Skip to main content

Grothendieck rings of a class of Hopf algebras of Kac-Paljutkin type

Abstract

We construct the Grothendieck rings of a class of 2n2 dimensional semisimple Hopf Algebras \(H_{2n^{2}}\), which can be viewed as a generalization of the 8 dimensional Kac-Paljutkin Hopf algebra H8. All irreducible \(H_{2n^{2}}\)-modules are classified. Furthermore, we describe the Grothendieck rings r(\(H_{2n^{2}}\)) by generators and relations explicitly.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Alaoui A E. The character table for a Hopf algebra arising from the Drinfel’d double. J Algebra, 2003, 265: 478–495

    MathSciNet  Article  Google Scholar 

  2. 2.

    Beattie M, Dăscălescu S, Grünenfelder L. Constructing pointed Hopf algebras by Ore extensions. J Algebra, 2000, 225: 743–770

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chen H, Oystaeyen F V, Zhang Y. The Green rings of Taft algebras. Proc Amer Math Soc, 2014, 142: 765–775

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cibils C. A quiver quantum groups. Comm Math Phys, 1993, 157: 459–477

    MathSciNet  Article  Google Scholar 

  5. 5.

    Huang H, Oystaeyen F V, Yang Y, Zhang Y. The Green rings of pointed tensor categories of finite type. J Pure Appl Algebra, 2014, 218: 333–342

    MathSciNet  Article  Google Scholar 

  6. 6.

    Huang H, Yang Y. The Green rings of minimal Hopf quivers. Proc Edinb Math Soc, 2014, 59: 107–141

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kac G I, Paljutkin V G. Finite ring groups. Trudy Moskov Mat Obshch, 1966, 15: 224–261 (in Russian)

    MathSciNet  Google Scholar 

  8. 8.

    Kassel C. Quantum Groups. Grad Texts in Math, Vol 155. New York: Springer-Verlag, 1995

    Book  Google Scholar 

  9. 9.

    Li L, Zhang Y. The Green rings of the Generalized Taft algebras. Contemp Math, 2013, 585: 275–288

    MathSciNet  Article  Google Scholar 

  10. 10.

    Li Y, Hu N. The Green rings of the 2-rank Taft algebra and its two relatives twisted. J Algebra, 2014, 410: 1–35

    MathSciNet  Article  Google Scholar 

  11. 11.

    Lorenz M. Representations of finite-dimensional Hopf algebras. J Algebra, 1997, 188: 476–505

    MathSciNet  Article  Google Scholar 

  12. 12.

    Majid S. Foundations of Quantum Group Theory. Cambridge: Cambridge Univ Press, 1995

    Book  Google Scholar 

  13. 13.

    Masuoka A. Semisimple Hopf algebras of dimension 6, 8. Israel J Math, 1995, 92: 361–373

    MathSciNet  Article  Google Scholar 

  14. 14.

    Montgomery S. Hopf Algebras and Their Actions on Rings. CBMS Reg Conf Ser Math, No 82. Providence: Amer Math Soc, 1993

    Book  Google Scholar 

  15. 15.

    Panov A N. Ore extensions of Hopf algebras. Math Notes, 2003, 74: 401–410

    MathSciNet  Article  Google Scholar 

  16. 16.

    Pansera D. A class of semisimple Hopf algebras acting on quantum polynomial algebras. In: Leroy A, Lomp C, López-Permouth S, Oggier F, eds. Rings, Modules and Codes. Contemp Math, Vol 727. Providence: Amer Math Soc, 2019, 303–316

    Chapter  Google Scholar 

  17. 17.

    Shi Y. Finite dimensional Hopf algebras over Kac-Paljutkin algebra H8. Rev Un Mat Argentina, 2019, 60: 265–298

    MathSciNet  Article  Google Scholar 

  18. 18.

    Su D, Yang S. Automorphism group of representation ring of the weak Hopf algebra H8. Czechoslovak Math J, 2018, 68: 1131–1148

    MathSciNet  Article  Google Scholar 

  19. 19.

    Su D, Yang S. Green rings of weak Hopf algebras based on generalized Taft algebras. Period Math Hunger, 2018, 76: 229–242

    MathSciNet  Article  Google Scholar 

  20. 20.

    Su D, Yang S. Representation ring of small quantum group \(\overline{U}_{q}(sl_{2})\). J Math Phys, 2017, 58: 091704

    MathSciNet  Article  Google Scholar 

  21. 21.

    Sweedler M E. Hopf Algebras. New York: Benjamin, 1969

    MATH  Google Scholar 

  22. 22.

    Wang D, Zhang J, Zhuang G. Primitive cohomology of Hopf algebras. J Algebra, 2016, 464: 36–96

    MathSciNet  Article  Google Scholar 

  23. 23.

    Wang Z, You L, Chen H. Representations of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one. Algebr Represent Theory, 2015, 18: 801–830

    MathSciNet  Article  Google Scholar 

  24. 24.

    Witherspoon S J. The representation ring of the quantum double of a finite group. J Algebra, 1996, 179: 305–329

    MathSciNet  Article  Google Scholar 

  25. 25.

    Xu Y, Wang D, Chen J. Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups. J Algebra Appl, 2016, 15: 1650179

    MathSciNet  Article  Google Scholar 

  26. 26.

    Yang S. Representation of simple pointed Hopf algebras. J Algebra Appl, 2004, 3: 91–104

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11671024, 11701019, 11871301) and the Science and Technology Project of Beijing Municipal Education Commission (Grant No. KM202110005012).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shilin Yang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, J., Yang, S. & Wang, D. Grothendieck rings of a class of Hopf algebras of Kac-Paljutkin type. Front. Math. China 16, 29–47 (2021). https://doi.org/10.1007/s11464-021-0893-x

Download citation

Keywords

  • Grothendieck ring
  • Hopf algebra
  • irreducible module

MSC2020

  • 16G10
  • 16D70
  • 16T05