González Rodríguez, RamónRodríguez Raposo, Ana Belén2024-02-062024-02-062022Ramón González Rodríguez, Ana Belén Rodríguez Raposo, Cohomological obstructions and weak crossed products over weak Hopf algebras, Journal of Algebra, Volume 610, 2022, Pages 491-526, ISSN 0021-8693, https://doi.org/10.1016/j.jalgebra.2022.07.030.0021-8693http://hdl.handle.net/10347/32429Let Hbe 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→Bwe define its obstruction θσas 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→Bsuch that H⊗Bcan be endowed with a weak crossed product structure with αkeeping a cohomological-like relation with σ. Then, as a consequence, the class of the obstruction of σvanish if, and only if, there exists a cleft extension of Bby H.engCC BY-NC-ND 4.0http://creativecommons.org/licenses/by-nc-nd/4.0/Weak Hopf algebrasSweedler cohomologyWeak crossed productsCleft extensionsObstruction18M0516T0520J06journal article10.1016/j.jalgebra.2022.07.0301090-266Xopen access