On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras

Abstract

For a class of neither pointed nor semisimple Hopf algebras $H_{4n}$ of dimension $4n$, it is shown that they are quasi-triangular, which universal $R$-matrices are described. The corresponding weak Hopf algebras $\mathfrak{w}H_{4n}$ and their representations are constructed. Finally, their duality and their Green rings are established by generators and relations explicitly. It turns out that the Green rings of the associated weak Hopf algebras are not commutative even if the Green rings of $H_{4n}$ are commutative.