Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension $p^3$ and $pq^2$

TL;DR

This paper classifies Galois objects and cocycle deformations of certain semisimple Hopf algebras of dimensions involving prime squares and cubes, extending previous classifications and showing some have no non-trivial deformations.

Contribution

It extends the classification of cocycle deformations and Galois objects for semisimple Hopf algebras of dimensions p^3 and pq^2, including new results on their rigidity.

Findings

p+1 self-dual Hopf algebras of dimension p^3 have no non-trivial cocycle deformations

Classification of categorical Morita equivalence classes among these Hopf algebras

Extension of previous results for the 8-dimensional Kac-Paljutkin algebra

Abstract

Let $p$ and $q$ be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $pq^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac-Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.