Coquasitriangular structures for extensions of Hopf algebras. Applications

TL;DR

This paper characterizes coquasitriangular structures on certain Hopf algebra extensions, providing a classification in terms of data related to unified products and applying it to quantum doubles.

Contribution

It establishes a bijective correspondence between coquasitriangular structures and specific data on unified products, extending understanding of quantum doubles.

Findings

Classifies coquasitriangular structures on extensions of Hopf algebras.

Provides necessary and sufficient conditions for quantum doubles to be coquasitriangular.

Includes detailed examples illustrating the theoretical results.

Abstract

Let $A \subseteq E$ be an extension of Hopf algebras such that there exists a normal left $A$-module coalgebra map $\pi : E \to A$ that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra $E$ in terms of the datum $(A, E, \pi)$ as follows: first, any such extension $E$ is isomorphic to a unified product $A \ltimes H$, for some unitary subcoalgebra $H$ of $E$ (\cite{am2}). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product $A \ltimes H$ and a certain set of datum $(p, \tau, u, v)$ related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite dimensional quantum double $D_{\lambda}(A, H) = A \bowtie_{\tau} H$ to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.