Cohomology of algebras over weak Hopf algebras

TL;DR

This paper develops Sweedler cohomology for cocommutative weak Hopf algebras, showing that the second cohomology group classifies weak crossed products with a common preunit.

Contribution

It introduces a cohomology theory for weak Hopf algebras and characterizes weak crossed products via the second cohomology group.

Findings

Second cohomology group classifies weak crossed products

Provides a cohomological framework for weak Hopf algebra extensions

Connects cohomology with algebraic structures in weak Hopf settings

Abstract

In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra H. We show that the second cohomology group classifies completely the weak crossed products, having a common preunit, of H with a commutative left H-module algebra A.