Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Kac-Paljutkin Hopf algebra; Sweedler's Hopf algebra; bicrossed product; factorization problem
Kac-Paljutkin Hopf algebra; Sweedler's Hopf algebra; bicrossed product; factorization problem
Summary:
Let $H_4$ and $H_8$ be the Sweedler's and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through $H_8$ and $H_4$ (equivalently, any bicrossed product between the Hopf algebras $H_8$ and $H_4$) must be isomorphic to one of the following four Hopf algebras: $H_8\otimes H_4,H_{32,1},H_{32,2},H_{32,3}$. The set of all matched pairs $(H_8,H_4,\triangleright ,\triangleleft )$ is explicitly described, and then the associated bicrossed product is given by generators and relations.
References:
[1] Agore, A. L.: Classifying bicrossed products of two Taft algebras. J. Pure Appl. Algebra 222 (2018), 914-930. DOI 10.1016/j.jpaa.2017.05.014 | MR 3720860 | Zbl 1416.16033
[2] Agore, A. L.: Hopf algebras which factorize through the Taft algebra $T_{m^{2}}(q)$ and the group Hopf algebra $K[C_{n}]$. SIGMA, Symmetry, Integrability Geom. Methods Appl. 14 (2018), Article ID 027. DOI 10.3842/SIGMA.2018.027 | MR 3778923 | Zbl 1414.16027
[3] Agore, A. L., Bontea, C. G., Militaru, G.: Classifying bicrossed products of Hopf algebras. Algebr. Represent. Theory 17 (2014), 227-264. DOI 10.1007/s10468-012-9396-5 | MR 3160722 | Zbl 1351.16031
[4] Agore, A. L., asitu, A. Chirv\v, Ion, B., Militaru, G.: Bicrossed products for finite groups. Algebr. Represent. Theory 12 (2009), 481-488. DOI 10.1007/s10468-009-9145-6 | MR 2501197 | Zbl 1187.20023
[5] Bontea, C. G.: Classifying bicrossed products of two Sweedler's Hopf algebras. Czech. Math. J. 64 (2014), 419-431. DOI 10.1007/s10587-014-0109-6 | MR 3277744 | Zbl 1322.16022
[6] Chen, Q., Wang, D.-G.: Constructing quasitriangular Hopf algebras. Commun. Algebra 43 (2015), 1698-1722. DOI 10.1080/00927872.2013.876036 | MR 3314638 | Zbl 1327.16017
[7] Kassel, C.: Quantum Group. Graduate Texts in Mathematics 155, Springer, New York (1995). DOI 10.1007/978-1-4612-0783-2 | MR 1321145 | Zbl 0808.17003
[8] Kats, G. I., Palyutkin, V. G.: Finite ring groups. Trans. Mosc. Math. Soc. 15 (1966), 251-294 translation from Tr. Mosk. Mat. O.-va 15 1966 224-261. MR 0208401 | Zbl 0218.43005
[9] Keilberg, M.: Automorphisms of the doubles of purely non-abelian finite groups. Algebr. Represent. Theory 18 (2015), 1267-1297. DOI 10.1007/s10468-015-9540-0 | MR 3422470 | Zbl 1354.16042
[10] Keilberg, M.: Quasitriangular structures of the double of a finite group. Commun. Algebra 46 (2018), 5146-5178. DOI 10.1080/00927872.2018.1461883 | MR 3923748 | Zbl 1414.16028
[11] Maillet, E.: Sur les groupes échangeables et les groupes décomposables. Bull. Soc. Math. Fr. 28 (1900), 7-16 French \99999JFM99999 31.0144.02. DOI 10.24033/bsmf.617 | MR 1504357
[12] Majid, S.: Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. J. Algebra 130 (1990), 17-64. DOI 10.1016/0021-8693(90)90099-A | MR 1045735 | Zbl 0694.16008
[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] Panov, A. N.: Ore extensions of Hopf algebras. Math. Notes 74 (2003), 401-410 translation from Mat. Zametki 74 2003 425-434. DOI 10.1023/A:1026115004357 | MR 2022506 | Zbl 1071.16035
[15] Pansera, D.: A class of semisimple Hopf algebras acting on quantum polynomial algebras. Rings, Modules and Codes Contemporary Mathematics 727, American Mathematical Society, Providence (2019), 303-316. DOI 10.1090/conm/727 | MR 3938158 | Zbl 07120022
[16] Takeuchi, M.: Matched pairs of groups and bismash products of Hopf algebras. Commun. Algebra 9 (1981), 841-882. DOI 10.1080/00927878108822621 | MR 0611561 | Zbl 0456.16011
[17] Wang, D.-G., Zhang, J. J., Zhuang, G.: Hopf algebras of GK-dimension two with vanishing Ext-group. J. Algebra 388 (2013), 219-247. DOI 10.1016/j.jalgebra.2013.03.032 | MR 3061686 | Zbl 1355.16033
[18] Wang, D.-G., Zhang, J. J., Zhuang, G.: Connected Hopf algebras of Gelfand-Kirillov dimension four. Trans. Am. Math. Soc. 367 (2015), 5597-5632. DOI 10.1090/S0002-9947-2015-06219-9 | MR 3347184 | Zbl 1330.16022
[19] Wang, D.-G., Zhang, J. J., Zhuang, G.: Primitive cohomology of Hopf algebras. J. Algebra 464 (2016), 36-96. DOI 10.1016/j.jalgebra.2016.07.003 | MR 3533424 | Zbl 1402.16019
[20] Xu, Y., Huang, H.-L., Wang, D.-G.: Realization of PBW-deformations of type $A_n$ quantum groups via multiple Ore extensions. J. Pure Appl. Algebra 223 (2019), 1531-1547. DOI 10.1016/j.jpaa.2018.06.017 | MR 3906516 | Zbl 06994941
[21] Zappa, G.: Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili tra loro. Atti 2. Congr. Un. Mat. Ital., Bologna 1940 (1942), 119-125 Italian. MR 0019090 | Zbl 0026.29104
Search
Partner of
