Quasi Filiform Lie Algebras

A new class of nilpotent algebras called completable nilpotent Lie algebras was found during the study of complete Lie algebras. Although some completable nilpotent Lie algebras, as nilpotent Lie algebras of maximal rank and Heisenberg algebra, were found, there are many completable nilpotent Lie algebras unknown. The purpose of this thesis is to look for completable nilpotent Lie algebras and investigate their structure.The first chapter is devoted to a class of 3-step nilpotent Lie algebras called quasi L₃-filiform Lie algebras. It is shown that every Lie algebra N of this class is completable and Der N is a solvable complete Lie algebra when dim c² N = 1.In chapter 2 we consider the quasi L_n-filiform Lie algebras. We prove that quasi L_n-filiform Lie algebras are completable, explicitly determine the derivation algebra, automorphism group of quasi L_n-filiform Lie algebras, and applying some properties of root vector decomposition we obtain isomorphism theorem of quasi L_n-filiform Lie algebras.In chapter 3 we consider the quasi Q_n-filiform Lie algebras. We prove that quasi Q_n-filiform Lie algebras are completable, explicitly determine the derivation algebra, automorphism group of quasi Q_n-filiform Lie algebras, and applying some properties of root vector decomposition we obtain isomorphism theorem of quasi Q_n-filiform Lie algebras.The final chapter is devoted to the automorphism groups of solvable complete Lie algebras. We determine the automorphism groups of solvable complete Lie algebras, whose nilpotent radical is a quasi Heisenberg algebra, a quasi L_n-filiform Lie algebra, respectively.