Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 70, No. 4, pp. 959-977, 2020

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

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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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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