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 s l 2 ( ℂ ) ⊕ s l 2 ( ℂ ) , s l 3 ( ℂ ) and s p 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.