Leibniz Algebras Constructed by Representations of General Diamond Lie Algebras

Abstract

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