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

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 ≥ 5) of these algebras are split.Mathematics Subject Classification 2010: 17A32, 17B30.

The Algebraic and Geometric Classification of Nilpotent Bicommutative Algebras

“…With this background, it comes as no surprise that the central extensions of Lie and non-Lie algebras have been exhaustively studied for years. It is interesting both to describe them and to use them to classify different varieties of algebras [2,28,29,34,39,40]. Firstly, Skjelbred and Sund devised a method for classifying nilpotent Lie algebras employing central extensions [39].…”

The algebraic and geometric classification of nilpotent anticommutative algebras

“…In later works, using the same method, all non-Lie central extensions of 4-dimensional Malcev algebras [31], all non-associative central extensions of 3-dimensional Jordan algebras [30], all anticommutative central extensions of 3-dimensional anticommutative algebras [10] and all central extensions of 2-dimensional algebras [12] were described, to mention but a few. Related work on central extensions can be found, for example, in [2,37,50].…”

The algebraic classification of nilpotent Tortkara algebras

“…The authors thank Pilar Páez-Guillán for some constructive comments. 2 Corresponding Author: kaygorodov.ivan@gmail.com 1 because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra. Kac-Moody algebras have been conjectured to be a symmetry groups of a unified superstring theory.…”

“…The algebraic study of central extensions of Lie and non-Lie algebras has a very long history [2,20,21,25,33,35]. Thus, Skjelbred and Sund used central extensions of Lie algebras for a classification of nilpotent Lie algebras [33].…”