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A geometric setting for quantum $ \mathfrak{osp}(1\Vert 2)$


Authors: Zhaobing Fan and Yiqiang Li
Journal: Trans. Amer. Math. Soc. 367 (2015), 7895-7916
MSC (2010): Primary 17B37, 14F43
DOI: https://doi.org/10.1090/S0002-9947-2015-06266-7
Published electronically: March 26, 2015
MathSciNet review: 3391903
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Abstract | References | Similar Articles | Additional Information

Abstract: A geometric categorification is given for arbitrary-large-finite-
dimensional quotients of quantum $ \mathfrak{osp}(1\vert 2)$ and tensor products of its simple modules. The modified quantum $ \mathfrak{osp}(1\vert 2)$ of Clark-Wang, a new version in this paper and the modified quantum $ \mathfrak{sl}(2)$ are shown to be isomorphic to each other over a field containing $ \mathbb{Q}(v)$ and $ \sqrt {-1}$.


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  • [BBD82] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5-171 (French). MR 751966 (86g:32015)
  • [BLM90] A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $ {\rm GL}_n$, Duke Math. J. 61 (1990), no. 2, 655-677. MR 1074310 (91m:17012), https://doi.org/10.1215/S0012-7094-90-06124-1
  • [BL94] Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527 (95k:55012)
  • [B03] Tom Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), no. 3, 209-216. MR 1996415 (2004f:14037), https://doi.org/10.1007/s00031-003-0606-4
  • [CW13] Sean Clark and Weiqiang Wang, Canonical basis for quantum $ \mathfrak{osp}(1\vert 2)$, Lett. Math. Phys. 103 (2013), no. 2, 207-231. MR 3010460, https://doi.org/10.1007/s11005-012-0592-3
  • [CFLW14] Sean Clark, Zhaobing Fan, Yiqiang Li, and Weiqiang Wang, Quantum supergroups III. Twistors, Comm. Math. Phys. 332 (2014), no. 1, 415-436. MR 3253707, https://doi.org/10.1007/s00220-014-2071-4
  • [D95] Jie Du, A note on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995), no. 4, 363-372. MR 1358684 (96m:17024)
  • [EKL14] 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
  • [EL13] Alexander P. Ellis and Aaron D. Lauda, An odd categorification of $ U_q(\mathfrak{sl}_2)$, arXiv:1307.7816.
  • [FL14] Z. Fan and Y. Li, Two-parameter quantum algebras, canonical bases and categorifications, Int. Math. Res. Not. 2014, DOI 10.1093/imrn/rnu159.
  • [GL92] I. Grojnowski and G. Lusztig, On bases of irreducible representations of quantum $ {\rm GL}_n$, Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 167-174. MR 1197834 (94a:20070), https://doi.org/10.1090/conm/139/1197834
  • [HW15] David Hill and Weiqiang Wang, Categorification of quantum Kac-Moody superalgebras, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1183-1216. MR 3280041, https://doi.org/10.1090/S0002-9947-2014-06128-X
  • [KKO13] Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh, Supercategorification of quantum Kac-Moody algebras, Adv. Math. 242 (2013), 116-162. MR 3055990, https://doi.org/10.1016/j.aim.2013.04.008
  • [KKO14] Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh, Supercategorification of quantum Kac-Moody algebras II, Adv. Math. 265 (2014), 169-240. MR 3255459, https://doi.org/10.1016/j.aim.2014.07.036
  • [KKT11] S.-J. Kang, M. Kashiwara, and S. Tsuchioka, Quiver Hecke superalgebras, arXiv:1107.1039.
  • [K91] M. Kashiwara, On crystal bases of the $ Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465-516. MR 1115118 (93b:17045), https://doi.org/10.1215/S0012-7094-91-06321-0
  • [KL09] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347. MR 2525917 (2010i:17023), https://doi.org/10.1090/S1088-4165-09-00346-X
  • [KL11] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685-2700. MR 2763732 (2012a:17021), https://doi.org/10.1090/S0002-9947-2010-05210-9
  • [Lan02] Emmanuel Lanzmann, The Zhang transformation and $ \mathcal {U}_q({\rm osp}(1,2l))$-Verma modules annihilators, Algebr. Represent. Theory 5 (2002), no. 3, 235-258. MR 1921760 (2003i:17020), https://doi.org/10.1023/A:1016550528593
  • [Lau10] Aaron D. Lauda, A categorification of quantum $ {\rm sl}(2)$, Adv. Math. 225 (2010), no. 6, 3327-3424. MR 2729010 (2012b:17036), https://doi.org/10.1016/j.aim.2010.06.003
  • [Li10] Y. Li, A geometric realization of modified quantum algebras, arXiv:1007.5384.
  • [L90] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. MR 1035415 (90m:17023), https://doi.org/10.2307/1990961
  • [L91] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365-421. MR 1088333 (91m:17018), https://doi.org/10.2307/2939279
  • [L92] G. Lusztig, Canonical bases in tensor products, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 17, 8177-8179. MR 1180036 (93j:17033), https://doi.org/10.1073/pnas.89.17.8177
  • [L93] George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston Inc., Boston, MA, 1993. MR 1227098 (94m:17016)
  • [R06] Raphaël Rouquier, Categorification of $ {\mathfrak{sl}}_2$ and braid groups, Trends in representation theory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., Providence, RI, 2006, pp. 137-167. MR 2258045 (2008f:17011), https://doi.org/10.1090/conm/406/07657
  • [R08] R. Rouquier, 2-Kac-Moody algebras, arXiv:0812.5023.
  • [VV11] M. Varagnolo and E. Vasserot, Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67-100. MR 2837011, https://doi.org/10.1515/CRELLE.2011.068
  • [W09] Weiqiang Wang, Double affine Hecke algebras for the spin symmetric group, Math. Res. Lett. 16 (2009), no. 6, 1071-1085. MR 2576694 (2011a:20009), https://doi.org/10.4310/MRL.2009.v16.n6.a14
  • [Z07] H. Zheng, A geometric categorification of representations of $ U_q({\rm sl}_2)$, Topology and physics, Nankai Tracts Math., vol. 12, World Sci. Publ., Hackensack, NJ, 2008, pp. 348-356. MR 2503405 (2010c:17017), https://doi.org/10.1142/9789812819116_0016
  • [Zou98] Yi Ming Zou, Integrable representations of $ U_q({\rm osp}(1,2n))$, J. Pure Appl. Algebra 130 (1998), no. 1, 99-112. MR 1632799 (99e:17028), https://doi.org/10.1016/S0022-4049(97)00088-1

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Additional Information

Zhaobing Fan
Affiliation: Department of Mathematics, 244 Mathematics Building, University of Buffalo, The State University of New York, Buffalo, New York 14260
Address at time of publication: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: zhaobing@buffalo.edu, fanz@math.ksu.edu

Yiqiang Li
Affiliation: Department of Mathematics, 244 Mathematics Building, University of Buffalo, The State University of New York, Buffalo, New York 14260
Email: yiqiang@buffalo.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06266-7
Keywords: Quantum $\mathfrak{osp}(1\|2)$, quantum modified algebra, tensor product module, categorification, perverse sheaf
Received by editor(s): May 1, 2013
Received by editor(s) in revised form: August 10, 2013, and August 14, 2013
Published electronically: March 26, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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