Biderivations and commutative post-Lie algebra structures on the Lie algebra W(a,b)

TL;DR

This paper classifies all biderivations of the Lie algebra W(a,b), revealing symmetric and skewsymmetric non-inner biderivations, and explores their implications for commutative post-Lie algebra structures.

Contribution

It determines all biderivations of W(a,b) and applies these results to construct commutative post-Lie algebra structures.

Findings

Existence of symmetric and skewsymmetric non-inner biderivations

Complete classification of biderivations for W(a,b)

Construction of commutative post-Lie algebra structures

Abstract

For $a,b\in \mathbb{C}$, the Lie algebra $\mathcal{W}(a,b)$ is the semidirect product of the Witt algebra and a module of the intermediate series. In this paper, all biderivations of $\mathcal{W}(a,b)$ are determined. Surprisingly, these Lie algebras have symmetric (and skewsymmetric) non-inner biderivations. As an applications, commutative post-Lie algebra structures on $\mathcal{W}(a,b)$ are obtained.