Biderivations and linear commuting maps on simple generalized Witt algebras over a field

Let W be a simple generalizedWitt algebras over a field of characteristic zero. In this paper, weprove that each anti-symmetric biderivation of W isinner. As an application of biderivations, we show that a linear mapψ on W is commuting if and only if ψ is ascalar multiplication map on W. The commutingautomorphisms and derivations of W are determined.