Weak Hopf algebras with projection and weak smash bialgebra structures

In this paper we study weak Hopf algebras with projection. If f :H→B, g :B→H are morphisms of weak Hopf algebras such that g f=idH, we prove that it is possible to find an object BH, in the new category of weak Yetter–Drinfeld modules, that verifies similar conditions to the ones include in the definition of weak Hopf algebra. Finally, we define weak smash bialgebra structures and prove that, under central and cocentral conditions, BH and H determine an example of them.