On a family of Hopf algebras of dimension 72

TL;DR

This paper studies a specific family of 72-dimensional Hopf algebras with a focus on their module structure, classification of simple modules, and properties like unimodularity and cocycle deformations.

Contribution

It classifies all simple modules of these Hopf algebras and analyzes their structural properties, including unimodularity and deformation relations.

Findings

Classified all simple modules of the Hopf algebras.

Proved the Hopf algebras are unimodular but not quasitriangular.

Showed they are cocycle deformations of each other.

Abstract

We investigate a family of Hopf algebras of dimension 72 whose coradical is isomorphic to the algebra of functions on S_3. We determine the lattice of submodules of the so-called Verma modules and as a consequence we classify all simple modules. We show that these Hopf algebras are unimodular (as well as their duals) but not quasitriangular; also, they are cocycle deformations of each other.