Cohomological obstructions and weak crossed products over weak Hopf algebras

Abstract

Let H be 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→B we define its obstruction θσ as 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 HφZ(B)3(H,Z(B)) if, and only if, there exists a twisted convolution invertible 2-cocycle α:H2→B such that H⊗B can be endowed with a weak crossed product structure with α keeping a cohomological-like relation with σ. Then, as a consequence, the class of the obstruction of σ vanish if, and only if, there exists a cleft extension of B by H.