We construct a class of non-commutative, non-cocommutative, semisimple Hopf algebras of dimension 2n 2 and present conditions to define an inner faithful action of these Hopf algebras on quantum polynomial algebras, providing, in this way, more examples of semisimple Hopf actions which do not factor through group actions. Also, under certain condition, we classify the inner faithful Hopf actions of the Kac-Paljutkin Hopf algebra of dimension 8, H8, on the quantum plane.1991 Mathematics Subject Classification. 12E15; 13A35; 16T05; 16W70. Key words and phrases. semisimple Hopf algebras, inner faithful action, quantum polynomial rings. I would like to thank Christian Lomp very much for many clarifications and helpful insights. The author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020, and also supported by CAPES, Coordination of Superior Level Staff Improvement -Brazil.Definition 2.1. Let R be a bialgebra and J be an invertible element in R ⊗ R. J is called a right twist (or a Drinfel'd twist) for R if J satisfies:If J is a left twist for R, then J −1 is a right twist for R.Definition 2.2 ([8, 2.1]). Let R be a bialgebra. Let J be a right twist for R and σ ∈ End(R). We say that the pair (σ, J) is a twisted homomorphism for R if σ satisfies: (i) J(σ ⊗ σ)∆(h) = ∆(σ(h))J for all h ∈ R;

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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.

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