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

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 $p{q}^2$. 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.

arXiv commentRemark 6.9 added; notation change in equation (5.7); some typos corrected and references updated. Accepted in J. Pure Appl. Algebra