Local and 2-local derivations of solvable Leibniz algebras

TL;DR

This paper investigates local and 2-local derivations in solvable Leibniz algebras, showing conditions under which they coincide with derivations and providing examples where they differ.

Contribution

It characterizes when local and 2-local derivations are actual derivations in solvable Leibniz algebras with specific nilradicals and dimensions.

Findings

Local derivations are derivations in certain maximal dimension cases.

Existence of local derivations that are not derivations in some abelian nilradical cases.

All 2-local derivations are derivations in a specific example.

Abstract

We show that any local derivation on the solvable Leibniz algebras with model or abelian nilradicals, whose the dimension of complementary space is maximal is a derivation. We show that solvable Leibniz algebras with abelian nilradicals, which have 1-dimension complementary space, admit local derivations which are not derivations. Moreover, similar problem concerning 1-local derivations of such algebras are investigated and an example of solvable Leibniz algebra given such that any 2-local derivation on it is a derivation, but which admit local derivations which are not derivations.