Cohomological obstructions and weak crossed products over weak Hopf algebras

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