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

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.