Leibniz Algebras Associated with Representations of Euclidean Lie Algebra

In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra 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 e(2) -module is associated to representations of e(2) in sl₂(ℂ) ⊕ sl₂(ℂ) , sl₃(ℂ) and sp₄(ℂ). Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra e(n) as its liezation I being an (n + 1)-dimensional right e(n) -module defined by transformations of matrix realization of e(n). Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra D_k and describe the structure of Leibniz algebras with corresponding Lie algebra D_k and with the ideal I considered as a Fock D_k-module.