Leibniz Algebras Associated with Representations of Euclidean Lie Algebra

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.