Cohomological aspects of Hopf algebra liftings

TL;DR

This paper develops a non-abelian cohomology theory to explicitly describe cocycles involved in Hopf algebra liftings, building on prior results that all such liftings are cocycle deformations.

Contribution

It introduces a new non-abelian cohomology framework for understanding Hopf algebra liftings and their cocycle structures.

Findings

Established a method for explicit cocycle description

Connected cohomology theory with Hopf algebra deformation classification

Extended previous results on cocycle deformations of Hopf algebras

Abstract

A recent result of ours [GM] shows that all Hopf algebra liftings of a given diagram in the sense of Andruskiewitsch and Schneider are cocycle deformations of each other. Here we develop a "non-abelian" cohomology theory, which gives a method for an explicit description of cocycles relevant to the lifting process.