On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras
Jialei Chen, Shilin Yang, Dingguo Wang, Yongjun Xu
TL;DR
The paper shows that a class of 4n-dimensional Hopf algebras $H_{4n}$ are quasitriangular with an explicit universal $R$-matrix, and constructs the associated weak Hopf algebras $\mathfrak{w}H_{4n}$ and $\mathfrak{w}H_{4n}^*$ along with their representation theories. It provides complete descriptions of indecomposable modules and tensor products for both Hopf and weak Hopf settings, and derives explicit presentations of their Green rings via generators and relations, including the noncommutativity that appears in the weak case. The duals $H_{4n}^*$ and $\mathfrak{w}H_{4n}^*$ are analyzed in parallel, yielding a decomposition $\mathfrak{w}H_{4n}^*\cong H_{4n}^*\oplus k[y]/(y^2-1)$ and explicit Green ring structures. Overall, the work advances understanding of quasi-triangular structures and representation rings in non-pointed, non-semisimple contexts, highlighting how weak Hopf algebra frameworks alter Green ring commutativity.
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.