ON CROSSED DOUBLE BIPRODUCT

Let H be a bialgebra. Let σ : H ⊗ H → A be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action. Let B be a right H-module algebra and also a comodule coalgebra. In this paper, we provide necessary and sufficient conditions for the one-sided crossed product algebra A#σ H # B and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which we call the crossed double biproduct. Majid's double biproduct is recovered from this. Moreover, necessary and sufficient conditions are given for Brzeziński's crossed product equipped with the smash coproduct coalgebra structure to be a bialgebra. The celebrated Radford's biproduct in [The structure of Hopf algebra with a projection, J. Algebra92 (1985) 322–347], the unified product defined by Agore and Militaru in [Extending structures II: The quantum version, J. Algebra336 (2011) 321–341] and the Wang–Jiao–Zhao's crossed product in [Hopf algebra structures on crossed products Comm. Algebra26 (1998) 1293–1303] are all derived as special cases.