Classifying Bicrossed Products of Hopf Algebras

Abstract

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 ⋈ 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 $CoZ^{1} (H, A) \times {\rm Aut}\,_{\rm CoAlg}^1 (H)$, where CoZ ¹ (H, A) is the group of unitary cocentral maps and ${\rm Aut}\,_{\rm CoAlg}^1 (H)$ is the group of unitary automorphisms of the coalgebra H. Among several applications and examples, all bicrossed products H ₄ ⋈ k[C _n ] are described by generators and relations and classified: they are quantum groups at roots of unity H _{4n, ω} which are classified by pure arithmetic properties of the ring 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 _Hopf(H _{4n, ω} ) of Hopf algebra automorphisms is fully described.