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

Abstract

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