Crossed products for weak Hopf algebras with coalgebra splitting

AbstractIn this paper we prove that if g:B→H is a morphism of weak Hopf algebras which is split as a coalgebra morphism, then there exists a subalgebra BH of B, morphisms φBH:H⊗BH→BH, σBH:H⊗H→BH and an isomorphism of algebras and right H-comodules bB:B→BH×H, being BH×H a subobject of BH⊗H with its algebra structure defined by a crossed product. Also, we obtain the dual results and as a consequence we prove the Radford's theorem for weak Hopf algebras