Semisimple Reflection Hopf Algebras of Dimension Sixteen

Abstract

For each nontrivial semisimple Hopf algebra H of dimension sixteen over \(\mathbb {C}\), the smallest dimension inner-faithful representation of H acting on a quadratic AS regular algebra A of dimension 2 or 3, homogeneously and preserving the grading, is determined. Each invariant subring AH is determined. When AH is also AS regular, thus providing a generalization of the Chevalley–Shephard–Todd Theorem, we say that H is a reflection Hopf algebra for A.

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Acknowledgments

We appreciate the careful reading of the manuscript by the referee who caught several mistakes and suggested a simplification of our arguments for inner-faithfulness, which we have adopted.

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Correspondence to Luigi Ferraro.

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Presented by: Michel Van den Bergh

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Ferraro, L., Kirkman, E., Moore, W.F. et al. Semisimple Reflection Hopf Algebras of Dimension Sixteen. Algebr Represent Theor (2021). https://doi.org/10.1007/s10468-021-10038-w

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Keywords

  • Reflection Hopf algebra
  • Artin-Schelter regular algebra
  • Invariant subring
  • Grothendieck ring
  • Inner faithful representation

Mathematics Subject Classification (2010)

  • Primary 16T05
  • 16E65
  • 16G10