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