Two-cocycles and cleft extensions in left braided categories

We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra [Formula: see text] a Yetter–Drinfeld module braids from the left with [Formula: see text]-modules. We will generalize classical results to this context and give some application for the categories of Yetter–Drinfeld modules and [Formula: see text]-modules. In particular, we will describe liftings of coradically graded Hopf algebras in the category of Yetter–Drinfeld modules with these techniques.

Comparison of two notions of weak crossed product

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in [Cleft extensions of Hopf algebroids, Appl. Categor. Struct. 14(5–6) (2006) 431–469] with the notion of crossed product with a weak Hopf algebra introduced in [Crossed products for weak Hopf algebras with coalgebra splitting, J. Algebra 281(2) (2004) 731–752].

Cyclic homology of cleft extensions of algebras

Let [Formula: see text] be a commutative algebra with [Formula: see text] and let [Formula: see text] be a cleft extension of [Formula: see text]. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of [Formula: see text] relative to [Formula: see text]. This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra.

Cleft extensions for quasi-entwining structures

In this paper we introduce the notions of quasi-entwining structure and cleft extension for a quasi-entwining structure. We prove that if (A, C, ψ) is a quasi-entwining structure and the associated extension to the submagma of coinvariants AC is cleft, there exists an isomorphism ωA between AC ⊗ C and A. Moreover, we define two unital but not necessarily associative products on AC ⊗ C. For these structures we obtain the necessary and sufficient conditions to assure that ωA is a magma isomorphism, giving some examples fulfilling these conditions.

Weak quasi-entwining structures

In this paper we introduce the notion of weak quasi-entwining structure as a generalization of quasi-entwining structures and weak entwining structures. Also, we formulate the notions of weak cleft extension, weak Galois extension, and weak Galois extension with normal basis associated to a weak quasientwining structure. Moreover, we prove that, under some suitable conditions, there exists an equivalence between weak Galois extensions with normal basis and weak cleft extensions. As particular instances, we recover some results previously proved for Hopf quasigroups, weak Hopf quasigroups and weak Hopf algebras.

Cleft extensions and Galois extensions for Hom-associative algebras

In this paper, we consider Hom-(co)modules associated to a Hom-(co)associative algebra and define the notion of Hom-triple. We introduce the definitions of cleft extension and Galois extension with normal basis in this setting and we show that, as in the classical case, these notions are equivalent in the Hom setting.