  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.
