Czechoslovak Mathematical Journal, Vol. 74, No. 4, pp. 1059-1082, 2024
Characterization of automorphisms of Radford's biproduct of Hopf group-coalgebra
Xing Wang, Daowei Lu, Ding-Guo Wang
Received October 10, 2023. Published online October 21, 2024.
Abstract: We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford's $\pi$-biproduct. Firstly, we discuss the endomorphism monoid ${\rm End}_{\pi\text{-Hopf}}(A\times H, p)$ and the automorphism group ${\rm Aut}_{\pi\text{-Hopf}}(A\times H, p)$ of Radford's $\pi$-biproduct $A \times H =\{A \times H_\alpha\}_{\alpha\in\pi}$, and prove that the automorphism has a factorization closely related to the factors $A$ and $H=\{H_\alpha\}_{\alpha\in\pi}$. What's more interesting is that a pair of maps $(F_L,F_R)$ can be used to describe a family of mappings $F=\{F_\alpha\}_{\alpha\in\pi}$. Secondly, we consider the relationship between the automorphism group ${\rm Aut}_{\pi\text{-Hopf}}(A\times H, p)$ and the automorphism group ${\rm Aut}_{\pi\text-\mathcal{Y}\mathcal{D}\text{-Hopf}}(A)$ of $A$, and a normal subgroup of the automorphism group ${\rm Aut}_{\pi\text{-Hopf}}(A\times H, p)$. Finally, we specifically describe the automorphism group of an example.
Affiliations: Xing Wang (corresponding author), Daowei Lu, School of Mathematics and Big Data, Jining University, 1 Xingtan Road, Qufu 273155, Shandong Province, P. R. China, e-mail: xwang17@126.com, ludaowei620@126.com; Ding-Guo Wang, School of Mathematical Sciences, Qufu Normal University, 57 Jingxuan West Road, Qufu 273165, Shandong Province, P. R. China, e-mail: dingguo95@126.com
