| Summary: | 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[superscript H] share similar regularity and Gorenstein properties as the invariant rings k[u,v] [superscript G] in the classical setting.We also present several questions and directions for expanding this work in noncommutative invariant theory.
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