Local and 2-Local \(\frac{1}{2}\)-Derivations on Finite-Dimensional Lie Algebras

In this work, we introduce the notion of local and 2-local \(\delta \) $\delta$ -derivations and describe local and 2-local \(\frac{1}{2}\) $\frac{1}{2}$ -derivation of finite-dimensional solvable Lie algebras with filiform, Heisenberg, and abelian nilradicals. Moreover, we describe the local \(\frac{1}{2}\) $\frac{1}{2}$ -derivation of oscillator Lie algebras, Schrödinger algebras, and the Lie algebra with the three-dimensional simple part, whose radical is an irreducible module. We prove that an algebra with only trivial \(\frac{1}{2}\) $\frac{1}{2}$ -derivation does not admit local and 2-local \(\frac{1}{2}\) $\frac{1}{2}$ -derivation, which is not \(\frac{1}{2}\) $\frac{1}{2}$ -derivation.