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

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.