Quantum supergroups II. Canonical basis

Clark, Sean; Hill, David; Wang, Weiqiang

doi:10.1090/S1088-4165-2014-00453-9

Quantum supergroups II. Canonical basis
HTML articles powered by AMS MathViewer

by Sean Clark, David Hill and Weiqiang Wang

Represent. Theory 18 (2014), 278-309

DOI: https://doi.org/10.1090/S1088-4165-2014-00453-9

Published electronically: September 9, 2014

PDF | Request permission

Abstract:

Following Kashiwara’s algebraic approach, we construct crystal bases and canonical bases for quantum supergroups of anisotropic type and for their integrable modules.

References

Georgia Benkart, Seok-Jin Kang, and Masaki Kashiwara, Crystal bases for the quantum superalgebra $U_q({\mathfrak {g}}{\mathfrak {l}}(m,n))$, J. Amer. Math. Soc. 13 (2000), no. 2, 295–331. MR 1694051, DOI 10.1090/S0894-0347-00-00321-0
Georgia Benkart, Seok-Jin Kang, and Duncan Melville, Quantized enveloping algebras for Borcherds superalgebras, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3297–3319. MR 1451594, DOI 10.1090/S0002-9947-98-02058-3
Sean Clark, Zhaobing Fan, Yiqiang Li, and Weiqiang Wang, Quantum supergroups III. Twistors, Comm. Math. Phys. 332 (2014), no. 1, 415–436. MR 3253707, DOI 10.1007/s00220-014-2071-4
Sean Clark, David Hill, and Weiqiang Wang, Quantum supergroups I. Foundations, Transform. Groups 18 (2013), no. 4, 1019–1053. MR 3127986, DOI 10.1007/s00031-013-9247-4
Sean Clark and Weiqiang Wang, Canonical basis for quantum $\mathfrak {osp}(1|2)$, Lett. Math. Phys. 103 (2013), no. 2, 207–231. MR 3010460, DOI 10.1007/s11005-012-0592-3
Alexander P. Ellis, Mikhail Khovanov, and Aaron D. Lauda, The odd nilHecke algebra and its diagrammatics, Int. Math. Res. Not. IMRN 4 (2014), 991–1062. MR 3168401, DOI 10.1093/imrn/rns240
A. Ellis and A. Lauda, An odd categorification of $U_q(\mathfrak {sl}_2)$, arXiv:1307.7816.
Z. Fan and Y. Li, Two-parameter quantum algebras, canonical bases and categorifications, IMRN (to appear), arXiv:1303.2429.
D. Hill and W. Wang, Categorification of quantum Kac-Moody superalgebras, Trans. Amer. Math. Soc. (to appear), arXiv:1202.2769v2.
Kyeonghoon Jeong, Crystal bases for Kac-Moody superalgebras, J. Algebra 237 (2001), no. 2, 562–590. MR 1816704, DOI 10.1006/jabr.2000.8590
Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh, Supercategorification of quantum Kac-Moody algebras, Adv. Math. 242 (2013), 116–162. MR 3055990, DOI 10.1016/j.aim.2013.04.008
S.-J. Kang, M. Kashiwara and 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), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
V. G. Kac, Infinite-dimensional algebras, Dedekind’s $\eta$-function, classical Möbius function and the very strange formula, Adv. in Math. 30 (1978), no. 2, 85–136. MR 513845, DOI 10.1016/0001-8708(78)90033-6
J.-H. Kwon, Super duality and crystal bases for quantum orthosymplectic superalgebras, arXiv:1301.1756.
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
Ian M. Musson and Yi Ming Zou, Crystal bases for $U_q(\textrm {osp}(1,2r))$, J. Algebra 210 (1998), no. 2, 514–534. MR 1662280, DOI 10.1006/jabr.1998.7591
Weiqiang Wang, Double affine Hecke algebras for the spin symmetric group, Math. Res. Lett. 16 (2009), no. 6, 1071–1085. MR 2576694, DOI 10.4310/MRL.2009.v16.n6.a14
Hiroyuki Yamane, Quantized enveloping algebras associated with simple Lie superalgebras and their universal $R$-matrices, Publ. Res. Inst. Math. Sci. 30 (1994), no. 1, 15–87. MR 1266383, DOI 10.2977/prims/1195166275
Yi Ming Zou, Integrable representations of $U_q(\textrm {osp}(1,2n))$, J. Pure Appl. Algebra 130 (1998), no. 1, 99–112. MR 1632799, DOI 10.1016/S0022-4049(97)00088-1