Coquasitriangular structures for extensions of Hopf algebras. Applications

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.