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Hall Algebras Associated to Complexes of Fixed Size

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Abstract

Let \(\cal{A}\) be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes of fixed size over projectives. Explicitly, we first give a relation between Hall algebras of complexes of fixed size and cyclic complexes. Second, we characterize the Hall algebra of complexes of fixed size by generators and relations, and relate it to the derived Hall algebra of \(\cal{A}\). Finally, we give the integration map on the Hall algebra of 2-term complexes over projectives.

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References

  1. Bautista, R.: The category of morphisms between projective modules. Comm. Algebra, 32(11), 4303–4331 (2004)

    Article  MathSciNet  Google Scholar 

  2. Bautista, R., Souto-Salorio, M., Zuazua, R.: Almost split sequences for complexes of fixed size. J. Algebra, 287, 140–168 (2005)

    Article  MathSciNet  Google Scholar 

  3. Berenstein, A., Zelevinsky, A.: Quantum cluster algebras. Adv. Math., 195(2), 405–455 (2005)

    Article  MathSciNet  Google Scholar 

  4. Bridgeland, T.: Quantum groups via Hall algebras of complexes. Ann. Math., 177, 1–21 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bühler, T.: Exact Categories. Expo. Math., 28(1), 1–69 (2010)

    Article  MathSciNet  Google Scholar 

  6. Chaio, C., Pratti, I., Souto-Salorio, M.: On sectional paths in a category of complexes of fixed size. Algebr. Represent. Theor., 20, 289–311 (2017)

    Article  MathSciNet  Google Scholar 

  7. Chaio, C., Chaio, G., Pratti, I.: On the type of a category of complexes of fixed size and the strong global dimension. Algebr. Represent. Theor., 22, 1343–1370 (2019)

    Article  MathSciNet  Google Scholar 

  8. Chen, Q., Deng, B.: Cyclic complexes, Hall polynomials and simple Lie algebras. J. Algebra, 440, 1–32 (2015)

    Article  MathSciNet  Google Scholar 

  9. Ding, M., Xu, F., Zhang, H.: Acyclic quantum cluster algebras via Hall algebras of morphisms. Math. Z., 296(3–4), 945–968 (2020)

    Article  MathSciNet  Google Scholar 

  10. Fomin, S., Zelevinsky, A.: Cluster algebras I: Foundations. J. Amer. Math. Soc., 15(2), 497–529 (2002)

    Article  MathSciNet  Google Scholar 

  11. Fu, C., Peng, L., Zhang, H.: Quantum cluster characters of Hall algebras revisited. arXiv:2005.10617

  12. Gorsky, M.: Semi-derived and derived Hall algebras for stable categories. Int. Math. Res. Notices, 2018(1), 138–159 (2018)

    Article  MathSciNet  Google Scholar 

  13. Hubery, A.: From triangulated categories to Lie Algebras: A theorem of Peng and Xiao. In: Trends in Representation Theory of Algebras and Related Topics, Contemp. Math., Vol. 406, Amer. Math. Soc., Providence, 2006, 51–66

    Chapter  Google Scholar 

  14. Lin, J., Peng, L.: Modified Ringel—Hall algebras, Green’s formula and derived Hall algebras. J. Algebra, 526, 81–103 (2019)

    Article  MathSciNet  Google Scholar 

  15. Peng, L.: Some Hall polynomials for representation-finite trivial extension algebras. J. Algebra, 197, 1–13 (1997)

    Article  MathSciNet  Google Scholar 

  16. Reineke, M.: The Harder—Narasimhan system in quantum groups and cohomology of quiver moduli. Invent. Math., 152(2), 349–368 (2003)

    Article  MathSciNet  Google Scholar 

  17. Riedtmann, C.: Lie algebras generated by indecomposables. J. Algebra, 170(2), 526–546 (1994)

    Article  MathSciNet  Google Scholar 

  18. Ringel, C. M.: Hall algebras. In: S. Balcerzyk, et al. (eds.), Topics in Algebra, Part 1, Banach Center Publ., Vol. 26, 1990, 433–447

  19. Ringel, C. M.: Hall algebras and quantum groups. Invent. Math., 101, 583–592 (1990)

    Article  MathSciNet  Google Scholar 

  20. Schiffmann, O.: Lectures on Hall algebras. In: Geometric methods in representation theory II, Sémin. Congr., Vol. 24-II, Soc. Math. France, Paris, 2012, 1–141

    Google Scholar 

  21. Toën, B.: Derived Hall algebras. Duke Math. J., 135, 587–615 (2006)

    Article  MathSciNet  Google Scholar 

  22. Xiao, J., Xu, F.: Hall algebras associated to triangulated categories. Duke Math. J., 143(2), 357–373 (2008)

    Article  MathSciNet  Google Scholar 

  23. Zhang, H.: Bridgeland’s Hall algebras and Heisenberg doubles. J. Algebra Appl., 17(6), 1850104, 12 pp. (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author is grateful to Shiquan Ruan for his suggestion to relate Hall algebra of m-cyclic complexes with Hall algebra of m-term complexes. He also would like to thank Changjian Fu and Fan Xu for the conversations on the paper, and thank the anonymous referees for their careful reading and helpful comments.

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  1. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, P. R. China

    Hai Cheng Zhang

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  1. Hai Cheng Zhang
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Correspondence to Hai Cheng Zhang.

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Supported by the National Natural Science Foundation of China (Grant No. 11801273) and Natural Science Foundation of Jiangsu Province of China (Grant No. BK20180722)

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Zhang, H.C. Hall Algebras Associated to Complexes of Fixed Size. Acta. Math. Sin.-English Ser. 38, 907–923 (2022). https://doi.org/10.1007/s10114-022-1057-y

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  • DOI: https://doi.org/10.1007/s10114-022-1057-y

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