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 $\mathfrak{ D}_m$.
Categories: math.RA