Local derivations on Solvable Lie algebras

TL;DR

This paper investigates local derivations on solvable Lie algebras, identifying conditions under which local derivations coincide with derivations and providing classifications for specific algebra classes.

Contribution

It establishes necessary and sufficient conditions for local derivations to be derivations in certain solvable Lie algebras, expanding understanding of their structure.

Findings

Existence of solvable Lie algebras with local derivations not being derivations.

Conditions under which all local derivations are derivations.

Classification results for algebras with abelian nilradical and model nilradical.

Abstract

We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation.