Biderivations, local and 2-local derivation and automorphism of simple $\omega$-Lie algebras

TL;DR

This paper investigates the structure of derivations, automorphisms, and related mappings in finite-dimensional simple complex $ ext{ extomega}$-Lie algebras, establishing that local and 2-local derivations and automorphisms are essentially derivations and automorphisms or anti-automorphisms.

Contribution

It characterizes local and 2-local derivations and automorphisms in simple $ ext{ extomega}$-Lie algebras, showing their equivalence to derivations and automorphisms or anti-automorphisms.

Findings

Every local and 2-local derivation is a derivation.

Every local and 2-local automorphism is an automorphism or anti-automorphism.

Biderivations and $rac{1}{2}$-derivations are characterized.

Abstract

Given a finite-dimensional complex simple $\omega$-Lie algebras $\mathfrak{}$ over $\mathbb{C}$. We prove that every local ,$2-$local derivation is a derivation and every local (resp. 2-local) automorphisms are automorphisms or an anti-automorphis (resp. automorphism). We characterize also biderivation, $\frac{1}{2}$-derivation and local (2-local) $\frac{1}{2}$-derivation of $\mathfrak{g}$.