Semisimple Reflection Hopf Algebras of Dimension Sixteen

TL;DR

This paper classifies semisimple Hopf algebras of dimension sixteen over complex numbers by analyzing their minimal faithful representations on quadratic Artin-Schelter regular algebras and characterizing their invariant subrings, extending classical reflection theorems.

Contribution

It determines the minimal inner-faithful representations of these Hopf algebras on specific regular algebras and identifies when their invariants are also regular, generalizing the Chevalley-Shephard-Todd theorem.

Findings

Classified all semisimple Hopf algebras of dimension sixteen over a9.

Identified minimal faithful representations on quadratic AS regular algebras.

Characterized invariant subrings and their regularity, defining reflection Hopf algebras.

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 $A^H$ is determined. When $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$.