Actions of Taft Algebras on Noetherian Down-Up Algebras
Simon Crawford, Jason Gaddis, Robert Won

TL;DR
This paper classifies actions of Taft algebras on noetherian graded down-up algebras, analyzing the properties of the invariant rings and revealing limitations of previous conjectures when semisimplicity is not assumed.
Contribution
It provides a complete classification of Taft algebra actions on these algebras and characterizes the invariant rings' properties, challenging existing assumptions in the literature.
Findings
Invariant rings can be non-commutative or non-Gorenstein.
Many conjectures fail without semisimplicity.
Conditions for Artin-Schelter regularity are explicitly identified.
Abstract
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 . We identify precisely when 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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Taxonomy
TopicsAdvanced Algebra and Logic · Algebraic structures and combinatorial models · Logic, programming, and type systems
