Leibniz algebras associated with representations of filiform Lie algebras

TL;DR

This paper classifies Leibniz algebras built upon naturally graded filiform Lie algebras by introducing a Fock module and analyzing the structure of Leibniz algebras with a specific quotient and module conditions.

Contribution

It introduces a Fock module for the algebra $n_{n,1}$ and classifies Leibniz algebras with this quotient and module structure, advancing understanding of their representations.

Findings

Classification of Leibniz algebras with quotient $n_{n,1}$

Introduction of Fock modules for $n_{n,1}$

Structural insights into Leibniz algebras with specific ideals

Abstract

In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra $n_{n,1}.$ We introduce a Fock module for the algebra $n_{n,1}$ and provide classification of Leibniz algebras $L$ whose corresponding Lie algebra $L/I$ is the algebra $n_{n,1}$ with condition that the ideal $I$ is a Fock $n_{n,1}$-module, where $I$ is the ideal generated by squares of elements from $L$.