Integrals and crossed products over weak Hopf algebras

الأصل · EN

In this paper we present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a weak left H-module algebra we obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions Aₕ A, where Aₕ is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over Aₕ. Finally, we 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 HᵩZ₍ₐₕ₎²(H, Z(Aₕ)), where Z(Aₕ) is the center of Aₕ.