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 $(A, H, \triangleleft, \triangleright)$ 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 $CoZ^{1} (H, A)$ is the group of unitary cocentral maps and $\Aut_{\rm CoAlg}^1(H)$ is the group of unitary automorphisms of the coalgebra $H$. Among several applications and examples, all bicrossed products $H_4 \bowtie k[C_n]$ 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 $\Aut_{\rm Hopf}(H_{4n, \omega})$ of Hopf algebra automorphisms is fully described.