Cleft extensions in braided categories

Majid in 14] and Bespalov in 2] obtain a braided interpretation of Radford’s theorem about Hopf algebras with projection (19]). In this paper we introduce the notion of H-cleft comodule (module) algebras (coalgebras) for a Hopf algebra H in a braided monoidal category, and we characterize it as crossed products (coproducts). This allows us give very short proofs for know results in our context, and to introduce others stated for the category of R-modules about of Hopf algebra extensions. In particular we give a proof of the result by Bespalov 2] for a braided monoidal category with co(equalizers).