# Frobenius-Perron dimensions of integral $\Bbb Z_+$-rings and applications

## TL;DR

This paper introduces the Frobenius-Perron dimension for integral _+-rings and applies it to classify certain finite-dimensional quasi-Hopf and Hopf algebras, especially those with prime dimension or a single nontrivial simple module.

## Contribution

It defines the Frobenius-Perron dimension for integral _+-rings and demonstrates its usefulness in classifying specific classes of quasi-Hopf and Hopf algebras.

## Key findings

- Classification of finite-dimensional quasi-Hopf algebras with a unique nontrivial simple module.
- Results on the structure of quasi-Hopf and Hopf algebras of prime dimension p.

## Abstract

We introduce the notion of the Frobenius-Perron dimension of an integral $\Bbb Z_+$-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of quasi-Hopf and Hopf algebras of prime dimension $p$.