Pointed Hopf algebras as cocycle deformations

TL;DR

This paper demonstrates that finite dimensional pointed Hopf algebras sharing the same diagram are related through cocycle deformations, providing a new perspective on their classification and structure.

Contribution

It establishes that such Hopf algebras are cocycle deformations of each other and introduces a method to explicitly describe the deforming cocycles.

Findings

All finite dimensional pointed Hopf algebras with the same diagram are cocycle deformations of each other.

A characterization of these Hopf algebras enables the application of Morita-Takeuchi and Hopf Galois extension results.

A method using exponential maps to describe the deforming cocycles is outlined.

Abstract

We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the application of results by Masuoka about Morita-Takeuchi equivalence and by Schauenburg about Hopf Galois extensions. We also outline a method to describe the deforming cocycles involved using the exponential map and its q-analogue.