Unions of Parafree Lie Algebras

We consider unions of parafree Lie algebras and we prove that such unions are again parafree under some conditions. 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 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. 2. Preliminaries 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