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Quantum supergroups II. Canonical basis

Authors: Sean Clark, David Hill and Weiqiang Wang
Journal: Represent. Theory 18 (2014), 278-309
MSC (2010): Primary 17B37
Published electronically: September 9, 2014
MathSciNet review: 3256709
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Abstract: Following Kashiwara's algebraic approach, we construct crystal bases and canonical bases for quantum supergroups of anisotropic type and for their integrable modules.

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  • [BKK] 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 (2000m:17015),
  • [BKM] 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 (99f:17014),
  • [CFLW] S. Clark, Z. Fan, Y. Li and W.  Wang, Quantum supergroups III. Twistors, Commun. Math. Phys. 322 (2014), 415-436. MR 3253707
  • [CHW] Sean Clark, David Hill, and Weiqiang Wang, Quantum supergroups I. Foundations, Transform. Groups 18 (2013), no. 4, 1019-1053. MR 3127986,
  • [CW] 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,
  • [EKL] 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
  • [ELa] A. Ellis and A. Lauda, An odd categorification of $ U_q(\mathfrak{sl}_2)$, arXiv:1307.7816.
  • [FLi] Z. Fan and Y. Li, Two-parameter quantum algebras, canonical bases and categorifications, IMRN (to appear), arXiv:1303.2429.
  • [HW] D.  Hill and W.  Wang, Categorification of quantum Kac-Moody superalgebras, Trans. Amer. Math. Soc. (to appear), arXiv:1202.2769v2.
  • [Jeo] Kyeonghoon Jeong, Crystal bases for Kac-Moody superalgebras, J. Algebra 237 (2001), no. 2, 562-590. MR 1816704 (2001m:17016),
  • [KKO] Seok-Jin Kang, Masaki Kashiwara, and Se-jin Oh, Supercategorification of quantum Kac-Moody algebras, Adv. Math. 242 (2013), 116-162. MR 3055990,
  • [KKT] S.-J. Kang, M. Kashiwara and S. Tsuchioka, Quiver Hecke superalgebras, arXiv:1107.1039.
  • [Ka] 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),
  • [Kac] 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 (83a:17014a),
  • [Kw] J.-H. Kwon, Super duality and crystal bases for quantum orthosymplectic superalgebras, arXiv:1301.1756.
  • [Lu1] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. MR 1035415 (90m:17023),
  • [Lu2] George Lusztig, Introduction to Quantum Groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098 (94m:17016)
  • [MZ] Ian M. Musson and Yi Ming Zou, Crystal bases for $ U_q({\rm osp}(1,2r))$, J. Algebra 210 (1998), no. 2, 514-534. MR 1662280 (99j:17024),
  • [Wa] Weiqiang Wang, Double affine Hecke algebras for the spin symmetric group, Math. Res. Lett. 16 (2009), no. 6, 1071-1085. MR 2576694 (2011a:20009),
  • [Ya] 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 (95d:17017),
  • [Zou] 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),

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Additional Information

Sean Clark
Affiliation: Department of Mathematics, 567 Lake Hall, Northeastern University, Boston, Massachusetts 02115
Email: se.clark@neu.ed

David Hill
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Weiqiang Wang
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Keywords: Quantum supergroups, crystal basis, canonical basis
Received by editor(s): April 16, 2013
Received by editor(s) in revised form: March 14, 2014
Published electronically: September 9, 2014
Additional Notes: The first author was partially supported by the Semester Fellowship from Department of Mathematics, University of Virginia (UVA)
The first and third authors gratefully acknowledge the support and stimulating environment at the Institute of Mathematics, Academia Sinica, Taipei, during their visits in Spring 2013
The third author was partially supported by NSF DMS-1101268 and the UVA Sesqui Fellowship
Article copyright: © Copyright 2014 American Mathematical Society

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