\((\mathcal{T}_{\textsf {Lie}})\)-Leibniz algebras and related properties

Abstract

In this paper, we study properties of Leibniz algebras of class \(\mathcal{T}_{\textsf {Lie}}\), a subclass of both the class \(\mathfrak{D}_{\textsf {Lie}}\) and the class of \({\textsf {Lie}}\)-stem Leibniz algebras. We determine necessary and sufficient conditions under which a non-Lie Leibniz algebra is of class \(\mathcal{T}_{\textsf {Lie}}\) and study their relationship with pseudo-abelian Leibniz algebras. We also show that Leibniz algebras of class \(\mathcal{T}_{\textsf {Lie}}\) have semi-simple central \({\textsf {Lie}}\)-derivations.