Local and 2-local 1/2-derivations on finite-dimensional Lie algebras

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