Some Leibniz bimodules of $\mathfrak{sl}_2$

TL;DR

This paper classifies certain Leibniz bimodules over rak{sl}_2, showing only two non-split types exist and providing explicit descriptions of their actions.

Contribution

It identifies and characterizes the only two non-split Leibniz rak{sl}_2-bimodules when the module decomposes into two simple rak{sl}_2-modules.

Findings

Only two non-split Leibniz rak{sl}_2-bimodules exist under the given conditions.

Explicit descriptions of the actions of these bimodules are provided.

Abstract

We study complex finite-dimensional Leibniz algebra bimodule over $\mathfrak{sl}_2$ that as a Lie algebra module is split into a direct sum of two simple $\mathfrak{sl}_2$-modules. We prove that in this case there are only two non-split Leibniz $\mathfrak{sl}_2$-bimodules and we describe the actions.