Categorification of quantum Kac-Moody superalgebras

Hill, David; Wang, Weiqiang

doi:10.1090/S0002-9947-2014-06128-X

Categorification of quantum Kac-Moody superalgebras
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by David Hill and Weiqiang Wang PDF

Trans. Amer. Math. Soc. 367 (2015), 1183-1216 Request permission

Abstract:

We introduce a non-degenerate bilinear form and use it to provide a new characterization of quantum Kac-Moody superalgebras of anisotropic type. We show that the spin quiver Hecke algebras introduced by Kang, Kashiwara and Tsuchioka provide a categorification of half the quantum Kac-Moody superalgebras, using the recent work of Ellis-Khovanov-Lauda. A new idea here is that a supersign is categorified as spin (i.e., the parity-shift functor).

References

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
Jonathan Brundan and Alexander Kleshchev, Hecke-Clifford superalgebras, crystals of type $A_{2l}^{(2)}$ and modular branching rules for $\hat S_n$, Represent. Theory 5 (2001), 317–403. MR 1870595, DOI 10.1090/S1088-4165-01-00123-6
Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009), no. 3, 451–484. MR 2551762, DOI 10.1007/s00222-009-0204-8
Jonathan Brundan, Alexander Kleshchev, and Weiqiang Wang, Graded Specht modules, J. Reine Angew. Math. 655 (2011), 61–87. MR 2806105, DOI 10.1515/CRELLE.2011.033
Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov’s diagram algebra III: category $\scr O$, Represent. Theory 15 (2011), 170–243. MR 2781018, DOI 10.1090/S1088-4165-2011-00389-7
Sabin Cautis, Joel Kamnitzer, and Anthony Licata, Coherent sheaves and categorical $\mathfrak {sl}_2$ actions, Duke Math. J. 154 (2010), no. 1, 135–179. MR 2668555, DOI 10.1215/00127094-2010-035
Joseph Chuang and Raphaël Rouquier, Derived equivalences for symmetric groups and $\mathfrak {sl}_2$-categorification, Ann. of Math. (2) 167 (2008), no. 1, 245–298. MR 2373155, DOI 10.4007/annals.2008.167.245
Sean Clark, David Hill, and Weiqiang Wang, Quantum supergroups II. Canonical basis, Represent. Theory 18 (2014), 278–309. MR 3256709, DOI 10.1090/S1088-4165-2014-00453-9
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
Nathan Geer, Etingof-Kazhdan quantization of Lie superbialgebras, Adv. Math. 207 (2006), no. 1, 1–38. MR 2264064, DOI 10.1016/j.aim.2005.11.005
David Hill and Joshua Sussan, The Khovanov-Lauda 2-category and categorifications of a level two quantum $\mathfrak {sl}_n$ representation, Int. J. Math. Math. Sci. , posted on (2010), Art. ID 892387, 34. MR 2669068, DOI 10.1155/2010/892387
Jun Hu and Andrew Mathas, Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type $A$, Adv. Math. 225 (2010), no. 2, 598–642. MR 2671176, DOI 10.1016/j.aim.2010.03.002
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
Seok-Jin Kang and Masaki Kashiwara, Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras, Invent. Math. 190 (2012), no. 3, 699–742. MR 2995184, DOI 10.1007/s00222-012-0388-1
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.
Seok-Jin Kang, Se-Jin Oh, and Euiyong Park, Categorification of quantum generalized Kac-Moody algebras and crystal bases, Internat. J. Math. 23 (2012), no. 11, 1250116, 51. MR 3005569, DOI 10.1142/S0129167X12501169
T. A. Khongsap and Weiqiang Wang, Hecke-Clifford algebras and spin Hecke algebras. I. The classical affine type, Transform. Groups 13 (2008), no. 2, 389–412. MR 2426136, DOI 10.1007/s00031-008-9012-2
Mikhail Khovanov, Categorifications from planar diagrammatics, Jpn. J. Math. 5 (2010), no. 2, 153–181. MR 2747932, DOI 10.1007/s11537-010-0925-x
M. Khovanov, How to categorify one-half of quantum $gl(1|2)$, arXiv:1007.3517.
Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309–347. 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. 5, 2685–2700. MR 2763732, DOI 10.1090/S0002-9947-2010-05210-9
Alexander Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics, vol. 163, Cambridge University Press, Cambridge, 2005. MR 2165457, DOI 10.1017/CBO9780511542800
Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 3, 419–481. MR 2651085, DOI 10.1090/S0273-0979-09-01277-4
Aaron D. Lauda, A categorification of quantum $\textrm {sl}(2)$, Adv. Math. 225 (2010), no. 6, 3327–3424. MR 2729010, DOI 10.1016/j.aim.2010.06.003
Aaron D. Lauda and Monica Vazirani, Crystals from categorified quantum groups, Adv. Math. 228 (2011), no. 2, 803–861. MR 2822211, DOI 10.1016/j.aim.2011.06.009
Johan W. van de Leur, A classification of contragredient Lie superalgebras of finite growth, Comm. Algebra 17 (1989), no. 8, 1815–1841. MR 1013470, DOI 10.1080/00927878908823823
G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1988), no. 2, 237–249. MR 954661, DOI 10.1016/0001-8708(88)90056-4
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
Maxim Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), no. 2, 190–257. MR 1448714, DOI 10.1006/aima.1997.1621
R. Rouquier, $2$-Kac-Moody algebras, arxiv:0812.5023.
Raphaël Rouquier, Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq. 19 (2012), no. 2, 359–410. MR 2908731, DOI 10.1142/S1005386712000247
Shunsuke Tsuchioka, Hecke-Clifford superalgebras and crystals of type $D^{(2)}_l$, Publ. Res. Inst. Math. Sci. 46 (2010), no. 2, 423–471. MR 2722783, DOI 10.2977/PRIMS/13
M. Varagnolo and E. Vasserot, Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67–100. 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. 6, 1071–1085. MR 2576694, DOI 10.4310/MRL.2009.v16.n6.a14
B. Webster, Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products, arXiv:1001.2020.
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