Sovereign and ribbon weak Hopf algebras

Abstract

In this paper, we study the Deligne's sovereign structure theorem on the finite dimensional weak Hopf algebras, and give a necessary and sufficient condition for a finite dimensional (co)quasitriangular weak Hopf algebra H with bijective antipode to admit a (co)ribbon structure. As an application we discuss the ribbon structures over the Drinfeld doubles of some weak Hopf algebras to verify our theory.