Module-Valued 2-Local Derivations on Reductive Lie Algebras

Let \(\F\) be an algebraically closed field of characteristic zero, \(\g=\s\oplus\z\) a finite-dimensional reductive Lie algebra over \(\F\), and \(V\) an arbitrary finite-dimensional \(\g\)-module. We classify all 2-local derivations of \(\g\) on \(V\), and show that every 2-local derivation is a derivation if and only if \(\dim\z\leq1\) or \(V^\g=0\). If \(\dim\z\geq2\) and \(V^\g\ne0\), the nonlinear homogeneous maps give all exceptional 2-local derivations.