Leibniz Algebras Associated with Representations of the Diamond Lie Algebra

In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra D and the ideal generated by the squares of elements (further denoted by I) is a right D-module. Using description (Casati et al J. Math. Phys. 51: 033515 (2010)) of representations of algebra D in ????(3 , C) and ????(4 , F) where F= R or C we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra D. Classification of Leibniz algebras with corresponding Lie algebra D and with the ideal I as a Fock right D-module is presented. The linear integrable deformations in terms of the second cohomology groups of obtained finite-dimensional Leibniz algebras are described. Two computer programs in Mathematica 10 which help to calculate for a given Leibniz algebra the general form of elements of spaces BL² and ZL² are constructed, as well.