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.

中文翻译：

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欧几里德李代数表示的莱布尼兹代数

在本文中，我们用3维欧几里德李代数\（\ mathfrak {e}（2）\）来描述莱布尼兹代数。此外，假设由代数元素的平方（由I表示）作为右\（\ mathfrak {e}（2）\）-模块生成的理想与\（\ mathfrak {e }（2）\）位于\（\ mathfrak {sl} _ {2}（{\ mathbb {C}}）\ oplus \ mathfrak {sl} _ {2}（{\ mathbb {C}}），\ mathfrak中{sl} _ {3}（{\ mathbb {C}}）\）和\（\ mathfrak {sp} _ {4}（\ mathbb {C}）\）。此外，我们用普通的欧几里德李代数\（{\ mathfrak {e（n）}} \\）来分类莱布尼兹代数I是一个（n +1）维权\（{\ mathfrak {e（n）}} \）-通过\（\ mathfrak {e（n）} \）的矩阵实现转换定义的模块。最后，我们将Heisenberg Lie代数上的Fock模块的概念扩展到Diamond Lie代数\（\ mathfrak {D} _ {k} \）的情况，并用相应的Lie代数\（\ mathfrak { D} _ {k} \），理想情况下，我将其视为Fock \（\ mathfrak {D} _ {k} \）模块。