Skip to main content
SpringerLink
Log in

Minimal Triangular Structures on Abelian Extensions

  • Research
  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We study minimal triangular structures on abelian extensions. In particular, we construct a family of minimal triangular semisimple Hopf algebras and prove that the Hopf algebra \(H_{b:y}\) in the semisimple Hopf algebras of dimension 16 classified by Y. Kashina in 2000 is minimal triangular Hopf algebra with smallest dimension among non-trivial semisimple triangular Hopf algebras (i.e. not group algebras or their dual).

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

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Abella, A.: Some advances about the existence of compact involutions in semisimple Hopf algebras. São Paulo J. Math. Sci. 13(2), 628–651 (2019)

    Article  MathSciNet  Google Scholar 

  2. Andruskiewitsch, N., Etingof, P., Gelaki, S.: Triangular Hopf algebras with the Chevalley property. Michigan Mathematical Journal 49(2), 277–298 (2001)

    Article  MathSciNet  Google Scholar 

  3. Andruskiewitsch, N., Natale, S.: Counting arguments for Hopf algebras of low dimension. Tsukuba J. Math. 25(1), 187–201 (2001)

    Article  MathSciNet  Google Scholar 

  4. Davydov, A.A.: Quasitriangular structures on cocommutative Hopf algebras, arXiv:q-alg/9706007 (1997)

  5. Drinfel’d, V.G.: Quantum groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986)

  6. Drinfel’d, V.G.: On almost cocommutative Hopf algebras. Leningrad Mathematical Journal. 321-342 (1990)

  7. Etingof, P., Gelaki, S.: Some properties of finite-dimensional semisimple Hopf algebras. Math. Res. Lett. 5(1–2), 191–197 (1998)

    Article  MathSciNet  Google Scholar 

  8. Etingof, P., Gelaki, S.: The classification of triangular semisimple and cosemisimple Hopf algebras over an algebraically closed field. Internat. Math. Res. Notices 5, 223–234 (2000)

    Article  MathSciNet  Google Scholar 

  9. Etingof, P., Gelaki, S.: Semisimple Hopf algebras of dimension \(pq\) are trivial. J. Algebra 210(2), 664–669 (1998)

    Article  MathSciNet  Google Scholar 

  10. Etingof, P., Gelaki, S.: A method of construction of finite-dimensional triangular semisimple Hopf algebras. Mathematical Research Letters 5, 551–561 (1998)

    Article  MathSciNet  Google Scholar 

  11. Gelaki, S.: Quantum groups of dimension \(pq^2\). Israel J. Math. 102, 227–267 (1997)

    Article  MathSciNet  Google Scholar 

  12. Gelaki, S.: Some properties and examples of triangular pointed Hopf algebras. Math. Res. Lett. 6, 563–572 (1999)

    Article  MathSciNet  Google Scholar 

  13. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)

    Article  MathSciNet  Google Scholar 

  14. Kashina, Y.: Classification of semisimple Hopf algebras of dimension 16. J. Algebra 232(2), 617–663 (2000)

    Article  MathSciNet  Google Scholar 

  15. Masuoka, A.: Extensions of Hopf algebras and Lie bialgebras. Trans. Amer. Math. Soc. 352(8), 3837–3879 (2000)

    Article  MathSciNet  Google Scholar 

  16. Masuoka, A.: Hopf algebra extensions and cohomology. New directions in Hopf algebras, 167–209, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge (2002)

  17. Montgomery, S.: Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 82. American Mathematical Society, Providence, RI (1993)

    Book  Google Scholar 

  18. Natale, S.: On quasitriangular structures in Hopf algebras arising from exact group factorizations. Comm. Algebra 39(12), 4763–4775 (2011)

    Article  MathSciNet  Google Scholar 

  19. Radford, D.E.: Hopf Algebras, World Scientific, Series on Knots and Everything, 49. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2012)

  20. Radford, D.E.: Minimal quasitriangular Hopf algebras. J. Algebra 157(2), 285–315 (1993)

    Article  MathSciNet  Google Scholar 

  21. Suzuki, S.: A family of braided cosemisimple Hopf algebras of finite dimension. Tsukuba J. Math. 22(1), 1–29 (1998)

    Article  MathSciNet  Google Scholar 

  22. Zhou, K., Liu, G.: On the quasitriangular structures of abelian extensions of \(\mathbb{Z} _{2}\). Comm. Algebra 49(11), 4755–4762 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Funding

No funding.

Author information

Authors and Affiliations

  1. Department of Sport, Gdansk University of Physical Education and Sport, 80-336, Gdansk, Poland

    Hong Fei Zhang

  2. Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China

    Kun Zhou

Authors
  1. Hong Fei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kun Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Kun Zhou wrote the main manuscript text, HongFei Zhang prepared the introduction and checked the article. All authors reviewed the manuscript.

Corresponding author

Correspondence to Kun Zhou.

Ethics declarations

Ethical Approval

Not applicable.

Competing interests

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Presented by: Kenneth Goodearl

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, H.F., Zhou, K. Minimal Triangular Structures on Abelian Extensions. Algebr Represent Theor 27, 1121–1136 (2024). https://doi.org/10.1007/s10468-023-10250-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-023-10250-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation