Weak Hopf algebras and weak Yang–Baxter operators

In this paper we introduce the notions of weak Yang–Baxter operator and weak braided Hopf algebra. We prove that it is possible to obtain examples of these notions working with Yetter–Drinfeld modules associated to a weak Hopf algebra H with invertible antipode. Finally, we complete the study of the structure of weak Hopf algebras with a projection obtaining a categorical equivalence between the category of weak Hopf algebra projections associated to H and the category of Hopf algebras in the non-strict braided monoidal category of left–left Yetter–Drinfeld modules over H.