Leibniz algebras constructed by representations of General Diamond Lie algebras

TL;DR

This paper constructs faithful representations of the complex and real general Diamond Lie algebras and describes Leibniz algebras associated with these structures, expanding understanding of their algebraic properties.

Contribution

It introduces minimal faithful representations of the general Diamond Lie algebras within special linear and symplectic Lie algebras, and characterizes related Leibniz algebras.

Findings

Faithful representations of complex and real general Diamond Lie algebras constructed.

Description of Leibniz algebras with specific ideal structures.

Representation theory links to subalgebras of classical Lie algebras.

Abstract

In this paper we construct a minimal faithful representation of the $(2m+2)$-dimensional complex general Diamond Lie algebra, $\mathfrak{D}_m(\mathbb{C})$, which is isomorphic to a subalgebra of the special linear Lie algebra $\mathfrak{sl}(m+2,\mathbb{C})$. We also construct a faithful representation of the general Diamond Lie algebra $\mathfrak{D}_m$ which is isomorphic to a subalgebra of the special symplectic Lie algebra $\mathfrak{sp}(2m+2,\mathbb{R})$. Furthermore, we describe Leibniz algebras with corresponding $(2m+2)$-dimensional general Diamond Lie algebra $\mathfrak{D}_m$ and ideal generated by the squares of elements giving rise to a faithful representation of $\mathfrak{D}_m$.