A class of double crossed biproducts - Dong, L. H.; Li, H. Y.; Ma, T. S. (Ukrains’kyi Matematychnyi Zhurnal, 2018)
Let H be a bialgebra, let A be an algebra and a left H-comodule coalgebra, let B be an algebra and a right H-comodule coalgebra. Also let f : H H → A H, R : H A → A H, and T : B H → H B be linear maps. We present necessary and sufficient conditions for the one-sided Brzezi´nski’s crossed product algebra A#^f_RH_T#B and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which generalizes the main results from [On Ranford biproduct // Communs Algebra. – 2015. – 43, № 9. – P. 3946 – 3966]. It is clear that both Majid’s double biproduct [Double-bosonization of braided groups and the construction of U_q(g) // Math. Proc. Cambridge Phil. Soc. – 1999. – 125, № 1. – P. 151 – 192] and the Wang – Jiao – Zhao’s crossed product [Hopf algebra structures on crossed products // Communs Algebra. – 1998. – 26. – P. 1293 – 1303] are obtained as special cases.