On some null-filiform algebras and solvable Leibniz algebras

Abstract

From the theory of Lie algebras it is known that any finite-dimensional Lie algebra can be represented as the sum of the semi-simple subalgebra and its maximal solvable ideal (Levi-Malcev’s theorem). The semi-simple Lie algebra is a direct sum of simple ideals that were fully classified by Cartan. Moreover, A.I. Malcev showed that the study of solvable algebras can be reduced to the study of nilpotent ones. However, the problem of describing nilpotent Lie algebras is boundless, so their study should be carried out with additional conditions. One of these conditions is a restriction on the nilpotency index. One of the distinctive features of Leibniz algebras to the case of Lie algebras is that there are one-generated Leibniz algebras. Moreover, one-generated n-dimensional nilpotent Leibniz algebras have nilpotency index equal to n+1. Such algebras are called null-filiform Leibniz algebras. We are going to consider algebras defined by some specific identities and having a maximum index of nilpotency. Although the class of nilpotent filiform Leibniz algebras has a relatively simple structure, however, the problem of describing them is problematic, which is why they are studied with the imposition of conditions on graduation. During the classification of naturally graded filiform Leibniz algebras it became clear that a decrease of nilindex the problem of describing algebras becomes much more complicated. Therefore, such algebras are considered with additional restrictions, such as restrictions on the characteristic sequence of algebra. We plan to get the description of the n-dimensional naturally graded Leibniz algebras with a characteristic sequence equal to (n-m, m).