Czechoslovak Mathematical Journal, first online, pp. 1-19


The bicrossed products of $H_4$ and $H_8$

Daowei Lu, Yan Ning, Dingguo Wang

Received February 25, 2019.   Published online March 30, 2020.

Abstract:  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: $H_8øtimes H_4,H_{32,1},H_{32,2},H_{32,3}$. 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.
Keywords:  Kac-Paljutkin Hopf algebra; Sweedler's Hopf algebra; bicrossed product; factorization problem
Classification MSC:  16T10, 16T05, 16S40
DOI:  10.21136/CMJ.2020.0079-19

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References:
[1] A. L. Agore: 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] A. L. Agore: 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] A. L. Agore, C. G. Bontea, G. Militaru: 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] A. L. Agore, A. Chirvasitu, B. Ion, G. Militaru: 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] C. G. Bontea: 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] Q. Chen, D.-G. Wang: Constructing quasitriangular Hopf algebras. Commun. Algebra 43 (2015), 1698-1722. DOI 10.1080/00927872.2013.876036 | MR 3314638 | Zbl 1327.16017
[7] C. Kassel: 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] G. I. Kats, V. G. Palyutkin: 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] M. Keilberg: 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] M. Keilberg: 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] E. Maillet: Sur les groupes échangeables et les groupes décomposables. Bull. Soc. Math. Fr. 28 (1900), 7-16. (In French.) DOI 10.24033/bsmf.617 | MR 1504357 | JFM 31.0144.02
[12] S. Majid: 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] A. Masuoka: Semisimple Hopf algebras of dimension 6, 8. Isr. J. Math. 92 (1995), 361-373. DOI 10.1007/BF02762089 | MR 1357764 | Zbl 0839.16036
[14] A. N. Panov: 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] D. Pansera: 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] M. Takeuchi: 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] D.-G. Wang, J. J. Zhang, G. Zhuang: 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] D.-G. Wang, J. J. Zhang, G. Zhuang: 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] D.-G. Wang, J. J. Zhang, G. Zhuang: 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] Y. Xu, H.-L. Huang, D.-G. Wang: 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] G. Zappa: Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili tra loro. Atti 2. Congr. Un. Mat. Ital., Bologna 1940 (1942), 119-125. (In Italian.) MR 0019090 | Zbl 0026.29104

Affiliations:   Daowei Lu, School of Mathematical Sciences, Qufu Normal University, No. 57 Jingxuan West Road, Qufu 273165, Shandong, P. R. China and Department of Mathematics, Jining University, No. 1 Xingtan Road, Qufu 273155, Shandong, P. R. China, e-mail: ludaowei620@126.com, Yan Ning, Department of Mathematics, Jining University, No. 1 Xingtan Road, Qufu 273155, Shandong, P. R. China, e-mail: ningkegood@126.com, Dingguo Wang (corresponding author), School of Mathematical Sciences, Qufu Normal University, No. 57 Jingxuan West Road, Qufu 273165, Shandong, P. R. China, e-mail: dgwang@qfnu.edu.cn


 
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