2021/05/06 by R. González Rodrı́guez, Rodríguez, Ramón González, Ana Belén Rodríguez Raposo +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2105.02528
openalex publication_date 2021/05/06 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
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:H⊗ H→ B we define its obstruction θσ 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 H3_φZ(B)(H, Z(B)) if, and only if, there exists a twisted convolution invertible 2-cocycle α:H⊗ H→ 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.