# The unrolled quantum group inside Lusztig's quantum group of divided powers

## TL;DR

This paper demonstrates that the unrolled small quantum group is a Hopf subalgebra of Lusztig's quantum group of divided powers, providing new insights into quantum topology and conformal field theory applications.

## Contribution

It establishes the embedding of the unrolled small quantum group into Lusztig's quantum group and constructs primitive elements with specific adjoint actions.

## Key findings

- Unrolled small quantum group is a Hopf subalgebra of Lusztig's quantum group.
- Provides a realization of the unrolled quantum group as operators on conformal field theory.
- Explains the origin of a weight shift observed in previous work.

## Abstract

In this letter we prove that the unrolled small quantum group, appearing in quantum topology, is a Hopf subalgebra of Lusztig's quantum group of divided powers. We do so by writing down non-obvious primitive elements with the correct adjoint action. As application we explain how this gives a realization of the unrolled quantum group as operators on a conformal feld theory and match some calculations on this side. In particular our results explain a prominent weight shift that appears in [FT10]. Our result extends to other Nichols algebras of diagonal type, including super Lie algebras.