Finite-dimensional Pointed or Copointed Hopf algebras over affine racks

TL;DR

This paper classifies finite-dimensional pointed and copointed Hopf algebras over affine racks, providing new examples and completing classifications for certain cases, with all resulting Hopf algebras being cocycle deformations.

Contribution

It completes the classification of finite-dimensional pointed and copointed Hopf algebras over affine racks, introducing new examples over non-abelian groups.

Findings

Complete classification for six affine racks in copointed case.

Finished four cases in pointed case.

All Hopf algebras are cocycle deformations of graded versions.

Abstract

We study the pointed or copointed liftings of Nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal YD-realization of these racks. In the copointed case we complete the classification for the six affine racks whose Nichols algebra is known to be of finite dimension. In the pointed case our method allows us to finish four of them. In all of the cases the Hopf algebras obtained turn out to be cocycle deformations of their associated graded Hopf algebras. All of them are new examples of finite-dimensional copointed or pointed Hopf algebras over non-abelian groups.