On some classification of finite-dimensional Hopf algebras over the Hopf algebra \(H_{b:1}^\) of Kashina

Let \(H\) be the dual of \(16\)-dimensional nontrivial semisimple Hopf algebra \(H_{b:1}\) in the classification work of Kashina \cite{K00}. We completely determine all finite-dimensional Nichols algebras satisfying \(\mathcal{B}(N)\cong \bigotimes_{i\in I}\mathcal{B}(N_i)\), where \(N=\bigoplus_{i\in I}N_i\), each \(N_i\) is a simple object in \(_H^H\mathcal{YD}\). Under this assumption, we classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra \(H\) via the relevant Nichols algebras \(\mathcal B(N)\).