Skip to main content
SpringerLink
Log in

Hall Algebras and Quantum Symmetric Pairs II: Reflection Functors

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Recently the authors initiated an \(\imath \)Hall algebra approach to (universal) \(\imath \)quantum groups arising from quantum symmetric pairs. In this paper we construct and study BGP type reflection functors which lead to isomorphisms of the \(\imath \)Hall algebras associated to acyclic \(\imath \)quivers. For Dynkin quivers, these symmetries on \(\imath \)Hall algebras induce automorphisms of universal \(\imath \)quantum groups, which are shown to satisfy the braid group relations associated to the restricted Weyl group of a symmetric pair. This leads to a conceptual construction of q-root vectors and PBW bases for (universal) quasi-split \(\imath \)quantum groups of ADE type.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Auslander, M., Platzeck, M., Reiten, I.: Coxeter functors without diagrams. Trans. Am. Math. Soc. 250, 1–46 (1979)

    Article  MathSciNet  Google Scholar 

  2. Assem, I., Simson, D., Skowroński, A.: Elements of representation theory of associative algebras, Volume 1, Techniques of Representation Theory. London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge, New York (2004)

  3. Bao, H., Wang, W.: A new approach to Kazhdan-Lusztig theory of type \(B\) via quantum symmetric pairs. Astérisque 402, vii+134 (2018). arXiv:1310.0103v2

    MathSciNet  MATH  Google Scholar 

  4. Bao, H., Wang, W.: Canonical bases arising from quantum symmetric pairs. Invent. Math. 213, 1099–1177 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  5. Baseilhac, P., Belliard, S.: The half-infinite XXZ chain in Onsager’s approach. Nucl. Phys. B 873, 550–583 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  6. Baseilhac, P., Kolb, S.: Braid group action and root vectors for the \(q\)-Onsager algebra. Transform. Groups 25, 363–389 (2020)

    Article  MathSciNet  Google Scholar 

  7. Bernstein, I.N., Gelfand, I.M., Ponomarev, V.A.: Coxeter functors, and Gabriel’s theorem. Uspehi. Mat. Nauk 28(2 (170)), 19–33 (1973)

    MathSciNet  Google Scholar 

  8. Bongartz, K., Gabriel, P.: Covering spaces in representation theory. Invent. Math. 65, 331–378 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  9. Bridgeland, T.: Quantum groups via Hall algebras of complexes. Ann. Math. 177, 739–759 (2013)

    Article  MathSciNet  Google Scholar 

  10. Butler, M.C.R., Ringel, C.M.: Auslander–Reiten sequences with few middle terms and applications to string algebras. Commun. Algebra 15(1–2), 145–179 (1987)

    Article  MathSciNet  Google Scholar 

  11. Chekhov, L.O.: Teichmüller theory of bordered surfaces. SIGMA Symmetry Integrability Geom. Methods Appl. 3, Paper 066, 37 (2007)

  12. Chen, X., Lu, M., Wang, W.: A Serre presentation of \(\imath \)quantum groups. Transform. Groups (to appear). https://doi.org/10.1007/s00031-020-09581-5, arXiv:1810.12475

  13. Deng, B., Du, J., Parshall, B., Wang, J.: Finite Dimensional Algebras and Quantum Groups, Mathematical Surveys and Monographs, vol. 150. AMS, Providence (2008)

    Book  Google Scholar 

  14. Dlab, V., Ringel, C.M.: Representations of graphs and algebras. Carleton Mathematical Lecture Notes, No. 8. Department of Mathematics, Carleton University, Ottawa (1974)

  15. Dobson, L.: Braid group actions for quantum symmetric pairs of type AIII/AIV. J. Algebra 564, 151–198 (2020)

    Article  MathSciNet  Google Scholar 

  16. Geiss, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices I: foundations. Invent. Math. 209, 61–158 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  17. Gorsky, M.: Semi-derived Hall algebras and tilting invariance of Bridgeland–Hall algebras, arXiv:1303.5879v2

  18. Gorsky, M.: Semi-derived and derived Hall algebras for stable categories. IMRN 1, 138–159 (2018). arXiv:1409.6798

    Article  MathSciNet  Google Scholar 

  19. Green, J.A.: Hall algebras, hereditary algebras and quantum groups. Invent. Math. 120, 361–377 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  20. Happel, D., Ringel, C.M.: Tilted algebras. Trans. Am. Math. Soc. 274(2), 399–443 (1982)

    Article  MathSciNet  Google Scholar 

  21. Kolb, S.: Quantum symmetric Kac–Moody pairs. Adv. Math. 267, 395–469 (2014)

    Article  MathSciNet  Google Scholar 

  22. Kolb, S., Pellegrini, J.: Braid group actions on coideal subalgebras of quantized enveloping algebras. J. Algebra 336, 395–416 (2011)

    Article  MathSciNet  Google Scholar 

  23. Letzter, G.: Symmetric pairs for quantized enveloping algebras. J. Algebra 220, 729–767 (1999)

    Article  MathSciNet  Google Scholar 

  24. Letzter, G.: Coideal subalgebras and quantum symmetric pairs. New directions in Hopf algebras (Cambridge), MSRI publications,vol. 43. Cambridge University Press, pp. 117–166 (2002)

  25. Li, F.: Modulation and natural valued quiver of an algebra. Pac. J. Math. 256, 105–128 (2012)

    Article  MathSciNet  Google Scholar 

  26. Lu, M., Peng, L.: Semi-derived Ringel–Hall algebras and Drinfeld doubles. arXiv:1608.03106v2

  27. Lu, M.: Appendix A to Lu and Wang [LW19]. arXiv:1901.11446

  28. Lu, M., Wang, W.: Hall algebras and quantum symmetric pairs I: foundations. arXiv:1901.11446

  29. Lu, M., Wang, W.: Hall algebras and quantum symmetric pairs of Kac–Moody type. arXiv:2006.06904

  30. Lusztig, G.: Finite dimensional Hopf algebras arising from quantized universal enveloping algebras. J. Am. Math. Soc. 3, 257–296 (1990)

    MathSciNet  MATH  Google Scholar 

  31. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447–498 (1990)

    Article  MathSciNet  Google Scholar 

  32. Lusztig, G.: Introduction to Quantum Groups. Birkhäuser, Boston (1993)

    MATH  Google Scholar 

  33. Molev, A., Ragoucy, E.: Symmetries and invariants of twisted quantum algebras and associated Poisson algebras. Rev. Math. Phys. 20, 173–198 (2008)

    Article  MathSciNet  Google Scholar 

  34. Ringel, C.M.: Hall algebras and quantum groups. Invent. Math. 101, 583–591 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  35. Ringel, C.M.: PBW-bases of quantum groups. J. Reine Angrew. Math. 470, 51–88 (1996)

    MathSciNet  MATH  Google Scholar 

  36. Ringel, C.M., Zhang, P.: Representations of quivers over the algebra of dual numbers. J. Algebra 475, 237–360 (2017)

    Article  MathSciNet  Google Scholar 

  37. Sevenhant, B., Van den Bergh, M.: On the double of the Hall algebra of a quiver. J. Algebra 221, 135–160 (1999)

    Article  MathSciNet  Google Scholar 

  38. Xiao, J.: Drinfeld double and Ringel–Green theory of Hall algebras. J. Algebra 190, 100–144 (1997)

    Article  MathSciNet  Google Scholar 

  39. Xiao, J., Yang, S.: BGP-reflection functors and Lusztig’s symmetries: a Ringel–Hall approach to quantum groups. J. Algebra 241, 204–246 (2001)

    Article  MathSciNet  Google Scholar 

  40. Xiao, J., Zhao, M.: BGP-reflection functors and Lusztig’s symmetries of modified quantized enveloping algebras. Acta Math. Sin. Engl. Ser. 29, 1833–1856 (2013)

    Article  MathSciNet  Google Scholar 

  41. Terwilliger, P.: The Lusztig automorphism of the \(q\)-Onsager algebra. J. Algebra 506, 56–75 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Institute of Mathematics at Academia Sinica (Taipei) and East China Normal University (Shanghai) for hospitality and support where part of this work was carried out. WW is partially supported by NSF Grant DMS-1702254 and DMS-2001351. We thank 2 anonymous referees for their careful readings and suggestions which help make this paper more readable.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610064, People’s Republic of China

    Ming Lu

  2. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    Weiqiang Wang

Authors
  1. Ming Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weiqiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weiqiang Wang.

Additional information

Communicated by Y. Kawahigashi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, M., Wang, W. Hall Algebras and Quantum Symmetric Pairs II: Reflection Functors. Commun. Math. Phys. 381, 799–855 (2021). https://doi.org/10.1007/s00220-021-03965-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-021-03965-8

Search

Navigation