Classifying bicrossed products of Hopf algebras

Let $A$ and $H$ be two Hopf algebras. We shall classify up to an isomorphism that stabilizes $A$ all Hopf algebras $E$ that factorize through $A$ and $H$ by a cohomological type object ${\mathcal{H}}^2(A,H)$. Equivalently, we classify up to a left $A$-linear Hopf algebra isomorphism, the set of all bicrossed products $A\bowtie H$ associated to all possible matched pairs of Hopf algebras $(A,H,◃,▹)$ that can be defined between $A$ and $H$. In the construction of ${\mathcal{H}}^2(A,H)$ the key role is played by special elements of $CoZ^{1} (H, A) \times \Aut_{\rm CoAlg}^1 (H)$,where$CoZ^{1} (H, A)$isthegroupofunitarycocentralmapsand$\Aut_{\rm CoAlg}^1(H)$isthegroupofunitaryautomorphismsofthecoalgebra$H$\mathrm{.}Amongseveralapplicationsandexamples,allbicrossedproducts$H_4 \bowtie k[C_n]$ are described by generators and relations and classified: they are quantum groups at roots of unity $H_{4n,\omega }$ which are classified by pure arithmetic properties of the ring ${\mathbb{Z}}_n$. The Dirichlet’s theorem on primes is used to count the number of types of isomorphisms of this family of $4n$-dimensional quantum groups. As a consequence of our approach the group ${\text{Aut}}_{\mathrm{H}\mathrm{o}\mathrm{p}\mathrm{f}}(H_{4n,\omega })$ of Hopf algebra automorphisms is fully described.