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 solve two conjectures of Ceken-Palmieri-Wang-Zhang concerning discriminants and give some applications.

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.

We continue our study of the McKay Correspondence for grading preserving actions of semisimple Hopf algebras H on (noncommutative) Artin-Schelter regular algebras A. Here, we establish correspondences between module categories over A H , over A#H, and over End A H (A). We also study homological properties of (endomorphism rings of) maximal Cohen-Macaulay modules over A H .