  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 classic setting. We also present several
questions and directions for expanding this work in noncommutative invariant
theory.
