ABSTRACT
In this chapter, the authors aim to study complex non Lie filiform Leibniz algebras. They provide some equivalent conditions for Leibniz algebra to be filiform and describe naturally graded complex filiform Leibniz algebras. The authors provide classification of solvable Leibniz algebras with given nilradicals. They describes the irreducible components of the class of nilpotent Leibniz algebras containing algebra of maximal nilindex. The authors clarify the situation when the nilradical is represented as a direct sum of its two null-filiform ideals and the complementary space to the nilradical is one-dimensional. In fact, when the nilradical of a solvable Leibniz algebra is a direct sum of s copies of null-filiform ideals, the complementary vector space has lesser dimension.
