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

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 field theory and match some calculations on this side. In particular, our results explain a prominent weight shift that appears in Feigin and Tipunin (Logarithmic CFTs connected with simple Lie algebras, preprint, 2010. arXiv:1002.5047 ). Our result extends to other Nichols algebras of diagonal type, including super-Lie algebras.