Central extensions of null-filiform and naturally graded filiform non-Lie Leibniz algebras

TL;DR

This paper classifies central extensions of certain nilpotent Leibniz algebras, focusing on those with maximal nilpotency index and naturally graded filiform non-Lie Leibniz algebras, revealing conditions for split extensions.

Contribution

It provides a classification of central extensions for specific nilpotent Leibniz algebras, including non-split cases and conditions for split extensions.

Findings

Classified central extensions of Leibniz algebras with maximal nilpotency.

Described non-split central extensions of naturally graded filiform non-Lie Leibniz algebras.

Showed that k-dimensional central extensions (k≥5) are split.

Abstract

In this paper we describe central extensions of some nilpotent Leibniz algebras. Namely, central extensions of the Leibniz algebra with maximal index of nilpotency are classified. Moreover, non-split central extensions of naturally graded filiform non-Lie Leibniz algebras are described up to isomorphism. It is shown that $k$-dimensional central extensions ($k\geq 5$) of these algebras are split.