# Weak c-ideals of Leibniz algebras

## TL;DR

This paper introduces the concept of weak c-ideals in Leibniz algebras, explores their properties, and uses them to characterize solvability and supersolvability, extending known results from Lie algebra theory.

## Contribution

It defines weak c-ideals in Leibniz algebras and provides new characterizations of their solvable and supersolvable structures, generalizing previous Lie algebra results.

## Key findings

- Weak c-ideals have specific properties that relate to solvability.
- One-dimensional weak c-ideals are c-ideals.
- A classification result for Leibniz algebras with all one-dimensional subalgebras as c-ideals does not hold generally, but does for symmetric cases.

## Abstract

A subalgebra $B$ of a Leibniz algebra $L$ is called a weak c-ideal of $L$ if there is a subideal $C$ of $L$ such that $L=B+C$ and $B\cap C\subseteq B_{L}$ where $B_{L}$ is the largest ideal of $L$ contained in $B.$ This is analogous to the concept of a weakly c-normal subgroup, which has been studied by a number of authors. We obtain some properties of weak c-ideals and use them to give some characterisations of solvable and supersolvable Leibniz algebras generalising previous results for Lie algebras. We note that one-dimensional weak c-ideals are c-ideals, and show that a result of Turner classifying Leibniz algebras in which every one-dimensional subalgebra is a c-ideal is false for general Leibniz algebras, but holds for symmetric ones.