A note on a paper by Cuadra, Etingof and Walton

Abstract

Abstract. We analyse the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebra H on the nth Weyl algebra A = An(K) over a field K of characteristic 0 factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebra H on an iterated Ore extension of derivation typeThe 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.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.We will verify part of their outlined program in Theorem 6 and will show that any action of a semisimple Hopf algebra H on an enveloping algebra of a finite dimensional Lie algebra or on an iterated Ore extension of derivation typein characteristic zero factors through a group algebra. Apart from this we give an alternative proof for the reduction step in Proposition 2.

A class of semisimple Hopf algebras acting on quantum polynomial algebras

“…In [14], by using and analyzing the results obtained by Cuadra, Etingof and Walton, it was showed that any semisimple Hopf action over an algebraically closed field of characteristic zero on an skew polynomial ring of derivation type must factor through a group action. Although there are examples in literature of Hopf actions on quantum polynomial algebras that do not factor through group actions (see [13, 7.4-7.6]), in this paper we give some conditions to define an action of a Hopf algebra on an skew polynomial of automorphism type which does not factor through a group action (Theorem 3.1).…”