Cohomological obstructions and weak crossed productus over weak Hopf algebras

Abstract

Let 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.