Quantum binary polyhedral groups
and their actions on quantum planes

Kenneth Chan; Ellen Kirkman; Chelsea Walton; James J. Zhang

doi:10.1515/crelle-2014-0047

Published by De Gruyter July 4, 2014

Quantum binary polyhedral groups and their actions on quantum planes

Kenneth Chan , Ellen Kirkman , Chelsea Walton and James J. Zhang

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2014-0047

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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 2. Remarkably, the corresponding invariant rings RH 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.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1102548

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-0855743

Funding source: Simons Foundation

Award Identifier / Grant number: #208314

Funding statement: Chelsea Walton and James J. Zhang were supported by the US National Science Foundation: NSF grants DMS-1102548 and DMS-0855743, respectively. Ellen Kirkman was partially supported by grant #208314 from the Simons Foundation.

Acknowledgements

The authors are grateful to the referee for pointing out several typographical errors and for making suggestions that improved greatly the exposition of this manuscript. The authors also thank Jacques Alev, Jim Kuzmanovich, and Graham Leuschke for several useful discussions and valuable comments. We thank especially Michael Wemyss for pointing out corrections for Section 7.

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Received: 2013-9-5

Revised: 2014-3-21

Published Online: 2014-7-4

Published in Print: 2016-10-1

Quantum binary polyhedral groups and their actions on quantum planes

Abstract

Acknowledgements

References

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