Cohomological Obstructions and Weak Crossed Products over Weak Hopf Algebras

TL;DR

This paper investigates the cohomological obstructions to constructing weak crossed products over weak Hopf algebras, linking the vanishing of certain cohomology classes to the existence of cleft extensions.

Contribution

It introduces a cohomological framework for understanding obstructions in forming weak crossed products over weak Hopf algebras, connecting these obstructions to Sweedler cohomology.

Findings

Obstruction class vanishes iff a compatible 2-cocycle exists

Existence of a cleft extension corresponds to vanishing obstruction

Provides cohomological conditions for weak crossed product structures

Abstract

Let $H$ be a cocommutative weak Hopf algebra and let $(B, \varphi_{B})$ a weak left $H$-module algebra. In this paper, for a twisted convolution invertible morphism $\sigma:H\otimes H\rightarrow B$ we define its obstruction $\theta_{\sigma}$ 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 $\mathcal{H}^3_{\varphi_{Z(B)}}(H, Z(B))$ if, and only if, there exists a twisted convolution invertible 2-cocycle $\alpha:H\otimes H\rightarrow B$ such that $H\otimes B$ 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 $H$.