Semisimple Hopf algebras of dimension $9q^2$ and high-dimensional semisimple Hopf algebras of Frobenius type

TL;DR

This paper classifies semisimple Hopf algebras of dimension 9q^2 and shows that odd-dimensional semisimple Hopf algebras under 600 are of Frobenius type, advancing understanding of their structure and properties.

Contribution

It provides structure theorems for semisimple Hopf algebras of specific dimensions and establishes Frobenius type results for low-dimensional cases.

Findings

Classified semisimple Hopf algebras of dimension 9q^2.

Proved odd-dimensional semisimple Hopf algebras under 600 are of Frobenius type.

Enhanced understanding of the structure of high-dimensional semisimple Hopf algebras.

Abstract

Let $k$ be an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $9q^2$ over $k$, where $q$ is a prime number. We also prove that odd-dimensional semisimple Hopf algebras over $k$ of dimension less than 600 are of Frobenius type.