Derivations of two-step nilpotent algebras

TL;DR

This paper investigates the structure of derivation Lie algebras of two-step nilpotent algebras, revealing new classes with specific properties and analyzing the nature of almost inner derivations in complex nilpotent Leibniz algebras.

Contribution

It introduces a new class of derivation Lie algebras with trivial center and abelian inner derivations, and characterizes almost inner derivations in complex nilpotent Leibniz algebras.

Findings

Identified a class of Lie algebras with trivial center and abelian inner derivations.

Described relations between complex and real indecomposable Heisenberg Leibniz algebras.

Showed that almost inner derivations are inner in most cases for complex nilpotent Leibniz algebras.

Abstract

In this paper we study the Lie algebras of derivations of two-step nilpotent algebras. We obtain a class of Lie algebras with trivial center and abelian ideal of inner derivations. Among these, the relations between the complex and the real case of the indecomposable Heisenberg Leibniz algebras are thoroughly described. Finally we show that every almost inner derivation of a complex nilpotent Leibniz algebra with one-dimensional commutator ideal, with three exceptions, is an inner derivation.