Abstract. Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if A arises as an H-module algebra via an inner faithful H-action, then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen.The main results of this article extend to working over k of positive characteristic. On the other hand, we obtain results on Hopf actions on Weyl algebras as a consequence of the main theorem.
Abstract. We classify quantum analogues of actions of finite subgroups G of SL 2 (k) on commutative polynomial rings k [u, v]. More precisely, we produce a classification of pairs (H, R), where H is a finite dimensional Hopf algebra that acts inner faithfully and preserves the grading of an Artin-Schelter regular algebra R of global dimension two. Remarkably, the corresponding invariant rings R H share similar regularity and Gorenstein properties as the invariant rings k [u, v] G in the classical setting. We also present several questions and directions for expanding this work in noncommutative invariant theory.
Abstract. Let H be a Hopf algebra with antipode S, and let A be an NKoszul Artin-Schelter regular algebra. We study connections between the Nakayama automorphism of A and S 2 of H when H coacts on A innerfaithfully. Several applications pertaining to Hopf actions on Artin-Schelter regular algebras are given. IntroductionThis article is a study in noncommutative invariant theory, particularly on the actions of finite dimensional Hopf algebras on Artin-Schelter (AS) regular algebras. Our results also lay the groundwork for other studies on Hopf actions on (filtered) AS regular algebras, namely for both [CKWZ] and [CWWZ14]. To begin, we discuss the vital role of Nakayama automorphisms.Let k be a base field and let B be either a connected graded AS regular algebra or a noetherian AS regular Hopf algebra. An algebra automorphism µ B of B is called a Nakayama automorphism of B if there is an integer d ≥ 0 such thatas B-bimodules, where B e = B ⊗ B op [BZ08, Definition 4.4(b)]. The algebra B is called Calabi-Yau if µ B = Id. Also, the quantity d is the global dimension of B when B is as given above. The definition of µ B is motivated by the classical notion of the Nakayama automorphism of a Frobenius algebra; see Section 1 for details. The Nakayama automorphism µ B is also unique up to inner automorphism of B. Further, if B is connected graded, then the Nakayama automorphism can be chosen to be a graded algebra automorphism, and in this case, it is unique since B has no non-trivial graded inner automorphism.Fairly recently, Brown and third-named author proved that the Nakayama automorphism of a noetherian AS regular Hopf algebra K can be written as follows:Here, S is the antipode of K and Ξ l l is the left winding automorphism of K associated to the left homological integral l of K [BZ08, Theorem 0.3]. This illustrates how one can express homological invariants (e.g., the Nakayama automorphism) in terms of other invariants (e.g., S 2 and l ) of such an AS regular Hopf algebra K.
Abstract. We study actions of semisimple Hopf algebras H on Weyl algebras A over an algebraically closed field of characteristic zero. We show that the action of H on A must factor through a group action; 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.
In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin-Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring A H under such an action arises as an analogue of a coordinate ring of a Kleinian singularity.
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