𝚤Hall algebra of the projective line and 𝑞-Onsager algebra

Lu, Ming; Ruan, Shiquan; Wang, Weiqiang

doi:10.1090/tran/8798

$\imath$Hall algebra of the projective line and $q$-Onsager algebra
HTML articles powered by AMS MathViewer

by Ming Lu, Shiquan Ruan and Weiqiang Wang HTML | PDF

Trans. Amer. Math. Soc. 376 (2023), 1475-1505 Request permission

Abstract:

The $\imath$Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the projective line. This $\imath$Hall algebra is shown to realize the universal $q$-Onsager algebra (i.e., $\imath$quantum group of split affine $A_1$ type) in its Drinfeld type presentation. The $\imath$Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $q$-Onsager algebra under the two presentations.

References

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
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
A. A. Beĭlinson, Coherent sheaves on $\textbf {P}^{n}$ and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69 (Russian). MR 509388
I. N. Bernšteĭn, I. M. Gel′fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33 (Russian). MR 0393065
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, 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
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
A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121–138 (French). MR 87176, DOI 10.2307/2372388
Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
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
Bernhard Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551–581. MR 2184464
Stefan Kolb, Quantum symmetric Kac-Moody pairs, Adv. Math. 267 (2014), 395–469. MR 3269184, DOI 10.1016/j.aim.2014.08.010
Gail Letzter, Coideal subalgebras and quantum symmetric pairs, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 117–165. MR 1913438
J. Lin and L. Peng, Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories, submitted.
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
M. Lu, S. Ruan and W. Wang, $\imath$Hall algebra of Jordan quiver and $\imath$Hall-Littlewood functions, arXiv:2104.12336.
M. Lu and S. Ruan, $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs, arXiv:2110.02575.
M. Lu and W. Wang, Hall algebras and quantum symmetric pairs of Kac-Moody type, arXiv:2006.06904
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
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