Letters in Mathematical Physics

, Volume 109, Issue 7, pp 1665–1682 | Cite as

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

  • Simon LentnerEmail author


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.


Quantum group Unrolled quantum group Conformal field theory 

Mathematics Subject Classification

16T05 17B69 



The author thanks C. Schweigert for interesting discussions, input and support. The author receives additional support by the DFG Graduiertenkolleg RTG 1670 at the University of Hamburg.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Algebra and Number TheoryUniversity HamburgHamburgGermany

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