  Let k be an algebraically closed field of characteristic zero. In joint work
with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], we showed
that a semisimple Hopf action on a Weyl algebra over a polynomial algebra
k[z_1,...,z_s] factors through a group action, and this in fact holds for any
finite dimensional Hopf action if s=0. We also generalized these results to
finite dimensional Hopf actions on algebras of differential operators. In this
work we establish similar results for Hopf actions on other algebraic
quantizations of commutative domains. This includes universal enveloping
algebras of finite dimensional Lie algebras, spherical symplectic reflection
algebras, quantum Hamiltonian reductions of Weyl algebras (in particular,
quantized quiver varieties), finite W-algebras and their central reductions,
quantum polynomial algebras, twisted homogeneous coordinate rings of abelian
varieties, and Sklyanin algebras. The generalization in the last three cases
uses a result from algebraic number theory, due to A. Perucca.
