Classifying bicrossed products of Hopf algebras

TL;DR

This paper classifies all Hopf algebras that factorize through two given Hopf algebras using cohomological methods, explicitly describing bicrossed products and their automorphisms, with applications to quantum groups at roots of unity.

Contribution

It provides a classification framework for bicrossed products of Hopf algebras via cohomological objects and describes automorphism groups explicitly, including applications to quantum groups.

Findings

Classified bicrossed products $A owtie H$ up to isomorphism.

Explicit description of quantum groups $H_{4n, ext{ω}}$ at roots of unity.

Determined automorphism groups of these quantum groups.

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 $A \bowtie H$ 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.