On Hopf algebras of dimension 4p

TL;DR

This paper classifies non-semisimple Hopf algebras of certain dimensions, showing they are pointed if they contain enough group-like elements, thereby completing classifications for specific dimensions.

Contribution

It proves that non-semisimple Hopf algebras of dimensions 20, 28, and 44 are pointed or duals are pointed, advancing the classification of these algebras.

Findings

Non-semisimple Hopf algebras of dimension 20 are pointed or duals are pointed.

Non-semisimple Hopf algebras of dimension 28 are pointed or duals are pointed.

Non-semisimple Hopf algebras of dimension 44 are pointed or duals are pointed.

Abstract

In this paper, we prove that a non-semisimple Hopf algebra H of dimension 4p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided H contains more than two group-like elements. In particular, we prove that non-semisimple Hopf algebras of dimensions 20, 28 and 44 are pointed or their duals are pointed, and this completes the classification of Hopf algebras in these dimensions.