Leibniz algebras associated with representations of the Diamond Lie algebra

Abstract

In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra $\mathfrak{D}$ and the ideal generated by the squares of elements (further denoted by $I$) is a right $\mathfrak{D}$-module. Using description \cite{Cas} of representations of algebra $\mathfrak{D}$ in $\mathfrak{sl}(3,\mathbb{C})$ and $\mathfrak{sp}(4,\mathbb{F})$ where $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra $\mathfrak{D}.$ Classification of Leibniz algebras with corresponding Lie algebra $\mathfrak{D}$ and with the ideal $I$ as a Fock right $\mathfrak{D}$-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 $B{L}^2$ and $Z{L}^2$ are constructed, as well.

18 pages

References in corpus (3)

• On some basic properties of Leibniz algebras
• Leibniz algebras associated with representations of filiform Lie algebras
• Leibniz algebras of Heisenberg type

Cited by in corpus (2)

• Leibniz algebras constructed by representations of General Diamond Lie algebras
• Minimal linear representations of filiform Lie algebras and their application for construction of Leibniz algebras