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

Abstract

In this work, we introduce the notion of local and 2-local \(\delta \)-derivations and describe local and 2-local \(\frac{1}{2}\)-derivation of finite-dimensional solvable Lie algebras with filiform, Heisenberg, and abelian nilradicals. Moreover, we describe the local \(\frac{1}{2}\)-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 \(\frac{1}{2}\)-derivation does not admit local and 2-local \(\frac{1}{2}\)-derivation, which is not \(\frac{1}{2}\)-derivation.