Semisimple Reflection Hopf Algebras of Dimension Sixteen

For each nontrivial semisimple Hopf algebra H of dimension sixteen over ℂ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {C}$\end{document}, 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.