Cohomological Obstructions and Weak Crossed Products over Weak Hopf Algebras

Abstract

Let $H$ be a cocommutative weak Hopf algebra and let $(B,{\phi }_B)$ a weak left $H$-module algebra. In this paper, for a twisted convolution invertible morphism $\sigma :H\otimes H\to B$ 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 ${\mathcal{H}}_{{\phi }_{Z(B)}}^3(H,Z(B))$ if, and only if, there exists a twisted convolution invertible 2-cocycle $\alpha :H\otimes H\to B$ 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 $H$.