Crossed products over weak Hopf algebras related to cleft extensions and cohomology

The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions A (H) a dagger(a) A, where A (H) is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over A (H) . Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group (H,Z(A (H) )), where Z(A (H) ) is the center of A (H) .