Biderivations and Commutative Post-Lie Algebra Structures on the Lie Algebra $\mathcal{W}(a,b)$

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 skew-symmetric) non-inner biderivations. As an application, commutative post-Lie algebra structures on $\mathcal{W}(a,b)$ are obtained.