We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ℕ) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ℤ and collapsing H 0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring...

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...

We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products $H_4\bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4,H_4,▹,◃)$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for...

Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products $(A{♮}_{R}B,\alpha \otimes \beta )$. Moreover, necessary and sufficient conditions for $(A{♮}_{R}B,\alpha \otimes \beta )$ to be a cobraided Hom-Hopf algebra are given.