Leibniz algebras associated with representations of Euclidean Lie algebra

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.