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 sl(2)(C) circle plus sl(2)(C), sl(3)(C) and sp(4)(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.