Lie algebras of vector fields on smooth affine varieties

We show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. This completes the result of Jordan on the simplicity of the derivation algebra \cite{Jo}. Given proof is self-contained and does not depend on the results of Jordan. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.