A characterization of perfect Leibniz algebras

Abstract: Leibniz algebras are non-antisymmetric generalizations of Lie algebras that have attracted substantial interest due to their close relation with the latter class. A Leibniz algebra $A$ is called perfect if it coincides with its derived subalgebra $A^2$. As a generalization of an analogous result for Lie algebras, we show that perfect Leibniz algebras, of arbitrary dimension and over any field, are characterized among Leibniz algebras by the property that they are ideals whenever they are embedded as subideals. Equivalently, we prove that perfect Leibniz algebras are precisely those Leibniz algebras such that whenever they are embedded as ideals, they are characteristic ideals, i.e., they are invariant under all derivations of the ambient algebra. We apply the above to prove certain inclusion relations for derivation algebras of perfect Leibniz algebras.