Parafree lie algebras

Let P be a Lie algebra over a ring k of characteristic zero and X be a non-empty set. The Lie algebra P is called parafree over a set X, if P is residually nilpotent and has the same lower central sequence as a free Lie algebra genereted by the set X. The cardinality of X is called the rank of P.In this thesis we investigate subalgebras and ideals of parafree Lie algebras. Furthermore we show that the direct sum of two parafree Lie algebras is again parafree. Also we prove that free factors of a parafree Lie algebra are parafree.By using direct limit of algebras, we show that the union of two generated free Lie algebras is parafree of two rank and based on this result, we obtain several properites. We consider the ascending series of parafree Lie algebras of finite rank r and we prove that the union of this series is parafree but not free. Moreover we investigate some examples of nonfree parafree Lie algebras and we give an example of free parafree Lie algebra.