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For a class of neither pointed nor semisimple Hopf algebras $H_{4n}$ ofdimension $4n$, 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, theirduality and their Green rings are established by generators and relationsexplicitly. It turns out that the Green rings of the associated weak Hopfalgebras are not commutative even if the Green rings of $H_{4n}$ arecommutative.

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