Skip to main content
SpringerLink
Log in

Differential Operator Approach to ıquantum Groups and Their Oscillator Representations

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

For a quasi-split Satake diagram, we define a modified q-Weyl algebra, and show that there is an algebra homomorphism between it and the corresponding ıquantum group. In other words, we provide a differential operator approach to ıquantum groups. Meanwhile, the oscillator representations of ıquantum groups are obtained. The crystal basis of the irreducible subrepresentations of these oscillator representations are constructed.

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

Similar content being viewed by others

References

  1. Bao, H. C., Kujawa, J., Li, Y. Q., et al.: Geometric Schur duality of classical type. Transform. Groups, 23, 329–389 (2018)

    Article  MathSciNet  Google Scholar 

  2. Bao, H. C., Shan, P., Wang, W. Q., et al.: Categorification of quantum symmetric pairs I. Quantum Topol., 9, 643–714 (2018)

    Article  MathSciNet  Google Scholar 

  3. Bao, H. C., Wang, W. Q.: A new approach to Kazhdan—Lusztig theory of type B via quantum symmetric pairs. Astérisque, 402, vii+134 pp. (2018)

  4. Beilinson, A., Lusztig, G., MacPherson, R.: A geometric setting for the quantum deformation of GLn. Duke Math. J., 61, 655–677 (1990)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Chen, X. H., Lu, M., Wang, W. Q.: A Serre presentation for the ıquantum groups. Transform. Groups, 26, 827–857 (2021)

    Article  MathSciNet  Google Scholar 

  7. Drinfeld, V.: Quantum groups. Proc. Int. Congr. Math. Berkeley, 1, 798–820 (1987)

    Google Scholar 

  8. Du, J., Fu, Q.: Quantum affine \({\mathfrak{gl}}_n\) via Hecke algebras. Adv. Math., 282, 23–46 (2015)

    Article  MathSciNet  Google Scholar 

  9. Du, J., Lin, Y. N., Zhou, Z. G.: Quantum queer supergroups via v-differential operators. J. Algebra, 599, 48–103 (2022)

    Article  MathSciNet  Google Scholar 

  10. Du, J., Wu, Y. D.: A new realisation of the i-quantum groups \(\mathbf{U}^{\cal{J}}(n)\). J. Pure Appl. Algebra, 226, 106793 (2022)

    Article  Google Scholar 

  11. Du, J., Wu, Y. D.: The i-quantum group Uı(n). Pacific J. Math., 320, 61–101 (2022)

    Article  MathSciNet  Google Scholar 

  12. Du, J., Zhou, Z. G.: The regular representation of \(U_{v}({\mathfrak{gl}}_{m\vert n})\). Proc. Amer. Math. Soc., 148, 111–124 (2020)

    Article  MathSciNet  Google Scholar 

  13. Fan, Z. B., Lai, C. J., Li, Y. Q., et al.: Affine flag varieties and quantum symmetric pairs. Mem. Amer. Math. Soc., 265, (1285), v+123 pp. (2020)

  14. Fan, Z. B., Lai, C. J., Li, Y. Q., et al.: Affine Hecke algebras and quantum symmetric pairs. Mem. Amer. Math. Soc., 281(1386), ix+92 pp. (2023)

  15. Fan, Z. B., Lai, C. J., Li, Y. Q., et al.: Quantum Schur duality of affine type C with three parameters. Math. Res. Lett., 27, 79–114 (2020)

    Article  MathSciNet  Google Scholar 

  16. Fan, Z. B., Ma, H. T., Xiao, H. S. L.: Equivariant K-theory approach to ı-quantum groups. Publ. Res. Inst. Math. Sci., 58, 635–668 (2022)

    Article  MathSciNet  Google Scholar 

  17. Ginzburg, V., Vasserot, E.: Langlands reciprocity for affine quantum groups of type An. Internat. Math. Res. Not., 3, 67–85 (1993)

    Article  MathSciNet  Google Scholar 

  18. Hayashi, T.: Q-analogues of Clifford and Weyl algebras–spinor and oscillator representations of quantum enveloping algebras. Commun. Math. Phys., 127, 129–144 (1990)

    Article  MathSciNet  Google Scholar 

  19. Hong, J., Kang, S. J.: Introduction to Quantum Groups and Crystal Bases, Grad. Stud. Math., Vol. 42, Amer. Math. Soc., Providence, RI, 2002

    Google Scholar 

  20. Hu, N. H.: Quantum divided power algebra, q-derivatives, and some new quantum groups. J. Algebra, 232, 507–540 (2000)

    Article  MathSciNet  Google Scholar 

  21. Jimbo, M.: A q-analogue of U (gl(N+1)), Hecke algebra, and the Yang–Baxter equation. Lett. Math. Phys., 11, 247–252 (1986)

    Article  MathSciNet  Google Scholar 

  22. Kashiwara, M.: On crystal bases of the q-analogue of the universal enveloping algebra. Duke Math. J., 63, 465–516 (1991)

    Article  MathSciNet  Google Scholar 

  23. Lu, M., Wang, W. Q.: Hall algebras and quantum symmetric pairs I: foundations. Proc. Lond. Math. Soc., 124, 1–82 (2022)

    Article  MathSciNet  Google Scholar 

  24. Lu, M., Wang W. Q.: Hall algebras and quantum symmetric pairs II: reflection functors. Commun. Math. Phys., 381, 799–855 (2021)

    Article  MathSciNet  Google Scholar 

  25. Lu, M., Wang W. Q.: Hall algebras and quantum symmetric pairs III: quiver varieties. Adv. Math., 393, 108071 (2021)

    Article  MathSciNet  Google Scholar 

  26. Lu, M., Wang, W. Q.: Hall algebras and quantum symmetric pairs of Kac–Moody type. Adv. Math., 430, 109215 (2023)

    Article  MathSciNet  Google Scholar 

  27. Lusztig, G.: Introduction to Quantum Groups, Modern Birkhauser Classics, Reprint of the 1993 Edition, Birkhäuser, Boston, 2010

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Watanabe, H.: Crystal basis theory for a quantum symmetric pair (U, \(\mathbf{U}^{\cal{J}}\)). Int. Math. Res. Not., 22, 8292–8352 (2020)

    Google Scholar 

  30. Watanabe, H.: Global crystal bases for integrable modules over a quantum symmetric pair of type AIII. Represent. Theory, 25, 27–66 (2021)

    Article  MathSciNet  Google Scholar 

  31. Zhang, J., Hu, N. H.: Realization of Uq \(U_{q}({\mathfrak{sp}}_{2n})\) within the differential algebra on quantum symplectic space. Symmetry, Integrability and Geometry: Methods and Applications, SIGMA, 13, 084, 21 pp. (2017)

    Google Scholar 

Download references

Acknowledgements

We thank the referees for many useful comments and suggestions.

Author information

Authors and Affiliations

  1. College of Mathematical Science, Harbin Engineering University, Harbin, 150001, P. R. China

    Zhao Bing Fan, Ji Cheng Geng & Shao Long Han

Authors
  1. Zhao Bing Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji Cheng Geng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shao Long Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhao Bing Fan.

Ethics declarations

Conflict of Interest The authors declare no conflict of interest.

Additional information

The first author was partially supported by the NSF of China grant 12271120, the NSF of Heilongjiang Province grant JQ2020A001, and the Fundamental Research Funds for the Central Universities

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fan, Z.B., Geng, J.C. & Han, S.L. Differential Operator Approach to ıquantum Groups and Their Oscillator Representations. Acta. Math. Sin.-English Ser. 40, 1360–1374 (2024). https://doi.org/10.1007/s10114-024-2151-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-024-2151-0

Keywords

MR(2010) Subject Classification

Search

Navigation