A geometric setting for quantum $\mathfrak {osp}(1|2)$
HTML articles powered by AMS MathViewer
- by Zhaobing Fan and Yiqiang Li PDF
- Trans. Amer. Math. Soc. 367 (2015), 7895-7916 Request permission
Abstract:
A geometric categorification is given for arbitrary-large-finite- dimensional quotients of quantum $\mathfrak {osp}(1|2)$ and tensor products of its simple modules. The modified quantum $\mathfrak {osp}(1|2)$ 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 $\sqrt {-1}$.References
- A. A. Beฤญlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astรฉrisque, vol. 100, Soc. Math. France, Paris, 1982, pp.ย 5โ171 (French). MR 751966
- A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no.ย 2, 655โ677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
- Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527, DOI 10.1007/BFb0073549
- Tom Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), no.ย 3, 209โ216. MR 1996415, DOI 10.1007/s00031-003-0606-4
- Sean Clark and Weiqiang Wang, Canonical basis for quantum $\mathfrak {osp}(1|2)$, Lett. Math. Phys. 103 (2013), no.ย 2, 207โ231. MR 3010460, DOI 10.1007/s11005-012-0592-3
- Sean Clark, Zhaobing Fan, Yiqiang Li, and Weiqiang Wang, Quantum supergroups III. Twistors, Comm. Math. Phys. 332 (2014), no.ย 1, 415โ436. MR 3253707, DOI 10.1007/s00220-014-2071-4
- Jie Du, A note on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995), no.ย 4, 363โ372. MR 1358684
- Alexander P. Ellis, Mikhail Khovanov, and Aaron D. Lauda, The odd nilHecke algebra and its diagrammatics, Int. Math. Res. Not. IMRN 4 (2014), 991โ1062. MR 3168401, DOI 10.1093/imrn/rns240
- Alexander P. Ellis and Aaron D. 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, Int. Math. Res. Not. 2014, DOI 10.1093/imrn/rnu159.
- I. Grojnowski and G. Lusztig, On bases of irreducible representations of quantum $\textrm {GL}_n$, Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp.ย 167โ174. MR 1197834, DOI 10.1090/conm/139/1197834
- David Hill and Weiqiang Wang, Categorification of quantum Kac-Moody superalgebras, Trans. Amer. Math. Soc. 367 (2015), no.ย 2, 1183โ1216. MR 3280041, DOI 10.1090/S0002-9947-2014-06128-X
- Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh, Supercategorification of quantum Kac-Moody algebras, Adv. Math. 242 (2013), 116โ162. MR 3055990, DOI 10.1016/j.aim.2013.04.008
- Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh, Supercategorification of quantum Kac-Moody algebras II, Adv. Math. 265 (2014), 169โ240. MR 3255459, DOI 10.1016/j.aim.2014.07.036
- 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.ย 2, 465โ516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
- Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309โ347. MR 2525917, DOI 10.1090/S1088-4165-09-00346-X
- Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), no.ย 5, 2685โ2700. MR 2763732, DOI 10.1090/S0002-9947-2010-05210-9
- Emmanuel Lanzmann, The Zhang transformation and $\scr U_q(\textrm {osp}(1,2l))$-Verma modules annihilators, Algebr. Represent. Theory 5 (2002), no.ย 3, 235โ258. MR 1921760, DOI 10.1023/A:1016550528593
- Aaron D. Lauda, A categorification of quantum $\textrm {sl}(2)$, Adv. Math. 225 (2010), no.ย 6, 3327โ3424. MR 2729010, DOI 10.1016/j.aim.2010.06.003
- Y. Li, A geometric realization of modified quantum algebras, arXiv:1007.5384.
- G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no.ย 2, 447โ498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
- G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no.ย 2, 365โ421. MR 1088333, DOI 10.1090/S0894-0347-1991-1088333-2
- G. Lusztig, Canonical bases in tensor products, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no.ย 17, 8177โ8179. MR 1180036, DOI 10.1073/pnas.89.17.8177
- George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhรคuser Boston, Inc., Boston, MA, 1993. MR 1227098
- Raphaรซl 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.ย 137โ167. MR 2258045, DOI 10.1090/conm/406/07657
- R. Rouquier, 2-Kac-Moody algebras, arXiv:0812.5023.
- M. Varagnolo and E. Vasserot, Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67โ100. MR 2837011, DOI 10.1515/CRELLE.2011.068
- Weiqiang Wang, Double affine Hecke algebras for the spin symmetric group, Math. Res. Lett. 16 (2009), no.ย 6, 1071โ1085. MR 2576694, DOI 10.4310/MRL.2009.v16.n6.a14
- H. Zheng, A geometric categorification of representations of $U_q(\textrm {sl}_2)$, Topology and physics, Nankai Tracts Math., vol. 12, World Sci. Publ., Hackensack, NJ, 2008, pp.ย 348โ356. MR 2503405, DOI 10.1142/9789812819116_{0}016
- Yi Ming Zou, Integrable representations of $U_q(\textrm {osp}(1,2n))$, J. Pure Appl. Algebra 130 (1998), no.ย 1, 99โ112. MR 1632799, DOI 10.1016/S0022-4049(97)00088-1
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
- MR Author ID: 684558
- 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
- MR Author ID: 828279
- ORCID: 0000-0003-4608-3465
- Email: yiqiang@buffalo.edu
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3391903