Title: 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 right
adjoint action. We also construct a new larger Hopf algebra that contains the
full unrolled quantum group. In fact this Hopf algebra contains both the
enveloping of the Lie algebra and the ring of functions on the Lie group, and
it should be interesting in its own right. We finally explain how this gives a
realization of the unrolled quantum group as operators on a conformal field
theory and match some calculations on this side. Our result extends to other
Nichols algebras of diagonal type, including super Lie algebras.