  Let H be a semisimple (so, finite dimensional) Hopf algebra over an
algebraically closed field k of characteristic zero and let A be a commutative
domain over k. We show that if A arises as an H-module algebra via an inner
faithful H-action, then H must be a group algebra. This answers a question of
E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen.
  The main results of this article extend to working over k of positive
characteristic. On the other hand, we obtain results on Hopf actions on Weyl
algebras as a consequence of the main theorem.
