Cohomological obstructions and weak crossed products over weak Hopf algebras

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 : H2-+ B we define its obstruction theta Sigma as a Sweedler 3-co cycle with values in the center of B. We obtain that the class of this obstruction vanish in third Sweedler cohomology group H3 phi Z(B) (H, Z(B)) if, and only if, there exists a twisted convolution invertible 2-co cycle alpha : H2-+ B such that H circle times 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.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).