Leibniz algebras with associated Lie algebras sl_2\dot{+} R (dim R=2)

TL;DR

This paper classifies a specific class of finite-dimensional Leibniz algebras whose quotient by a certain ideal resembles a semidirect sum of sl_2 and a 2-dimensional solvable algebra, extending Lie algebra decomposition concepts.

Contribution

It provides a classification of Leibniz algebras with quotients isomorphic to sl_2 plus a 2-dimensional solvable ideal, a novel extension of Lie algebra decomposition theory.

Findings

Classified Leibniz algebras with quotient isomorphic to sl_2  plus a 2-dimensional solvable ideal.

Extended the understanding of Leibniz algebra structures beyond classical Lie algebra decompositions.

Abstract

From the theory of finite dimensional Lie algebras it is known that every finite dimensional Lie algebra is decomposed into a semidirect sum of semisimple subalgebra and solvable radical. Moreover, due to work of Mal'cev the study of solvable Lie algebras is reduced to the study of nilpotent ones. For the finite dimensional Leibniz algebras the analogues of the mentioned results are not proved yet. In order to get some idea how to establish the results we examine the Leibniz algebra for which the quotient algebra with respect to the ideal generated by squares elements of the algebra (denoted by $I$) is a semidirect sum of semisimple Lie algebra and the maximal solvable ideal. In this paper the class of complex Leibniz algebras, for which quotient algebras by the ideal $I$ are isomorphic to the semidirect sum of the algebra $sl_2$ and two-dimensional solvable ideal $R$, are described.