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

TL;DR

This paper characterizes local and 2-local half-derivations in infinite-dimensional Lie algebras, proving they are often actual half-derivations, with specific results for Witt and related algebras.

Contribution

It establishes that all local and 2-local half-derivations on several classes of infinite-dimensional Lie algebras are genuine half-derivations, providing new insights into their structure.

Findings

All local and 2-local 1/2-derivations of Witt algebras are 1/2-derivations.

Existence of an infinite-dimensional Lie algebra with local (2-local) 1/2-derivation not being a 1/2-derivation.

All local (2-local) 1/2-derivations on the (a,b) algebra are 1/2-derivations.

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.