zbMATH — the first resource for mathematics

The bicrossed products of \(H_4\) and \(H_8\). (English) Zbl 07285973
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.
16T10 Bialgebras
16T05 Hopf algebras and their applications
16S40 Smash products of general Hopf actions
Full Text: DOI
[1] Agore, A. L., Classifying bicrossed products of two Taft algebras, J. Pure Appl. Algebra 222 (2018), 914-930
[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
[3] Agore, A. L.; Bontea, C. G.; Militaru, G., Classifying bicrossed products of Hopf algebras, Algebr. Represent. Theory 17 (2014), 227-264
[4] Agore, A. L.; asitu, A. Chirv\v; Ion, B.; Militaru, G., Bicrossed products for finite groups, Algebr. Represent. Theory 12 (2009), 481-488
[5] Bontea, C. G., Classifying bicrossed products of two Sweedler’s Hopf algebras, Czech. Math. J. 64 (2014), 419-431
[6] Chen, Q.; Wang, D.-G., Constructing quasitriangular Hopf algebras, Commun. Algebra 43 (2015), 1698-1722
[7] Kassel, C., Quantum Group, Graduate Texts in Mathematics 155, Springer, New York (1995)
[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
[9] Keilberg, M., Automorphisms of the doubles of purely non-abelian finite groups, Algebr. Represent. Theory 18 (2015), 1267-1297
[10] Keilberg, M., Quasitriangular structures of the double of a finite group, Commun. Algebra 46 (2018), 5146-5178
[11] Maillet, E., Sur les groupes échangeables et les groupes décomposables, Bull. Soc. Math. Fr. 28 (1900), 7-16 French \99999JFM99999 31.0144.02
[12] Majid, S., Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 130 (1990), 17-64
[13] Masuoka, A., Semisimple Hopf algebras of dimension 6, 8, Isr. J. Math. 92 (1995), 361-373
[14] Panov, A. N., Ore extensions of Hopf algebras, Math. Notes 74 (2003), 401-410 translation from Mat. Zametki 74 2003 425-434
[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
[16] Takeuchi, M., Matched pairs of groups and bismash products of Hopf algebras, Commun. Algebra 9 (1981), 841-882
[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
[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
[19] Wang, D.-G.; Zhang, J. J.; Zhuang, G., Primitive cohomology of Hopf algebras, J. Algebra 464 (2016), 36-96
[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
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.