Biderivations, commuting mappings and (2-)local derivations of $\mathbb{N}$-graded Lie algebras of maximal class

TL;DR

This paper investigates the structure of biderivations, commuting mappings, and local derivations of three specific $ $-graded Lie algebras of maximal class, providing a detailed characterization of these mappings.

Contribution

It introduces a comprehensive analysis of biderivations and commuting mappings for these Lie algebras, and characterizes local and 2-local derivations based on their gradings.

Findings

Determined all biderivations for the three Lie algebras.

Characterized commuting mappings explicitly for these algebras.

Described local and 2-local derivations in terms of algebra gradings.

Abstract

In Fialowski's classification for algebras of maximal class, there are three Lie algebras of maximal class with 1-dimensional homogeneous components: $\mathfrak{m}_0$, $L_1$ and $\mathfrak{m}_2$. In this paper, we studied their biderivations by considering the embedded mapping to derivation algebras. Then we determined commuting mappings on these algebras as an application of biderivations. Finally, local and 2-local derivations for these three algebras were characterized as the given gradings.