Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
automorphism group; representation ring; weak Hopf algebra
automorphism group; representation ring; weak Hopf algebra
Summary:
Let $H_8$ be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra $\widetilde {H_8}$ based on $H_8$, then we investigate the structure of the representation ring of $\widetilde {H_8}$. Finally, we prove that the automorphism group of $r(\widetilde {H_8})$ is just isomorphic to $D_6$, where $D_6$ is the dihedral group with order 12.
References:
[1] Aizawa, N., Isaac, P. S.: Weak Hopf algebras corresponding to $U_q[sl_n]$. J. Math. Phys. 44 (2003), 5250-5267. DOI 10.1063/1.1616999 | MR 2014859 | Zbl 1063.16041
[2] Alperin, R. C.: Homology of the group of automorphisms of $k[x, y]$. J. Pure Appl. Algebra 15 (1979), 109-115. DOI 10.1016/0022-4049(79)90027-6 | MR 0535179 | Zbl 0415.13006
[3] Chen, H.-X.: The coalgebra automorphism group of Hopf algebra $k_q[x,x^{-1},y]$. J. Pure Appl. Algebra 217 (2013), 1870-1887. DOI 10.1016/j.jpaa.2013.01.013 | MR 3053522 | Zbl 1292.16021
[4] Chen, H., Oystaeyen, F. Van, Zhang, Y.: The Green rings of Taft algebras. Proc. Am. Math. Soc. 142 (2014), 765-775. DOI 10.1090/S0002-9939-2013-11823-X | MR 3148512 | Zbl 1309.16021
[5] Cheng, D.: The structure of weak quantum groups corresponding to Sweedler algebra. JP J. Algebra Number Theory Appl. 16 (2010), 89-99. MR 2662950 | Zbl 1217.16028
[6] Cheng, D., Li, F.: The structure of weak Hopf algebras corresponding to $U_q( sl_2)$. Commun. Algebra 37 (2009), 729-742. DOI 10.1080/00927870802243499 | MR 2503175 | Zbl 1166.16018
[7] Dicks, W.: Automorphisms of the polynomial ring in two variables. Publ. Secc. Mat. Univ. Autòn. Barc. 27 (1983), 155-162. DOI 10.5565/publmat_27183_04 | MR 0763864 | Zbl 0593.13005
[8] Drensky, V., Yu, J.-T.: Automorphisms of polynomial algebras and Dirichlet series. J. Algebra 321 (2009), 292-302. DOI 10.1016/j.jalgebra.2008.08.026 | MR 2469362 | Zbl 1157.13015
[9] Han, J., Su, Y.: Automorphism groups of Witt algebras. Available at ArXiv 1502.0144v1 [math.QA].
[10] Jia, T., Zhao, R., Li, L.: Automorphism group of Green ring of Sweedler Hopf algebra. Front. Math. China 11 (2016), 921-932. DOI 10.1007/s11464-016-0565-4 | MR 3531037 | Zbl 1372.16038
[11] Li, F.: Weak Hopf algebras and some new solutions of the quantum Yang-Baxter equation. J. Algebra 208 (1998), 72-100. DOI 10.1006/jabr.1998.7491 | MR 1643979 | Zbl 0916.16020
[12] Li, L., Zhang, Y.: The Green rings of the generalized Taft Hopf algebras. Hopf Algebras and Tensor Categories Contemporary Mathematics 585, Proc. of the International Conf., University of Almería, Almería, Amer. Math. Soc., Providence N. Andruskiewitsch et al. (2013), 275-288. DOI 10.1090/conm/585/11618 | MR 3077243 | Zbl 1309.19001
[13] Masuoka, A.: Semisimple Hopf algebras of dimension $6,8$. Isr. J. Math. 92 (1995), 361-373. DOI 10.1007/BF02762089 | MR 1357764 | Zbl 0839.16036
[14] Montgomery, S.: Hopf Algebras and Their Actions on Rings. Conference on Hopf algebras and Their Actions on Rings, Chicago, 1992, CBMS Regional Conference Series in Mathematics 82, AMS, Providence (1993). DOI 10.1090/cbms/082 | MR 1243637 | Zbl 0793.16029
[15] Ştefan, D.: Hopf algebras of low dimension. J. Algebra 211 (1999), 343-361. DOI 10.1006/jabr.1998.7602 | MR 1656583 | Zbl 0918.16031
[16] Sweedler, M. E.: Hopf Algebras. Mathematics Lecture Note Series, W. A. Benjamin, New York (1969). MR 0252485 | Zbl 0194.32901
[17] Kulk, W. van der: On polynomial rings in two variables. Nieuw Arch. Wiskd., III. Ser. 1 (1953), 33-41. MR 0054574 | Zbl 0050.26002
[18] Yang, S.: Representations of simple pointed Hopf algebras. J. Algebra Appl. 3 (2004), 91-104. DOI 10.1142/S021949880400071X | MR 2047638 | Zbl 1080.16043
[19] Yang, S.: Weak Hopf algebras corresponding to Cartan matrices. J. Math. Phys. 46 (2005), 073502, 18 pages. DOI 10.1063/1.1933063 | MR 2153558 | Zbl 1110.16047
[20] Yu, J.-T.: Recognizing automorphisms of polynomial algebras. Mat. Contemp. 14 (1998), 215-225. MR 1663646 | Zbl 0930.13016
[21] Zhao, K.: Automophisms of the binary polynomial algebras on integer rings. Chinese Ann. Math. Ser. A 4 (1995), 448-494 Chinese.
Search
Partner of
