  For a smooth manifold $X$ equipped with a volume form, let $\dL$ be the Lie
algebra of volume preserving smooth vector fields on $X$. A. Lichnerowicz
proved that the abelianization of $\dL$ is a finite-dimensional vector space,
and that its dimension depends only on the topology of $X$.
  In this paper we provide analogous results for some classical examples of
non-singular complex affine algebraic varieties that admit a nowhere-zero
algebraic form of top degree (which plays the role of a volume form).
