  We reprove the results of Jordan [18] and Siebert [31] and 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. Given proof is self-contained and
does not depend on papers mentioned above. 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.
