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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
I wish to thank David Jordan for pointing out this reference, which covers the general case.
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.
References
Andruskiewitsch, N., Angiono, I., Rossi Bertone, F.: The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type, preprint. arXiv:1501.04518 (2015)
Angiono, I.: Distinguished pre-Nichols algebras. Transform. Groups 21(1), 1–33 (2016)
Andruskiewitsch, N.: Note on extensions of Hopf algebras. Can. J. Math. 48(1), 3–42 (1996)
Andruskiewitsch, N., Heckenberger, I., Schneider, H.-J.: The Nichols algebra of a semisimple Yetter–Drinfeld module. Am. J. Math. 132(6), 1493–1547 (2010)
Andruskiewitsch, N., Schneider, H.-J.: Finite quantum groups and Cartan matrices. Adv. Math. 154(1), 1–45 (2000)
Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. 2(1), 375–417 (2010)
Andruskiewitsch, N., Schweigert, C.: On unrolled Hopf algebras. arXiv:1701.00153v1
Angiono, I., Yamane, H.: The \(R\)-matrix of quantum doubles of Nichols algebras of diagonal type, preprint. arXiv:1304.5752 (2013)
Costantino, F., Geer, N., Patureau-Mirand, B.: Some remarks on the unrolled quantum group of sl (2). J. Pure Appl. Algebra 219, 3238–3262 (2015)
Cuntz, M., Heckenberger, I.: Weyl groupoids with at most three objects. J. Pure Appl. Algebra 213(6), 1112–1128 (2009)
Feigin, B.L., Gainutdinov, A.M., Semikhatov, A.M., Tipunin, I.Y.: Kazhdan–Lusztig correspondence for the representation category of the triplet \(\cal{W}\)-algebra in logarithmic CFT. Teor. Mat. Fiz. 148, 398–427 (2006)
Feigin, B.L., Tipunin, I.Y.: Logarithmic CFTs connected with simple Lie algebras, preprint. arXiv:1002.5047 (2010)
Gauß, C.F.: Summatio quarumdam serierum singularium, comment. Soc. Reg. Sci. Gott. (1811). Available for example in Werke II, Georg Olms Verlag (1981)
Gainutdinov, A.M., Lentner, S.D., Ohrmann, T.: Modularization of small quantum groups, preprint. arXiv:1809.02116 (2018)
Garcia, G.A., Gavarini, F.: Multiparameter quantum groups at roots of unity, preprint. arXiv:1708.05760 (2017)
Geer, N., Patureau-Mirand, B.: The trace on projective representations of quantum groups. arXiv:1610.09129v1
Geer, N., Patureau-Mirand, B., Turaev, V.: Modified quantum dimensions and re-normalized link invariants. Compos. Math. 145, 196–212 (2009)
Gainutdinov, A.M., Runkel, I.: Symplectic fermions and a quasi-Hopf algebra structure on \(\bar{U}^-_i(\mathfrak{sl}_2)\), preprint. arXiv:1503.07695 (2015)
Heckenberger, I.: Classification of arithmetic root systems. Adv. Math. 220(1), 59–124 (2009)
Heckenberger, I.: Lusztig isomorphisms for Drinfel’d doubles of bosonizations of Nichols algebras of diagonal type. J. Algebra 323(8), 2130–2182 (2010)
Kondo, H., Saito, Y.: Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to \(\mathfrak{sl}_2\). J. Algebra 330(1), 103–129 (2011)
Lusztig, G.: Quantum groups at roots of 1. Geom. Ded. 35, 89–114 (1990)
Lusztig, G.: Introduction to Quantum Groups, Progress in Mathematics 110. Birkhäuser, Basel (1993)
Lentner, S.: A Frobenius homomorphism for Lusztig’s quantum groups over arbitrary roots of unity. Commun. Contemp. Math. 18/3, 1550040 (2015)
Lentner, S.: Quantum groups and Nichols algebras acting on conformal quantum field theories, preprint. arXiv:1702.06431 (2017)
Lentner, S., Ohrmann, T.: Factorizable \(R\)-matrices for small quantum groups. SIGMA 13, 76 (2017)
Nagatomo, K., Tsuchiya, A.: The triplet vertex operator algebra \(W(p)\) and the restricted quantum group \(U_q(\mathfrak{sl}_2)\) at \(1=e^{\frac{\pi i}{p}}\). Adv. Stud. Pure Math. 61, 1–49 (2011). arXiv:0902.4607
Reshetikhin, N.: Quasitriangularity of quantum groups at roots of 1. Commun. Math. Phys. 170(1), 79–99 (1995)
Reshetikhin, N., Turaev, V.G.: Invariants of \(3\)-manifolds via link polynomials and quantum groups. Invent. Math. 103, 547–597 (1991)
Tanisaki, T.: Killing forms, Harish–Chandra homomorphisms and universal \(R\)-matrices for quantum algebras. Int. J. Mod. Phys. A 07(supp01b), 941–962 (1992)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-019-01185-9