Semisimple Hopf actions on Weyl algebras

Abstract

Abstract. We study actions of semisimple Hopf algebras H on Weyl algebras A over an algebraically closed field of characteristic zero. We show that the action of H on A must factor through a group action; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.

A note on a paper by Cuadra, Etingof and Walton

“…By Proposition 2(3) one concludes that H has to be cocommutative, and hence a group algebra. Their key result of [3] extending [4] is the following: As a consequence from this Proposition and the reduction process to fields of positive characteristic, as described in sections 2 and 3, one deduces: …”

“…Given a left H-module algebra A that is free as an R-module, we extend the H-action on A to A/mA by h · a = h · a, for all a ∈ A, h ∈ H. Hence A/mA is a left H/mH-module algebra over a finite field F m . This reduction from a semisimple Hopf algebra action on a finitely presented algebra over a field of characteristic zero to an action of a semisimple and cosemisimple Hopf algebra over a finite field is the key step in [3]. In some cases the algebras A/mA over F m become finitely generated over their centre.…”

“…The main purpose in [3] was to show that any semisimple Hopf action on a Weyl algebra factors through a group algebra. However, Cuadra, Etingof and Walton's method can also be used to show that actions of semisimple Hopf algebras on enveloping algebras of finite dimensional Lie algebras or on iterated differential operator rings over a field K of characteristic 0, factor through group algebras.…”

“…It has already been outlined in [3, p.2] that these methods could be used to establish more general results on semisimple Hopf actions on quantized algebras and that the authors of [3] will do so in their future work. In particular it has been announced in [3, p.2] that their methods will apply to actions on module algebras A such that the resulting algebra A p when passing to a field of characteristic p, for large p, is PI and their PI-degree is a power of p. Such algebras include universal enveloping algebras of finite dimensional Lie algebras and algebras of differential operators of smooth irreducible affine varieties.…”

“…The purpose of this note is to analyse the main result of the paper [3] which says that any action of a semisimple Hopf algebra H on the nth Weyl algebra A = A n (K) over a field K of characteristic 0 factors through a group algebra. The central idea is to pass from algebras in characteristic 0 to algebras in positive characteristic by using the subring R of K, generated by all structure constants of H and the action on A and by passing to a finite field R/m.…”