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, log in via an institution to check access.
Similar content being viewed by others
References
Alaoui A E. The character table for a Hopf algebra arising from the Drinfel’d double. J Algebra, 2003, 265: 478–495
Beattie M, Dăscălescu S, Grünenfelder L. Constructing pointed Hopf algebras by Ore extensions. J Algebra, 2000, 225: 743–770
Chen H, Oystaeyen F V, Zhang Y. The Green rings of Taft algebras. Proc Amer Math Soc, 2014, 142: 765–775
Cibils C. A quiver quantum groups. Comm Math Phys, 1993, 157: 459–477
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
Huang H, Yang Y. The Green rings of minimal Hopf quivers. Proc Edinb Math Soc, 2014, 59: 107–141
Kac G I, Paljutkin V G. Finite ring groups. Trudy Moskov Mat Obshch, 1966, 15: 224–261 (in Russian)
Kassel C. Quantum Groups. Grad Texts in Math, Vol 155. New York: Springer-Verlag, 1995
Li L, Zhang Y. The Green rings of the Generalized Taft algebras. Contemp Math, 2013, 585: 275–288
Li Y, Hu N. The Green rings of the 2-rank Taft algebra and its two relatives twisted. J Algebra, 2014, 410: 1–35
Lorenz M. Representations of finite-dimensional Hopf algebras. J Algebra, 1997, 188: 476–505
Majid S. Foundations of Quantum Group Theory. Cambridge: Cambridge Univ Press, 1995
Masuoka A. Semisimple Hopf algebras of dimension 6, 8. Israel J Math, 1995, 92: 361–373
Montgomery S. Hopf Algebras and Their Actions on Rings. CBMS Reg Conf Ser Math, No 82. Providence: Amer Math Soc, 1993
Panov A N. Ore extensions of Hopf algebras. Math Notes, 2003, 74: 401–410
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
Shi Y. Finite dimensional Hopf algebras over Kac-Paljutkin algebra H8. Rev Un Mat Argentina, 2019, 60: 265–298
Su D, Yang S. Automorphism group of representation ring of the weak Hopf algebra H8. Czechoslovak Math J, 2018, 68: 1131–1148
Su D, Yang S. Green rings of weak Hopf algebras based on generalized Taft algebras. Period Math Hunger, 2018, 76: 229–242
Su D, Yang S. Representation ring of small quantum group \(\overline{U}_{q}(sl_{2})\). J Math Phys, 2017, 58: 091704
Sweedler M E. Hopf Algebras. New York: Benjamin, 1969
Wang D, Zhang J, Zhuang G. Primitive cohomology of Hopf algebras. J Algebra, 2016, 464: 36–96
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
Witherspoon S J. The representation ring of the quantum double of a finite group. J Algebra, 1996, 179: 305–329
Xu Y, Wang D, Chen J. Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups. J Algebra Appl, 2016, 15: 1650179
Yang S. Representation of simple pointed Hopf algebras. J Algebra Appl, 2004, 3: 91–104
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
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-021-0893-x