Finite dimensional Hopf actions on Weyl algebras (Q317291)

Let \(k\) be an algebraically closed field of characteristic zero and \(X\) a smooth affine irreducible variety over \(k\). Denote by \(D(X)\) the algebra of differential operators on \(X\). It is shown that an action of any finite dimensional Hopf algebra on \(D(X)\) factors through a group action. In particular, an action of any finite dimensional Hopf algebra on a Weyl algebra factors though a group action.NEWLINENEWLINEEarlier, similar results were proved by the reviewer for actions of pointed finite dimensional Hopf algebras on generic quantum polynomials [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 51, 29--60 (2005; Zbl 1184.16036)]. In [Adv. Math. 251, 47--61 (2014; Zbl 1297.16029)], the second and third authors proved similar results for actions of semisimple Hopf algebras on commutative domains.