The Grothendieck ring of quantum double of quaternion group

Sun, Hua ; Pang, Jia ; Shen, Yanxi

Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 881-896

Abstract (VO)

Let $\Bbbk $ be an algebraically closed field of characteristic $p\neq 2$, and let $Q_8$ be the quaternion group. We describe the structures of all simple modules over the quantum double $D(\Bbbk Q_8)$ of group algebra $\Bbbk Q_8$. Moreover, we investigate the tensor product decomposition rules of all simple $D(\Bbbk Q_8)$-modules. Finally, we describe the Grothendieck ring $G_0(D(\Bbbk Q_8))$ by generators with relations.

Détail

DOI : 10.21136/CMJ.2024.0113-24

Classification : 16G30, 16T99
Keywords: Grothendieck ring; simple module; quantum double; quaternion group

Bibliographie

[1] Auslander, M., Reiten, I., Smalø, S. O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics 36. Cambridge University Press, Cambridge (1995). | DOI | MR | JFM

[2] Chen, H.-X.: Irreducible representations of a class of quantum doubles. J. Algebra 225 (2000), 391-409. | DOI | MR | JFM

[3] Chen, H.-X.: Finite-dimensional representations of a quantum double. J. Algebra 251 (2002), 751-789. | DOI | MR | JFM

[4] Chen, H.-X.: Representations of a class of Drinfeld's doubles. Commun. Algebra 33 (2005), 2809-2825. | DOI | MR | JFM

[5] Dong, J.: Grothendieck ring of quantum double of finite groups. Czech. Math. J. 60 (2010), 869-879. | DOI | MR | JFM

[6] Dong, J., Chen, H.: The representations of quantum double of dihedral groups. Algebra Colloq. 20 (2013), 95-108. | DOI | MR | JFM

[7] Kassel, C.: Quantum Groups. Graduate Texts in Mathematics 155. Springer, New York (1995). | DOI | MR | JFM

[8] Kondo, H., Saito, Y.: Indecomposable decomposition of tensor products of mudules over the restricted quantum universal enveloping algebra associated to $\mathfrak{sl}_2$. J. Algebra 330 (2011), 103-129. | DOI | MR | JFM

[9] Liu, G., Oystaeyen, F. Van, Zhang, Y.: Reprensentations of the small quasi-quantum group $Qu_q(\mathfrak{sl}_2)$. Bull. Belg. Math. Soc. - Simon Stevin 23 (2016), 779-800. | DOI | MR | JFM

[10] Liu, G., Oystaeyen, F. Van, Zhang, Y.: Quasi-Frobenius-Lusztig kernels for simple Lie algebras. Trans. Am. Math. Soc. 369 (2017), 2049-2086. | DOI | MR | JFM

[11] Majid, S.: Doubles of quasitriangular Hopf algebras. Commun. Algebra 19 (1991), 3061-3073. | DOI | MR | JFM

[12] Mason, G.: The quantum double of a finite group and its role in conformal field theory. Groups '93 Galway/St. Andrews. Volume 2 London Mathematical Society Lecture Note Series 212. Cambridge University Press, Cambridge (1995). | DOI | MR | JFM

[13] Montgomery, S.: Hopf Algebras and Their Actions on Rings. Regional Conference Series in Mathematics 82. AMS, Providence (1993). | DOI | MR | JFM

[14] Suter, R.: Modules over $\mathfrak{U}_q(\mathfrak{sl}_2)$. Commun. Math. Phys. 163 (1994), 359-393. | DOI | MR | JFM

[15] Witherspoon, S. J.: The representation ring of the quantum double of a finite group. J. Algebra 179 (1996), 305-329. | DOI | MR | JFM

[16] Xiao, J.: Finite dimensional representations of $U_t$(sl(2)) at roots of unity. Can. J. Math. 49 (1997), 772-787. | DOI | MR | JFM