On a Description of Irreducible Component in the Set of Nilpotent Leibniz Algebras Containing the Algebra of Maximal Nilindex, and Classification of Graded Filiform Leibniz Algebras

This paper is devoted to the study of Leibniz algebras introduced by Loday in [1-2] as an analogue of zero ”noncommutative” Lie algebras. We define the notion of zero-filiform Leibniz algebras and study their properties. There is a notion of p-filiform Lie algebras for p≥ 1 [3], which loses a sense in case p = 0, since Lie algebra has at least two generators. In the case of Leibniz algebras for p = 0 this notion substantial, and thereby, introduction of a zero-filiform algebra is quite natural. We also investigate the complex non-Lie filiform Leibniz algebras. In particular, we give some equivalent conditions of filiformity of Leibniz algebras and describe complex Leibniz algebras, which were graded in natural way.