On pointed Hopf algebras over nilpotent groups

TL;DR

This paper classifies finite-dimensional Nichols algebras over finite nilpotent groups of odd order, revealing structural properties of conjugacy classes and proposing a new conjecture related to racks of type C, with implications for pointed Hopf algebras.

Contribution

It provides a classification of Nichols algebras over certain nilpotent groups and introduces a new conjecture on racks of type C, extending understanding of Hopf algebra structures.

Findings

Conjugacy classes in these groups are either abelian or of type C.

Bosonization over infinite conjugacy classes results in infinite GK-dimension.

Application to finite GK-dimensional pointed Hopf algebras over torsion-free nilpotent groups.

Abstract

We classify finite-dimensional Nichols algebras over finite nilpotent groups of odd order in group-theoretical terms. The main step is to show that the conjugacy classes of such finite groups are either abelian or of type C; this property also holds for finite conjugacy classes of finitely generated nilpotent groups whose torsion has odd order. To extend our approach to the setting of finite GK-dimension, we propose a new Conjecture on racks of type C. We also prove that the bosonization a of Nichols algebra of a Yetter-Drinfeld module over a group whose support is an infinite conjugacy class has infinite GK-dimension. We apply this to the study of the finite GK-dimensional pointed Hopf algebras over finitely generated torsion-free nilpotent groups.