Pointed Hopf algebras: a guided tour to the liftings

TL;DR

This paper surveys the classification of finite-dimensional pointed Hopf algebras with abelian coradical and provides a detailed guide for constructing their liftings, exemplified by the classification of type B2.

Contribution

It offers a comprehensive overview of the classification process and introduces a step-by-step method for constructing liftings of graded Hopf algebras.

Findings

Classification of pointed Hopf algebras of type B2 completed

Detailed methodology for constructing liftings provided

Survey of existing classification results included

Abstract

This article serves a two-fold purpose. On the one hand, it is a survey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construct the liftings. As an example, we conclude the work with the classification of pointed Hopf algebras of type $B_2$.