L-R-SMASH BIPRODUCTS, DOUBLE BIPRODUCTS AND A BRAIDED CATEGORY OF YETTER-DRINFELD-LONG BIMODULES

Let H be a bialgebra and D an-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 M endowed with left-left 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).