$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs
HTML articles powered by AMS MathViewer
- by Ming Lu and Shiquan Ruan;
- Represent. Theory 28 (2024), 112-188
- DOI: https://doi.org/10.1090/ert/669
- Published electronically: March 4, 2024
- HTML | PDF | Request permission
Abstract:
The $\imath$Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The $\imath$Hall algebra of the $\imath$quiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $\imath$quantum group of split affine type A under the two presentations.References
- Martina Balagović and Stefan Kolb, Universal K-matrix for quantum symmetric pairs, J. Reine Angew. Math. 747 (2019), 299–353. MR 3905136, DOI 10.1515/crelle-2016-0012
- Huanchen Bao and Weiqiang Wang, A new approach to Kazhdan-Lusztig theory of type $B$ via quantum symmetric pairs, Astérisque 402 (2018), vii+134 (English, with English and French summaries). MR 3864017
- Huanchen Bao and Weiqiang Wang, Canonical bases arising from quantum symmetric pairs, Invent. Math. 213 (2018), no. 3, 1099–1177. MR 3842062, DOI 10.1007/s00222-018-0801-5
- Pascal Baseilhac and Stefan Kolb, Braid group action and root vectors for the $q$-Onsager algebra, Transform. Groups 25 (2020), no. 2, 363–389. MR 4098883, DOI 10.1007/s00031-020-09555-7
- Pierre Baumann and Christian Kassel, The Hall algebra of the category of coherent sheaves on the projective line, J. Reine Angew. Math. 533 (2001), 207–233. MR 1823869, DOI 10.1515/crll.2001.031
- Jonathan Beck, Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), no. 3, 555–568. MR 1301623, DOI 10.1007/BF02099423
- Tom Bridgeland, Quantum groups via Hall algebras of complexes, Ann. of Math. (2) 177 (2013), no. 2, 739–759. MR 3010811, DOI 10.4007/annals.2013.177.2.9
- Igor Burban and Olivier Schiffmann, On the Hall algebra of an elliptic curve, I, Duke Math. J. 161 (2012), no. 7, 1171–1231. MR 2922373, DOI 10.1215/00127094-1593263
- Igor Burban and Olivier Schiffmann, Two descriptions of the quantum affine algebra $U_v(\widehat {\mathfrak {sl}}_2)$ via Hall algebra approach, Glasg. Math. J. 54 (2012), no. 2, 283–307. MR 2911369, DOI 10.1017/S0017089511000607
- Igor Burban and Olivier Schiffmann, The composition Hall algebra of a weighted projective line, J. Reine Angew. Math. 679 (2013), 75–124. MR 3065155, DOI 10.1515/crelle.2012.023
- Tim Cramer, Double Hall algebras and derived equivalences, Adv. Math. 224 (2010), no. 3, 1097–1120. MR 2628805, DOI 10.1016/j.aim.2009.12.021
- Ilaria Damiani, From the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras: injectivity, Publ. Res. Inst. Math. Sci. 51 (2015), no. 1, 131–171. MR 3367090, DOI 10.4171/PRIMS/150
- Rujing Dou, Yong Jiang, and Jie Xiao, The Hall algebra approach to Drinfeld’s presentation of quantum loop algebras, Adv. Math. 231 (2012), no. 5, 2593–2625. MR 2970461, DOI 10.1016/j.aim.2012.07.026
- V. G. Drinfel′d, A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13–17 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 2, 212–216. MR 914215
- Howard Garland, The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 5–136. MR 601519, DOI 10.1007/BF02684779
- Werner Geigle and Helmut Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 265–297. MR 915180, DOI 10.1007/BFb0078849
- Mikhail Gorsky, Semi-derived and derived Hall algebras for stable categories, Int. Math. Res. Not. IMRN 1 (2018), 138–159. MR 3800631, DOI 10.1093/imrn/rnv325
- James A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), no. 2, 361–377. MR 1329046, DOI 10.1007/BF01241133
- M. M. Kapranov, Eisenstein series and quantum affine algebras, J. Math. Sci. (New York) 84 (1997), no. 5, 1311–1360. Algebraic geometry, 7. MR 1465518, DOI 10.1007/BF02399194
- Stefan Kolb, Quantum symmetric Kac-Moody pairs, Adv. Math. 267 (2014), 395–469. MR 3269184, DOI 10.1016/j.aim.2014.08.010
- Stefan Kolb and Jacopo Pellegrini, Braid group actions on coideal subalgebras of quantized enveloping algebras, J. Algebra 336 (2011), 395–416. MR 2802552, DOI 10.1016/j.jalgebra.2011.04.001
- Bernhard Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551–581. MR 2184464, DOI 10.4171/dm/199
- Gail Letzter, Symmetric pairs for quantized enveloping algebras, J. Algebra 220 (1999), no. 2, 729–767. MR 1717368, DOI 10.1006/jabr.1999.8015
- Ming Lu and Liangang Peng, Semi-derived Ringel-Hall algebras and Drinfeld double, Adv. Math. 383 (2021), Paper No. 107668, 72. MR 4232541, DOI 10.1016/j.aim.2021.107668
- Ming Lu, Shiquan Ruan, and Weiqiang Wang, $\imath$Hall algebra of the projective line and $q$-Onsager algebra, Trans. Amer. Math. Soc. 376 (2023), no. 2, 1475–1505. MR 4531682, DOI 10.1090/tran/8798
- Ming Lu and Weiqiang Wang, Hall algebras and quantum symmetric pairs II: Reflection functors, Comm. Math. Phys. 381 (2021), no. 3, 799–855. MR 4218672, DOI 10.1007/s00220-021-03965-8
- Ming Lu and Weiqiang Wang, A Drinfeld type presentation of affine $\imath$quantum groups I: Split ADE type, Adv. Math. 393 (2021), Paper No. 108111, 46. MR 4340233, DOI 10.1016/j.aim.2021.108111
- Ming Lu and Weiqiang Wang, Hall algebras and quantum symmetric pairs I: Foundations, Proc. Lond. Math. Soc. (3) 124 (2022), no. 1, 1–82. With an appendix by Lu. MR 4389313, DOI 10.1112/plms.12423
- Ming Lu and Weiqiang Wang, Braid group symmetries on quasi-split $\imath$quantum groups via $\imath$Hall algebras, Selecta Math. (N.S.) 28 (2022), no. 5, Paper No. 84, 64. MR 4486230, DOI 10.1007/s00029-022-00800-3
- Ming Lu and Weiqiang Wang, Hall algebras and quantum symmetric pairs of Kac-Moody type, Adv. Math. 430 (2023), Paper No. 109215, 56. MR 4619448, DOI 10.1016/j.aim.2023.109215
- 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
- Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
- Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. MR 1062796, DOI 10.1007/BF01231516
- Olivier Schiffmann, Noncommutative projective curves and quantum loop algebras, Duke Math. J. 121 (2004), no. 1, 113–168. MR 2031167, DOI 10.1215/S0012-7094-04-12114-1
- Olivier Schiffmann, Lectures on Hall algebras, Geometric methods in representation theory. II, Sémin. Congr., vol. 24, Soc. Math. France, Paris, 2012, pp. 1–141 (English, with English and French summaries). MR 3202707
- Torkil Stai, The triangulated hull of periodic complexes, Math. Res. Lett. 25 (2018), no. 1, 199–236. MR 3818620, DOI 10.4310/mrl.2018.v25.n1.a9
- Weiqiang Wang, Quantum symmetric pairs, ICM—International Congress of Mathematicians. Vol. 4. Sections 5–8, EMS Press, Berlin, [2023] ©2023, pp. 3080–3102. MR 4680353
Bibliographic Information
- Ming Lu
- Affiliation: Department of Mathematics, Sichuan University, Chengdu 610064, People’s Republic of China
- Email: luming@scu.edu.cn
- Shiquan Ruan
- Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen 361005, People’s Republic of China
- MR Author ID: 1040853
- Email: sqruan@xmu.edu.cn
- Received by editor(s): July 2, 2023
- Received by editor(s) in revised form: December 18, 2023, and January 19, 2024
- Published electronically: March 4, 2024
- Additional Notes: The first author was partially supported by the National Natural Science Foundation of China (No. 12171333, 12261131498). The first author was supported by the University of Virginia. The second author was partially supported by the National Natural Science Foundation of China (No. 12271448) and the Fundamental Research Funds for Central Universities of China (No. 20720210006).
- © Copyright 2024 American Mathematical Society
- Journal: Represent. Theory 28 (2024), 112-188
- MSC (2020): Primary 17B37, 14A22, 16E60, 18G80
- DOI: https://doi.org/10.1090/ert/669
- MathSciNet review: 4712701