Skip to main content
SpringerLink
Log in

Remarks on Sekine Quantum Groups

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

We first describe the Sekine quantum groups \({{\cal A}_k}\) (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of \({{\cal A}_k}\) and describe their representation rings \(r({{\cal A}_k})\). Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of \(r({{\cal A}_k})\).

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. P. Etingof, V. Ostrik: Finite tensor categories. Mosc. Math. J. 4 (2004), 627–654.

    Article  MathSciNet  Google Scholar 

  2. U. Franz, A. Skalski: On idempotent states on quantum groups. J. Algebra 322 (2009), 1774–1802.

    Article  MathSciNet  Google Scholar 

  3. G. I. Kac, V. G. Paljutkin: Finite ring groups. Trans. Mosc. Math. Soc. 15 (1966), 251–294.

    MathSciNet  Google Scholar 

  4. M. Lorenz: Some applications of Frobenius algebras to Hopf algebras. Groups, Algebras and Applications. Contemporary Mathematics 537. AMS, Providence, 2011, pp. 269–289.

    MATH  Google Scholar 

  5. Y. Sekine: An example of finite-dimensional Kac algebras of Kac-Paljutkin type. Proc. Am. Math. Soc. 124 (1996), 1139–1147.

    Article  MathSciNet  Google Scholar 

  6. S. Vaes, L. Vainerman: Extensions of locally compact quantum groups and the bicrossed product construction. Adv. Math. 175 (2003), 1–101.

    Article  MathSciNet  Google Scholar 

  7. Z. Wang, L. Li, Y. Zhang: A criterion for the Jacobson semisimplicity of the Green ring of a finite tensor category. Glasg. Math. J. 60 (2018), 253–272.

    Article  MathSciNet  Google Scholar 

  8. H. Zhang: Idempotent states on Sekine quantum groups. Commun. Algebra 47 (2019), 4095–4113.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are particularly grateful to the referee for his/her carefully reading the manuscript and for many valuable comments which largely improved this paper.

Author information

Authors and Affiliations

  1. College of Mathematics, Faculty of Science, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing, 100124, P. R. China

    Jialei Chen & Shilin Yang

Authors
  1. Jialei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shilin Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shilin Yang.

Additional information

The work was supported by the National Natural Science Foundation of China (Grant Nos. 11701019, 11671024) and the Science and Technology Project of Beijing Municipal Education Commission (Grant No. KM202110005012).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, J., Yang, S. Remarks on Sekine Quantum Groups. Czech Math J 72, 695–707 (2022). https://doi.org/10.21136/CMJ.2022.0112-21

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2022.0112-21

Keywords

MSC 2020

Search

Navigation