Pointed and copointed Hopf algebras as cocycle deformations

TL;DR

This paper demonstrates that finite dimensional pointed Hopf algebras sharing the same diagram are interconnected through cocycle deformations, with explicit descriptions using Hochschild cohomology, extending to copointed cases.

Contribution

It provides a characterization of such Hopf algebras and shows they are all cocycle deformations of each other, applying advanced cohomological methods.

Findings

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

Explicit description of the deformation cocycle via Hochschild cohomology.

Results extend to copointed Hopf algebras.

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 a result of Masuoka about Morita-Takeuchi equivalence and of Schauenburg about Hopf Galois extensions. The "infinitesimal" part of the deforming cocycle and of the deformation determine the deformed multiplication and can be described explicitly in terms of Hochschild cohomology. Applications to, and results for copointed Hopf algebras are also considered.