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

Abstract

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

Keywords

Grothendieck ring / Hopf algebra / irreducible module

References

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

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

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

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

[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

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

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

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

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

[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

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

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

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

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

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

[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

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

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

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

[20] Su D, Yang S. Representation ring of small quantum group U‾q(sl2).J Math Phys, 2017, 58: 091704

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

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

[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

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

[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

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

RIGHTS & PERMISSIONS

2021 Higher Education Press