Quantum binary polyhedral groups and their actions on quantum planes

2013/03/28 by Kenneth Chan, Ellen Kirkman, Chan, Kenneth +5
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1303.7203

openalex publication_date 2013/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 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.

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