Abstract
We establish direct connections at several levels between quantum groups and supergroups associated to bar-consistent anisotropic super Cartan datum by constructing an automorphism (called twistor) in the setting of covering quantum groups. The canonical bases of the halves of quantum groups and supergroups are shown to match under the twistor up to powers of \({\sqrt{-1}}\). We further show that the modified quantum group and supergroup are isomorphic over the rational function field adjoined with \({\sqrt{-1}}\), by constructing a twistor on the modified covering quantum group. An equivalence of categories of weight modules for quantum groups and supergroups follows.
Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.
—Jacques Hadamard
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Clark S., Hill D., Wang W.: Quantum supergroups I. Foundations. Transform. Groups 18(4), 1019–1053 (2013)
Clark, S., Hill, D., Wang, W.: Quantum supergroups II. Canonical basis. arXiv:1304.7837v2
Clark S., Wang W.: Canonical basis for quantum \({\mathfrak{osp}(1|2)}\). Lett. Math. Phys. 103, 207–231 (2013)
Ellis A., Khovanov M., Lauda A.: The odd nilHecke algebra and its diagrammatics. Int. Math. Res. Not. 2014(4), 991–1062 (2014)
Ellis, A., Lauda, A.: An odd categorification of \({U_q(\mathfrak{sl}_2)}\). arXiv:1307.7816
Etingof P., Kazhdan D.: Quantization of Lie bialgebras. VI. Quantization of generalized Kac-Moody algebras. Transform. Groups 13, 527–539 (2008)
Fan, Z., Li, Y.: Two-parameter quantum algebras, canonical bases and categorifications (2012). arXiv:1303.2429
Fan, Z., Li, Y.: A geometric setting for quantum \({\mathfrak{osp}(1|2)}\), Trans. Amer. Math. Soc. (2014, to appear). arXiv:1305.0710
Geer N.: Etingof–Kazhdan quantization of Lie superbialgebras. Adv. Math. 207, 1–38 (2006)
Hill, D., Wang, W.: Categorification of quantum Kac-Moody superalgebras. Trans. Amer. Math. Soc. (2014, to appear). arXiv:1202.2769v2
Kac V.: Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius function and the very strange formula. Adv. Math. 30, 85–136 (1978)
Kashiwara K.M.: On crystal bases of the Q-analogue of universal enveloping algebras. Duke Math. J. 63, 456–516 (1991)
Kang, S.-J., Kashiwara, M., Tsuchioka, S.: Quiver Hecke superalgebras. J. Reine. Angew. Math. (2014, to appear). doi:10.1515/cralle-2013-0089; arXiv:1107.1039
Kang, S.-J., Kashiwara, M., Oh, S.-J.: Supercategorification of quantum Kac-Moody algebras II. arXiv:1303.1916
Lanzman E.: The Zhang transformation and U q (osp(1, 2l))-Verma modules annihilators. Algebra Represent. Theory 5, 235–258 (2002)
Li, Y.: A geometric realization of modified quantum algebras. arXiv:1007.5384
Lusztig G.: Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. 3, 447–498 (1990)
Lusztig, G.: Introduction to quantum groups. In: Progress in Mathematics, vol. 110. Birkhäuser, Basel (1993)
Wang W.: Double affine Hecke algebras for the spin symmetric group. Math. Res. Lett. 16, 1071–1085 (2009)
Yamane H.: Quantized enveloping algebras associated with simple Lie superalgebras and their universal R-matrices. Publ. Res. Inst. Math. Sci. 30, 15–87 (1994)
Yamane H.: On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras. Pub. Res. Inst. Math. Sci. 35, 321–390 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
About this article
Cite this article
Clark, S., Fan, Z., Li, Y. et al. Quantum Supergroups III. Twistors. Commun. Math. Phys. 332, 415–436 (2014). https://doi.org/10.1007/s00220-014-2071-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2071-4