Weak Hopf algebras arising from weak matched pairs

This work extends the idea of matched pairs presented by Majid in {Majid} and Takeuchi in {Takeuchi} for the context of weak bialgebras and weak Hopf algebras. We introduce, also inspired by partial matched pairs {matchedpair}, the notion of weak matched pairs and establish conditions for a subspace of the smash product be a weak bialgebra/Hopf algebra. Further, some new examples of (co)actions of weak bialgebras over algebras and some results about integral elements are presented.