The authors [Adv. Math. 251, 47–61 (2014; Zbl 1297.16029)] first proved that any semisimple Hopf action on a commutative domain over a algebraic closed field \(k\) of characteristic zero factors through a group action; this result was extended by J. Cuadra and the authors [ibid. 302, 25–39 (2016; Zbl 1356.16025)] to finite dimensional Hopf algebra actions on a Weyl algebra, or more generally on an algebra of differential operators of a smooth algebraic variety. In such a situation, one says that there is no quantum symmetry.
It is proven that there is no quantum symmetry on certain algebraic quantizations, including enveloping algebras of finite-dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras, finite \(W\)-algebras, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, Sklynanin algebras.