Quantum binary polyhedral groups and their actions on quantum planes

We classify quantum analogues of actions of finite subgroups G of SL₂(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 2. 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.