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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.
MSC:
 16T10 Bialgebras 16T05 Hopf algebras and their applications 16S40 Smash products of general Hopf actions
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References:
 [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
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