On unrolled Hopf algebras

TL;DR

This paper extends the concept of unrolled Hopf algebras to Nichols algebras associated with Yetter-Drinfeld modules, connecting them to quantum groups of various types.

Contribution

It introduces a natural extension of unrolled Hopf algebras to Nichols algebras with Lie algebra actions, including specific cases of quantum groups.

Findings

Unrolled Hopf algebra structures are constructed for Nichols algebras of diagonal type.

Special cases include unrolled versions of small, De Concini-Procesi, and Lusztig quantum groups.

The framework links Nichols algebras with classical quantum group structures.

Abstract

We show that the definition of unrolled Hopf algebras can be naturally extended to the Nichols algebra $\mathcal{B}$ of a Yetter-Drinfeld module $V$ on which a Lie algebra $\mathfrak g$ acts by biderivations. Specializing to Nichols algebras of diagonal type, we find unrolled versions of the small, the De Concini-Procesi and the Lusztig divided power quantum group, respectively.