Local derivations on finite-dimensional Lie algebras

TL;DR

This paper proves that all local derivations on finite-dimensional semisimple Lie algebras over algebraically closed fields of characteristic zero are derivations, but provides counterexamples for certain nilpotent Lie algebras.

Contribution

It establishes the equivalence of local derivations and derivations for semisimple Lie algebras and presents examples where this does not hold for some nilpotent Lie algebras.

Findings

All local derivations on finite-dimensional semisimple Lie algebras are derivations.

Existence of local derivations that are not derivations in certain nilpotent Lie algebras.

Counterexamples for nilpotent Lie algebras with dimension ≥ 3.

Abstract

We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of finite-dimensional nilpotent Lie algebras $\mathcal{L}$ with $\dim\mathcal{L}\geq 3$ which admit local derivations which are not derivations.