Local derivations on the Euclidean algebras and the Diamond Lie algebra

The present paper is devoted to studying local derivations on the Euclidean algebras e(2), e(3), and the Diamond Lie algebra D. We prove that every local derivation on e(2) and e(3) is a derivation. On the other hand, the Diamond Lie algebra D, as a central extension of e(2), admits local derivations which are not derivations. Furthermore, we determine all local derivations on D.