Czechoslovak Mathematical Journal, Vol. 71, No. 1, pp. 253-267, 2021


Drinfeld doubles via derived Hall algebras and Bridgeland's Hall algebras

Fan Xu, Haicheng Zhang

Received July 16, 2019.   Published online October 20, 2020.

Abstract:  Let ${\cal A}$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of ${\cal A}$ via its derived Hall algebra and Bridgeland's Hall algebra of $m$-cyclic complexes.
Keywords:  Heisenberg double; Drinfeld double; derived Hall algebra; Bridgeland's Hall algebra
Classification MSC:  16G20, 17B20, 17B37
DOI:  10.21136/CMJ.2020.0313-19

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References:
[1] T. Bridgeland: Quantum groups via Hall algebras of complexes. Ann. Math. (2) 177 (2013), 739-759. DOI 10.4007/annals.2013.177.2.9 | MR 3010811 | Zbl 1268.16017
[2] Q. Chen, B. Deng: Cyclic complexes, Hall polynomials and simple Lie algebras. J. Algebra 440 (2015), 1-32. DOI 10.1016/j.jalgebra.2015.04.043 | MR 3373385 | Zbl 1328.16007
[3] J. A. Green: Hall algebras, hereditary algebras and quantum groups. Invent. Math. 120 (1995), 361-377. DOI 10.1007/BF01241133 | MR 1329046 | Zbl 0836.16021
[4] M. Kapranov: Eisenstein series and quantum affine algebras. J. Math. Sci., New York 84 (1997), 1311-1360. DOI 10.1007/BF02399194 | MR 1465518 | Zbl 0929.11015
[5] M. Kapranov: Heisenberg doubles and derived categories. J. Algebra 202 (1998), 712-744. DOI 10.1006/jabr.1997.7323 | MR 1617651 | Zbl 0910.18005
[6] R. M. Kashaev: The Heisenberg double and the pentagon relation. St. Petersbg. Math. J. 8 (1997), 585-592; translation from Algebra Anal. 8 (1996), 63-74. MR 1418255 | Zbl 0870.16023
[7] J. Lin: Modified Ringel-Hall algebras, naive lattice algebras and lattice algebras. Sci. China, Math. 63 (2020), 671-688. DOI 10.1007/s11425-017-9377-8 | MR 4079890 | Zbl 07184422
[8] C. M. Ringel: Hall algebras. Topics in Algebra. Part 1. Rings and Representations of Algebras. Banach Center Publications 26. PWN, Warsaw (1990), 433-447. DOI 10.4064/-26-1-433-447 | MR 1171248 | Zbl 0778.16004
[9] C. M. Ringel: Hall algebras and quantum groups. Invent. Math. 101 (1990), 583-591. DOI 10.1007/BF01231516 | MR 1062796 | Zbl 0735.16009
[10] O. Schiffmann: Lectures on Hall algebras. Geometric Methods in Representation Theory II. Société Mathématique de France, Paris (2012), 1-141. MR 3202707 | Zbl 1309.18012
[11] J. Sheng, F. Xu: Derived Hall algebras and lattice algebras. Algebra Colloq. 19 (2012), 533-538. DOI 10.1142/S1005386712000399 | MR 2999262 | Zbl 1250.18013
[12] B. Toën: Derived Hall algebras. Duke Math. J. 135 (2006), 587-615. DOI 10.1215/s0012-7094-06-13536-6 | MR 2272977 | Zbl 1117.18011
[13] J. Xiao: Drinfeld double and Ringel-Green theory of Hall algebras. J. Algebra 190 (1997), 100-144. DOI 10.1006/jabr.1996.6887 | MR 1442148 | Zbl 0874.16026
[14] J. Xiao, F. Xu: Hall algebras associated to triangulated categories. Duke Math. J. 143 (2008), 357-373. DOI 10.1215/00127094-2008-021 | MR 2420510 | Zbl 1168.18006
[15] S. Yanagida: A note on Bridgeland's Hall algebra of two-periodic complexes. Math. Z. 282 (2016), 973-991. DOI 10.1007/s00209-015-1573-x | MR 3473652 | Zbl 1403.16011
[16] H. Zhang: A note on Bridgeland Hall algebras. Commun. Algebra 46 (2018), 2551-2560. DOI 10.1080/00927872.2017.1388812 | MR 3778411 | Zbl 06900764
[17] H. Zhang: Bridgeland's Hall algebras and Heisenberg doubles. J. Algebra Appl. 17 (2018), Article ID 1850104, 12 pages. DOI 10.1142/S0219498818501049 | MR 3805715 | Zbl 1391.16016

Affiliations:   Fan Xu, Department of Mathematical Sciences, Tsinghua University, Zhongguancun N St, Haidian District, Beijing 100084, P. R. China, e-mail: fanxu@mail.tsinghua.edu.cn; Haicheng Zhang (correspondig author), Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, No.1 Wenyuan Road, Qixia District, Nanjing 210023, P. R. China, e-mail: zhanghc@njnu.edu.cn


 
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