Weak Hopf Algebras Explained By Corings

Weak bialgebras and Hopf algebras are generalizations of ordinary bial-gebras and Hopf algebras in the following sense: the defining axioms are the same, but the multiplicativity of the counit and comultiplicativity of the unit, are replaced by weaker axioms. The aim of this note is to develop the theory of corings and Galois theory for weak Hopf algebras, then we get some corresponding results about weak Hopf algebras. Weak Hopf modules can be viewed as comodules over a coring and this implies that the general theory of Galois corings can be applied to weak Hopf algebras. Finally, we look at the dual situation, where the weak Hopf algebra is the dual of a finite groupoid algebra.