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Let $H$ be a finite-dimensional bialgebra. In this paper, we prove that the category $\mathrm{ℒℛ}\left(H\right)$ of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category ${}_{H\otimes H^{*}}^{H\otimes H^{*}}\mathrm{𝒴𝒟}$ over the tensor product bialgebra $H\otimes H^{*}$ as monoidal categories. Moreover if $H$ is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.

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 $(H_8,H_4,▹,◃)$ is explicitly described, and then the associated bicrossed product is given by generators and relations.

Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m,d}\left(q\right)\bowtie k\left[G\right]$ between the generalized Taft algebra $H_{m,d}\left(q\right)$ and group algebra $k\left[G\right]$ is actually the smash product $H_{m,d}\left(q\right)♯k\left[G\right]$. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$. As an application, the classification of $H_{m,d}\left(q\right)\bowtie k[C_{n_1}\times C_{n_2}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.

We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford’s $\pi$-biproduct. Firstly, we discuss the endomorphism monoid ${\mathrm{End}}_{\pi \text{-Hopf}}(A\times H,p)$ and the automorphism group ${\mathrm{Aut}}_{\pi \text{-Hopf}}(A\times H,p)$ 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 $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$. What’s more interesting is that a pair of maps $(F_L,F_R)$ can be used to describe a family of mappings $F={\left\{F_{\alpha }\right\}}_{\alpha \in \pi }$. Secondly, we consider the relationship between the automorphism group ${\mathrm{Aut}}_{\pi \text{-Hopf}}(A\times H,p)$ and the automorphism group ${\mathrm{Aut}}_{\pi \text{-}𝒴𝒟\text{-Hopf}}\left(A\right)$ of $A$, and...