Coquasitriangular structures for extensions of Hopf algebras. Applications

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 $A⋉H$, 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 $A⋉H$ 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.

16 pages, to appear in Glasgow Math. J

References in corpus (1)

• Crossed product of Hopf algebras