For a weak entwining structure (A, C, ψ) we formulate the notion of weak C-Galois extension with normal basis and we show that these Galois extensions are equivalent to the weak C-cleft extensions introduced in [J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, A.B. Rodríguez Raposo, Weak C-cleft extensions, weak entwining structures and weak Hopf algebras, J. Algebra 284 (2005) 679-704].

We formulate the concept of weak cleft extension for a weak entwining structure in a braided monoidal category C with equalizers and coequalizers. We prove that if A is a weak C-cleft extension, then there is an isomorphism of algebras between A and a subobject of the tensor product of A C and C where A C is a subalgebra of A. Also, we prove the corresponding dual results and linking the information of this two parts we obtain a general property for a pair morphisms f : C → A and g : A → C of algebras and coalgebras satisfying certain conditions. Finally, as particular instances, we get the results of Fernández and Rodríguez, the theorems of Radford, Majid and Bespalov (in the case of Hopf algebras with projection) and the ones obtained by Alonso and González for weak Hopf algebras living in a symmetric category with split idempotents, for example, the weak theorem of Blattner, Cohen and Montgomery for weak Hopf algebras with coalgebra splitting is one of them.  2004 Elsevier Inc. All rights reserved.

No abstract

In this paper we study weak Hopf algebras with projection. If f : H → B, g : B → H are morphisms of weak Hopf algebras such that g • f = id H , we prove that it is possible to find an object B H , in the new category of weak Yetter-Drinfeld modules, that verifies similar conditions to the ones include in the definition of weak Hopf algebra. Finally, we define weak smash bialgebra structures and prove that, under central and cocentral conditions, B H and H determine an example of them.

In this paper we introduce the notion of weak Hopf quasigroup as a generalization of weak Hopf algebras and Hopf quasigroups. We obtain its main properties and we prove the fundamental theorem of Hopf modules for these algebraic structures.