Leibniz algebras associated with representations of Euclidean Lie algebra

Abstract

In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra $\mathfrak{e}(2)$ 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 $\mathfrak{e}(2)$-module is associated to representations of $\mathfrak{e}(2)$ in ${\mathfrak{sl}}_2(\mathbb{C})\oplus {\mathfrak{sl}}_2(\mathbb{C}),{\mathfrak{sl}}_3(\mathbb{C})$ and ${\mathfrak{sp}}_4(\mathbb{C})$. Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra $\mathfrak{e}(\mathfrak{n})$ as its liezation $I$ being an $(n+1)$-dimensional right $\mathfrak{e}(\mathfrak{n})$-module defined by transformations of matrix realization of $\mathfrak{e}(\mathfrak{n}).$ Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra ${\mathfrak{D}}_k$ and describe the structure of Leibniz algebras with corresponding Lie algebra ${\mathfrak{D}}_k$ and with the ideal $I$ considered as a Fock ${\mathfrak{D}}_k$-module.

12 pages

References in corpus (1)

• On some basic properties of Leibniz algebras