Categorical Morita equivalence and monoidal Morita equivalence of semisimple Hopf algebras of dimension pqr

TL;DR

This paper classifies semisimple Hopf algebras of dimension pqr by their cocycle deformations, Galois objects, and Morita equivalence classes, revealing their twist inequivalence and categorically Morita equivalence structure.

Contribution

It determines the categorically and monoidally Morita equivalent classes of semisimple Hopf algebras of dimension pqr, including their Galois objects and cocycle deformations.

Findings

All such Hopf algebras have only one trivial Galois object.

They are pairwise twist inequivalent.

Categorically Morita equivalent classes are fully determined.

Abstract

In this paper, we determine the cocycle deformations and Galois objects for semisimple Hopf algebras of dimension pqr, and decide the categorically Morita equivalent classes and monoidally Morita equivalent classes of them. We show that all of them only have one trivial Galois objects, therefore these Hopf algebras are pairwise twist inequivalent, equivalently they are not monoidally Morita equivalent to each other, moreover, all the categorically Morita equivalent classes are determined.