L-R-smash biproducts, double biproducts and a braided category of Yetter-Drinfeld-Long bimodules

Let H be a bialgebra and D an H-bimodule algebra and H-bicomodule coalgebra. We find sufficient conditions on D for the L-R-smash product algebra and coalgebra structures on D ⊗ H to form a bialgebra (in this case we say that (H,D) is an L-R-admissible pair), called L-R-smash biproduct. The Radford biproduct is a particular case, and so is, up to isomorphism, a double biproduct with trivial pairing. We construct a prebraided monoidal category LR(H), whose objects are H-bimodules H-bicomodules M endowed with leftleft and right-right Yetter-Drinfeld module as well as left-right and right-left Long module structures over H , with the property that, if (H,D) is an L-R-admissible pair, then D is a bialgebra in LR(H).