On Hopf algebras whose coradical is a cocentral abelian cleft extension (Q6561421)

In [Math. USSR, Sb. 5, 451--474 (1969; Zbl 0205.03301); translation from Mat. Sb., N. Ser. 76(118), 473--496 (1968)], \textit{G. I. Kac} introduced a family of semisimple finite-dimensional Hopf algebras. In the paper under review, the authors consider certain subfamily \(K_n\), \(n\in\mathbb{N}\). Every \(K_n\) can be constructed as a double crossed product \(\Bbbk^{\mathbb{Z}_n\times\mathbb{Z}_n}\rtimes_\beta\Bbbk\mathbb{Z}_2\) or alternatively, as an abelian extension \(\Bbbk^\mathbb{Z}_n\hookrightarrow K_n\twoheadrightarrow\Bbbk\mathbb{D}_n\) of Hopf algebras, where \(\mathbb{D}_n\) is the dihedral group of order \(2n\).\N\NThe present work is a first step toward the ultimate aim of the authors of classifying the finite-dimensional Hopf algebras whose coradical is isomorphic to \(K_n\) following the lifting method. With this in mind, they begin by classifying the simple objects in the category \({}_{K_n}^{K_n}\mathcal{YD}\) of Yetter-Drinfeld modules over \(K_n\) and describing their braidings. They also compute the fusion ring of \({}_{K_n}^{K_n}\mathcal{YD}\).\N\NIt turns out that the dimensions of the simple objects are \(1\), \(2\) or \(n\). Those of dimension \(1\) or \(2\) are braided vector space of diagonal type. Instead, those of dimension \(n\) are \(t\)-equivalent to braided vector spaces associated to the rack of involutions in \(\mathbb{D}_n\) (equivalently, transpositions in \(\mathbb{S}_n\)) and cocycles, and hence the corresponding Nichols algebras are isomorphic as graded vector spaces. Then, they appeal to the literature for deciding when their Nichols algebras are finite-dimensional or not.\N\NThe authors analyze in detail the Nichols algebras of the \(n\)-dimensional simple objects for \(n=3\). In this case, they find three finite-dimensional Nichols algebras: \(\mathfrak{B}_{1}\), \(\mathfrak{B}_{\xi}\) and \(\mathfrak{B}_{\xi^2}\) (here \(\xi\) is a 3rd root of unity involved in the structure of the simple objects). They are isomorphic as graded vector spaces to the well-known Fomin-Kirillov algebra \(\mathcal{E}_3\) and hence of dimension \(12\). Moreover, \(\mathfrak{B}_1\simeq\mathcal{E}_3\) as graded algebras but \(\mathfrak{B}_{\xi}\) are \(\mathfrak{B}_{\xi^2}\) new examples.