Some remarks on Leibniz algebras whose semisimple part related with $sl_2$

TL;DR

This paper classifies complex finite-dimensional Leibniz algebras with semisimple parts related to multiple copies of sl_2 and a solvable radical, providing detailed structures for specific dimensions and conditions.

Contribution

It offers new classifications of Leibniz algebras with semisimple parts involving multiple sl_2 components, extending previous understanding of their structure.

Findings

Classified Leibniz algebras with semisimple parts involving multiple sl_2 components.

Provided explicit structures for cases with dim R=2, 3, and specific ideal conditions.

Extended the classification to Leibniz algebras with quotient isomorphic to sl_2^1  sl_2^2.

Abstract

In this paper we identify the structure of complex finite-dimensional Leibniz algebras with associated Lie algebras $sl_2^1\oplus sl_2^2\oplus \dots \oplus sl_2^s\oplus R,$ where $R$ is a solvable radical. The classifications of such Leibniz algebras in the cases $dim R=2, 3$ and $dim I\neq 3$ have been obtained. Moreover, we classify Leibniz algebras with $L/I\cong sl_2^1\oplus sl_2^2$ and some conditions on ideal $I=id<[x,x] \ | \ x\in L>.$