Quantum binary polyhedral groups and their actions on quantum planes

Abstract

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.

Noncommutative cyclic isolated singularities

“…As a consequence, the Milnor number of A G is 5. Note that the McKay quiver corresponding to (A, G) is of type L 1 , see [CKWZ1,. This is slightly different from the classical A, D, E types.…”

“…Definitions of other standard concepts such as Artin-Schelter regularity, Auslander regularity, Cohen-Macaulay property are omitted as these can be found in many papers such as [Le,CKWZ1,MSm].…”

Actions of quantum linear spaces on quantum algebras

“…The natural next step, then, is to study quantum symmetries, or actions by Hopf algebras. Semisimple Hopf actions on quantum planes and quantum Weyl algebras are well-understood [8,9]. Our goal is to better understand non-semisimple Hopf actions, specifically actions by pointed Hopf algebras, which themselves have attracted much recent interest [11,17].…”

Maximal Cohen-Macaulay modules over a noncommutative 2-dimensional singularity

“…The noncommutative singularities (or equivalently, their associated algebras) studied in [CKWZ1,CKWZ2,CKWZ3] are usually far from commutative and do not satisfy a polynomial identity. For these noncommutative singularities we introduced the concept of a noncommutative quasi-resolution [QWZ,Definition 0.5] which generalizes Van den Bergh's noncommutative crepant resolution [VdB1,VdB2].…”

“…For these noncommutative singularities we introduced the concept of a noncommutative quasi-resolution [QWZ,Definition 0.5] which generalizes Van den Bergh's noncommutative crepant resolution [VdB1,VdB2]. The smash product constructions used in [CKWZ1,CKWZ2,CKWZ3] are examples of noncommutative quasi-resolutions. Recently Reyes-Rogalski proved that the Gabriel quivers of non-connected, N-graded, Artin-Schelter regular algebras of global dimension two are twisted versions (which are called pretzeled quivers in this paper) of the A D E graphs.…”