Weak C-cleft extensions, weak entwining structures and weak Hopf algebras

Abstract

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.

H-Galois Extensions With Normal Basis for Weak Hopf Algebras

“…It is a well-known fact in classical Galois theory that if B ⊂ A is a finite Galois extension of fields with Galois group H, then A/B has a normal basis, i.e., there exists a ∈ A such that the set {x.a ; x ∈ H} is a basis for A over B. Generalizing finite Galois extension of fields, Kreimer and Takeuchi introduce in [13] the notion of H-Galois extension with normal basis, associated to a Hopf algebra H in a category of modules over a commutative ring, and in [10] Doi and Takeuchi show that there exists an equivalence between the notion of H-Galois extension with normal basis and the one of H-cleft extension for H. This result can be generalized to symmetric closed categories [11] and in [7] we find a more general formulation in the context of entwining structures that was extended to the weak setting in [2] by using the notion of weak C-cleft extensions defined in [1]. On the other hand, being A an algebra, C a coalgebra and Γ A H : C ⊗ A → A ⊗ C a morphism in a strict monoidal category with equalizers and coequalizers, such that (A, C, Γ A H ) is a weak entwining structure, we have introduced in [2] the notion of weak C-Galois extension with Grant MTM2013-43687-P: Homología, homotopía e invariantes categóricos en grupos yálgebras no asociativas.…”

“…In [5] we introduce the notion of H-cleft extension for a weak Hopf algebra H and we prove that this kind of extensions are examples of weak H-cleft extensions like the ones introduced in [1] and satisfying the classical notion of cleftness when particularizing to the Hopf setting. Assuming cocommutativity for H, we give in [5] a bijective correspondence between the equivalence classes of H-cleft extensions A H → B and the equivalence classes of crossed systems for H over A H where A H denotes the subalgebra of coinvariants of the H-comodule algebra (A, ρ A ) in the weak context.…”

“…We assume that the reader is familiar with the notions of (co)algebra and (co)module and morphisms between them in this monoidal setting (see [1], [2]). …”

“…More precisely, the main result in [1] establishes that, if A ⊗ − preserves coequalizers, A H → A is a weak H-cleft extension if and only if A H → A is a weak H-Galois extension with normal basis. For clarity we briefly review the proof:…”

Cleft Extensions of Hopf Algebroids

“…Subsequently, the notion of a cleft coalgebra extension was introduced in [15, p. 293], and most comprehensively studied in terms of cleft entwining structures in [1], [22]. The latter were extended further to weak entwining structures in [2], [3].…”