Finite-dimensional Hopf actions on the Weyl Poisson algebra and related filtered Poisson algebras that preserve the filtration must factor through group algebras.
Actions of Taft Algebras on Noetherian Down-Up Algebras
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We consider actions of Taft algebras on noetherian graded down-up algebras. We classify all such actions and determine properties of the corresponding invariant rings $A^T$. We identify precisely when $A^T$ is commutative, when it is Artin-Schelter regular, and give sufficient conditions for it to be Artin-Schelter Gorenstein. Our results show that many results and conjectures in the literature concerning actions of semisimple Hopf algebras on Artin-Schelter regular algebras can fail when the semisimple hypothesis is omitted.
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Hopf actions on Poisson algebras
Finite-dimensional Hopf actions on the Weyl Poisson algebra and related filtered Poisson algebras that preserve the filtration must factor through group algebras.