Local derivations and automorphisms of nilpotent Lie algebra

TL;DR

This paper investigates local derivations and automorphisms in nilpotent Lie algebras, revealing cases where local derivations are not actual derivations and establishing conditions for their existence.

Contribution

It provides new insights into when local derivations and automorphisms differ from true derivations and automorphisms in nilpotent Lie algebras.

Findings

Nilpotent Lie algebras with nilpotency indices 3 and 4 admit non-derivation local derivations.

A sufficient condition for the existence of non-derivation local derivations is established.

An example of a non-associative algebra where local derivations coincide with derivations is presented.

Abstract

The paper is devoted to the study of local derivations and automorphisms of nilpotent Lie algebras. Namely, we proved that nilpotent Lie algebras with indices of nilpotency $3$ and $4$ admit local derivation (local automorphisms) which is not a derivation (automorphisms). Further, it is presented a sufficient condition under which a nilpotent Lie algebra admits a local derivation which is not a derivation. With the same condition, it is proved the existence of pure local automorphism on a nilpotent Lie algebra. Finally, we present an $n$-dimensional non-associative algebra for which the space of local derivations coincides with the space of derivations.