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 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.

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Notes

  1. 1.

    I wish to thank David Jordan for pointing out this reference, which covers the general case.

  2. 2.

    Morally the reader should have in mind \(H_\alpha =\frac{K_{\alpha }^{2\ell _\alpha }-1}{v_\alpha ^{2\ell _\alpha }-1}\), which differs in the specialization by a nonzero scalar.

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Acknowledgements

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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Correspondence to Simon Lentner.

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Lentner, S. The unrolled quantum group inside Lusztig’s quantum group of divided powers. Lett Math Phys 109, 1665–1682 (2019). https://doi.org/10.1007/s11005-019-01185-9

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Keywords

  • Quantum group
  • Unrolled quantum group
  • Conformal field theory

Mathematics Subject Classification

  • 16T05
  • 17B69