Classifying Hopf algebras of a given dimension

TL;DR

This paper advances methods for classifying finite-dimensional Hopf algebras, applying these techniques to specific dimensions involving prime factors, and summarizes the current classification status up to dimension 100.

Contribution

It introduces new classification tools building on previous work and applies them to dimensions rpq and 8p, providing a comprehensive classification overview.

Findings

Classification of Hopf algebras of dimension rpq and 8p achieved

New techniques improve understanding of sub- and quotient Hopf algebras

Summary table of classification status up to dimension 100

Abstract

Classifying all Hopf algebras of a given finite dimension over the complex numbers is a challenging problem which remains open even for many small dimensions, not least because few general approaches to the problem are known. Some useful techniques include counting the dimensions of spaces related to the coradical filtration, studying sub- and quotient Hopf algebras, especially those sub-Hopf algebras generated by a simple subcoalgebra, working with the antipode, and studying Hopf algebras in Yetter-Drinfeld categories to help to classify Radford biproducts. In this paper, we add to the classification tools in our previous work [arXiv:1108.6037v1] and apply our results to Hopf algebras of dimension rpq and 8p where p,q,r are distinct primes. At the end of this paper we summarize in a table the status of the classification for dimensions up to 100 to date.