Abstract
A quantum covering group is an algebra with parameters q and \(\pi \) subject to \(\pi ^2=1\), and it admits an integral form; it specializes to the usual quantum group at \(\pi =1\) and to a quantum supergroup of anisotropic type at \(\pi =-1\). In this paper we establish the Frobenius–Lusztig homomorphism and Lusztig–Steinberg tensor product theorem in the setting of quantum covering groups at roots of 1. The specialization of these constructions at \(\pi =1\) recovers Lusztig’s constructions for quantum groups at roots of 1.
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References
Andersen, H., Jantzen, J., Soergel, W.: Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Astérisque 220, 321 (1994)
Brundan, J., Ellis, A.: Super Kac-Moody 2-categories. Proc. Lond. Math. Soc. 115, 925–973 (2017)
Benkart, G., Kang, S.-J., Melville, D.: Quantized enveloping algebras for Borcherds superalgebras. Trans. Am. Math. Soc. 350, 3297–3319 (1998)
Clark, S., Hill, D., Wang, W.: Quantum supergroups I. Foundations. Transform. Groups 18, 1019–1053 (2013)
Clark, S., Hill, D., Wang, W.: Quantum supergroups II. Canonical basis. Represent. Theory 18, 278–309 (2014)
Clark, S., Fan, Z., Li, Y., Wang, W.: Quantum supergroups III. Twistors. Commun. Math. Phys. 332, 415–436 (2014)
Clark, S.: Quantum supergroups IV: the modified form. Math. Z. 278, 493–528 (2014)
Clark, S., Hill, D.: Quantum supergroups V. Braid group action. Commun. Math. Phys. 344, 25–65 (2016)
Egilmez, I., Lauda, A.: DG structures on odd categorified quantum \(sl(2)\). arXiv:1808.04924
Ellis, A., Lauda, A.: An odd categorification of \(U_q({{\mathfrak{s}}}{{\mathfrak{l}}}_2)\). Quantum Topol. 7, 329–433 (2016)
Fan, Z., Li, Y.: A geometric setting for quantum \({\mathfrak{osp}}(1|2)\). Trans. Am. Math. Soc. 367, 7895–7916 (2015)
Hill, D., Wang, W.: Categorification of quantum Kac-Moody superalgebras. Trans. Am. Math. Soc. 367, 1183–1216 (2015)
Kang, S.-J., Kashiwara, M., Oh, S.-J.: Supercategorification of quantum Kac-Moody algebras II. Adv. Math. 265, 169–240 (2014)
Kang, S.-J., Kashiwara, M., Tsuchioka, S.: Quiver Hecke superalgebras. J. Reine Angew. Math. 711, 1–54 (2016)
Lusztig, G.: Finite dimensional Hopf algebras arising from quantum groups. J. Am. Math. Soc. 3, 257–296 (1990)
Lusztig, G.: Quantum groups at roots of \(1\). Geom. Dedicata 35, 89–114 (1990)
Lusztig, G.: Introduction to Quantum Groups. Birkhäuser, Basel (1994)
Acknowledgements
This research is partially supported by Wang’s NSF Grant DMS-1702254 (including GRA supports for the two junior authors). WW thanks Adacemia Sinica Institute of Mathematics (Taipei) for the hospitality and support during a past visit, where some of the work was carried out.
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Chung, C., Sale, T. & Wang, W. Quantum supergroups VI: roots of 1. Lett Math Phys 109, 2753–2777 (2019). https://doi.org/10.1007/s11005-019-01209-4
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DOI: https://doi.org/10.1007/s11005-019-01209-4