Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension $16$

TL;DR

This paper classifies cocycle deformations and Galois objects of certain semisimple Hopf algebras of dimension 16, revealing their twist inequivalence and deformation properties through Frobenius-Schur indicators.

Contribution

It provides a complete classification of cocycle deformations and Galois objects for semisimple Hopf algebras of dimension 16, identifying which are twist inequivalent and deformable.

Findings

Most Hopf algebras are twist inequivalent.

Only three are cocycle deformations of dual group algebras.

Most do not admit non-trivial cocycle deformations.

Abstract

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension $16$. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.