Cohomological obstructions and weak crossed products over weak Hopf algebras

Abstract

Let H be a cocommutative weak Hopf algebra and let (B,φB)a weak left H-module algebra. In this paper, for a twisted convolution invertible morphism σ : H2 → B we define its obstruction θσ as a Sweedler 3-cocycle with values in the center of B. We obtain that the class of this obstruction vanish in third Sweedler cohomology group H3φZ(B) (H, Z(B)) if, and only if, there exists a twisted convolution invertible 2-cocycle α : H2 → B such that H ⊗ B can be endowed with a weak crossed product structure with α keeping a cohomological-like relation with σ. Then, as a consequence, the class of the obstruction of σ vanish if, and only if, there exists a cleft extension of B by H.