Abstract
In this part one of a series of papers, we introduce a new version of quantum covering and super groups with no isotropic odd simple root, which is suitable for the study of integrable modules, integral forms, and the bar involution. A quantum covering group involves parameters q and π with π2 = 1, and it specializes at π = −1 to a quantum supergroup. Following Lusztig, we formulate and establish various structural results of the quantum covering groups, including a bilinear form, quasi-\( \mathcal{R} \)-matrix, Casimir element, character formulas for integrable modules, and higher Serre relations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Benkart, S.-J. Kang, D. Melville, Quantized enveloping algebras for Borcherds superalgebras, Trans. AMS. 350 (1998), 3297–3319.
S. Clark, D. Hill, W. Wang, Quantum supergroups II. Canonical basis, arxiv:1304.7837.
S. Clark, W. Wang, Canonical basis for quantum \( \mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {\left. 1 \right|2} \right) \), Lett. Math. Phys. 103 (2013), 207–231.
V. Drinfeld, Quantum groups, in: Proceedings of the ICM (Berkeley, 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798-820.
A. Ellis, M. Khovanov, A. Lauda, The odd nilHecke algebra and its diagrammatics, IMRN (2013), arXiv:1111.1320.
N. Geer, Etingof-Kazhdan quantization of Lie superbialgebras. Adv. Math. 207 (2006), 1–38.
D. Hill, W. Wang, Categorification of quantum Kac-Moody superalgebras, Trans. Amer. Math. Soc. (to appear), arXiv:1202.2769v2.
J. C. Jantzen, Lectures on Quantum Groups, Graduate Studies in Mathematics, Vol. 6, Amer. Math. Soc., Providence, RI, 1996.
M. Jimbo, A q-analogue of U \( \left( {\mathfrak{gl}\left( {N+1} \right)} \right) \), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247–252.
V. Kac, Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius function and the very strange formula, Adv. Math. 30 (1978), 85–136.
S.-J. Kang, M. Kashiwara, S. Tsuchioka, Quiver Hecke superalgebras, arXiv:1107.1039.
M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516.
G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1990), 447–498.
G. Lusztig, Introduction to Quantum Groups, Progress in Mathematics, Vol. 110, Birkhäuser Boston, Boston, MA, 1993.
I. Musson, Y.-M. Zou, Crystal Bases for Uq (osp(1, 2r)), J. Algebra 210 (1998), 514–534.
W. Wang, Double affine Hecke algebras for the spin symmetric group, Math. Res. Lett. 16 (2009), 1071–1085.
H. Yamane, Quantized enveloping algebras associated with simple Lie superalgebras and their universal R-matrices, Publ. Res. Inst. Math. Sci. 30 (1994), 15–87.
Y.-M. Zou, Integrable representations of Uq (osp(1, 2n)), J. Pure Appl. Alg. 130 (1998), 99–112.
Author information
Authors and Affiliations
Corresponding author
Additional information
(Weiqiang Wang) Supported by NSF DMS-1101268.
Rights and permissions
About this article
Cite this article
CLARK, S., HILL, D. & WANG, W. QUANTUM SUPERGROUPS I. FOUNDATIONS. Transformation Groups 18, 1019–1053 (2013). https://doi.org/10.1007/s00031-013-9247-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-013-9247-4