Leibniz algebras associated with representations of Euclidean Lie algebra

TL;DR

This paper classifies Leibniz algebras related to Euclidean Lie algebras and their representations, extending to general Euclidean algebras and introducing Fock modules over Diamond Lie algebras.

Contribution

It provides a classification of Leibniz algebras with Euclidean Lie algebra as their liezation, including representations and extensions to Diamond Lie algebras.

Findings

Classification of Leibniz algebras with $ ext{e}(2)$ as liezation.

Representation analysis in $ ext{sl}_2$, $ ext{sl}_3$, and $ ext{sp}_4$.

Extension of Fock modules to Diamond Lie algebras.

Abstract

In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra $\mathfrak{e}(2)$ as its liezation. Moreover, it is assumed that the ideal generated by the squares of elements of an algebra (denoted by $I$) as a right $\mathfrak{e}(2)$-module is associated to representations of $\mathfrak{e}(2)$ in $\mathfrak{sl}_2({\mathbb{C}})\oplus \mathfrak{sl}_2({\mathbb{C}}), \mathfrak{sl}_3({\mathbb{C}})$ and $\mathfrak{sp}_4(\mathbb{C})$. Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra ${\mathfrak{e(n)}}$ as its liezation $I$ being an $(n+1)$-dimensional right ${\mathfrak{e(n)}}$-module defined by transformations of matrix realization of $\mathfrak{e(n)}.$ Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra $\mathfrak{D}_k$ and describe the structure of Leibniz algebras with corresponding Lie algebra $\mathfrak{D}_k$ and with the ideal $I$ considered as a Fock $\mathfrak{D}_k$-module.