For two families of Yetter-Drinfel'd Hopf algebras considered earlier by the authors, we determine which of them are isomorphic. We also determine which of their Radford biproducts are isomorphic.

Let $K_n$ be Kiselman's semigroup. We show that the sequence $2^{-n/2}\cdot \log |K_n|$ admits finite nonzero limits as $n$ grows to infinity both on odd and even values.

We use Cartier's preadditive symmetric monoidal categories to study Lie bialgebras. We prove that bosonization can be done consistently in this framework. In the last part of the paper we present explicit examples and indicate a deep relationship between certain curved Lie bialgebras and Nichols algebras over abelian groups.

We describe the structure of the braided shuffle algebra associated to a symmetrically braided vector space of diagonal type, as a braided symmetric algebra generated by a precise set of words (described in terms of Lyndon words) associated to the braiding. The description of the set of generating words depends only on a super-vector space structure canonically associated to the original symmetrically braided vector space.

We prove two isomorphisms of some subcategories of fusion systems defined on $N$-interior $p$-permutation $G$-algebras by extending the main results obtained by M. E. Harris.

Let $k$ be a field. We study the free bialgebra $\mathcal{T}$ generated by the coalgebra $C=k\{1,g,h\}$ and its quotient bialgebras (or Hopf algebras) over $k$. We show that the free non-commutative Faà di Bruno bialgebra is a sub-bialgebra of $\mathcal{T}$, and the quotient bialgebra $\overline{\mathcal{T}}:=\mathcal{T}/(E_{\alpha} \mid \alpha(g)\geq 2)$ is an Ore extension of the well-known Faà di Bruno bialgebra. The image of the free non-commutative Faà di Bruno bialgebra in the quotient $\overline{\mathcal{T}}$ gives a more reasonable non-commutative (non-free) version of the commutative Faà di Bruno bialgebra from the PBW basis point view. If char$\, k=p>0$, we obtain a chain of quotient Hopf algebras of $\mathcal{T}$ with infinite or finite GK-dimension. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.

We exhibit a bijective correspondence between certain left ideal coideals in a Hopf algebroid for which the resulting quotient is a coequalizer and certain right coideal subrings which are themselves an equalizer, remarkably improving a recent result obtained in collaboration with L. El Kaoutit, A. Ghobadi, and J. Vercruysse. Interpreting this result in the Hopf algebra setting provides a bijective correspondence which extends the previously known cases.

Lie algebras are extended to the affine case using the heap operation, giving them a definition that is not dependent on the unique element $0$, such that they still adhere to antisymmetry and Jacobi properties. It is then looked at how Nijenhuis brackets function on these Lie affgebras and demonstrated how they fulfil the compatibility condition in the affine case.