Cohomological Obstructions and Weak Crossed Products over Weak Hopf Algebras

Let $H$ be a cocommutative weak Hopf algebra and let $(B, \varphi_{B})$ a weak left $H$-module algebra. In this paper, for a twisted convolution invertible morphism $\sigma:H\otimes H\rightarrow B$ we define its obstruction $\theta_{\sigma}$ as a degree three Sweedler 3-cocycle with values in the center of $B$. We obtain that the class of this obstruction vanish in third Sweedler cohomology group $\mathcal{H}^3_{\varphi_{Z(B)}}(H, Z(B))$ if, and only if, there exists a twisted convolution invertible 2-cocycle $\alpha:H\otimes H\rightarrow B$ such that $H\otimes B$ can be endowed with a weak crossed product structure with $\alpha$ keeping a cohomological-like relation with $\sigma$. Then, as a consequence, the class of the obstruction of $\sigma$ vanish if, and only if, there exists a cleft extension of $B$ by $H$.