Local and 2-local $\frac{1}2$-derivations of infinite-dimensional Lie algebras

Abstract

In this work, we describe local and 2-local $\frac12$-derivations of infinite-dimensional Lie algebras. We prove that all local and 2-local $\frac12$-derivations of the Witt algebra as well as of the positive Witt algebra and the classical one-sided Witt algebra are $\frac12$-derivations. We also give an example of an infinite-dimensional Lie algebra with a local (2-local) $\frac12$-derivation which is not a $\frac12$-derivation. Further we prove that all local (2-local) $\frac12$-derivations on the $\mathcal{W}(a,b)$ algebra are $\frac12$-derivations.