Some classifications of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:x^2y}$ of Kashina

Let $H$ be the $16$-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra $H_{b:x^2y}$ classified by Kashina. We figure out all simple Yetter-Drinfeld $H$-modules, and then determine all finite-dimensional Nichols algebras satisfying the constraint condition $\mathcal{B}(V)\cong \bigotimes_{i\in I}\mathcal{B}(V_i)$, where $V=\bigoplus_{i\in I}V_i$, each $V_i$ is a simple object in $_H^H\mathcal{YD}$. Finally, we describe some liftings of the corresponding Radford biproducts $\mathcal{B}(V)\sharp H$, which provide some classifications of finite dimensional Hopf algebras with $H$ as their coradical.

💡 Research Summary

The paper investigates finite‑dimensional Hopf algebras whose coradical is the 16‑dimensional non‑trivial semisimple Hopf algebra (H=H_{b:x^{2}y}) classified by Kashina. The authors follow the Andruskiewitsch–Schneider lifting method: first they determine all simple Yetter‑Drinfeld modules over (H), then they study Nichols algebras generated by these modules, and finally they classify all possible liftings of the corresponding Radford biproducts (\mathcal B(V)\sharp H).

1. Drinfeld double and simple Yetter‑Drinfeld modules.
The Drinfeld double (D=D(H^{\mathrm{cop}})) is presented explicitly by generators and relations. By a detailed representation‑theoretic analysis the authors prove (Theorem 3.7) that (D) possesses 32 one‑dimensional simple modules and 56 two‑dimensional simple modules. Using the well‑known monoidal equivalence ({}_H^H\mathcal{YD}\simeq D\text{-mod}), they translate this classification into a complete list of simple objects in the Yetter‑Drinfeld category ({}_H^H\mathcal{YD}). The one‑dimensional objects are denoted (M_i) (with indices in several subsets of ({1,\dots,12})), while the two‑dimensional ones are denoted (V_i).

2. Nichols algebras under the tensor‑product constraint.
For a semisimple Yetter‑Drinfeld module (N=\bigoplus_{i\in I}N_i) the authors impose the condition
\