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The bicrossed products of $H_4$ and $H_8$
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 959-977
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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.
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.
DOI : 10.21136/CMJ.2020.0079-19
Classification : 16S40, 16T05, 16T10
Keywords: Kac-Paljutkin Hopf algebra; Sweedler's Hopf algebra; bicrossed product; factorization problem 
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Lu, Daowei; Ning, Yan; Wang, Dingguo. The bicrossed products of $H_4$ and $H_8$. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 959-977. doi: 10.21136/CMJ.2020.0079-19

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