Weak Hopf algebras with projection and weak smash bialgebra structures.
The aim of this paper is to investigate weak Hopf algebras with projection. As a generalization of the corresponding theory of Hopf algebras, for a weak Hopf algebra \(H\), the authors introduce the category of weak Yetter-Drinfeld modules, denoted by \(^H_H\mathcal{WYD}\). For morphisms of weak Hopf algebras \(f\colon H\to B\), \(g\colon B\to H\) such that \(g\circ f=\text{id}_H\), it is verified that there exists an object \(B_H\) in \(^H_H\mathcal{WYD}\) as a weak Hopf algebra in this category with a weak Hopf algebra isomorphism \(\omega\) between \(B\) and \(B_H\bowtie H\). On the other hand, the authors give the definition of weak smash bialgebra structures and prove that, under central and cocentral conditions, the example \((B_H,H,R,S)\) can be constructed from \(B_H\) and \(H\).
scientific article; zbMATH DE number 2013036