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Register a new userExamples of non-semisimple Hopf algebra actions on Artin-Schelter regular algebras
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Let $Bbbk$ be a base field of characteristic $p>0$ and let $U$ be the restricted enveloping algebra of a 2-dimensional nonabelian restricted Lie algebra. We classify all inner-faithful $U$-actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three.
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We study semisimple Hopf algebra actions on Artin-Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds.
We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; 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.
We compute the Nakayama automorphism of a PBW-deformation of a Koszul Artin-Schelter Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be Calabi-Yau. The relations between the Calabi-Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this paper is elementary and based on linear algebra. The results obtained in this paper will be applied in a subsequent paper.
Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A#H$ is a dg algebra. For any dg $A#H$-module $M$, there is a quasi-isomorphism of dg algebras: $mathrm{RHom}_A(M,M)#Hlongrightarrow mathrm{RHom}_{A#H}(Mot H,Mot H)$. This result is applied to $d$-Koszul algebras, Calabi-Yau algebras and AS-Gorenstein dg algebras
We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum symmetries. This improves a previous result by the authors, where the statement was established for semisimple H. The proof relies on a refinement of the method previously used: namely, considering reductions of the action of H on A modulo prime powers rather than primes. We also show that the result holds, more generally, for algebras of differential operators. This gives an affirmative answer to a question posed by the last two authors.
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