BIPRODUCT BIALGEBRAS WITH A PROJECTION ONTO A HOPF ALGEBRA

Let (D;B) be an admissible pair. Then recall that B £LHD À¼DiDD are bialgebra maps satisfying ¼D ± iD = I: We have solved a converse in case D is a Hopf algebra. Let D be a Hopf algebra with antipodesD and be a left H-comodule algebra and a left H-module coalgebra over a¯eld k: Let A be a bialgebra over k: Suppose A À¼ i D are bialgebra maps satisfying ¼ ± i = ID: Set ¦ = ID ¤ (i ± sD ± ¼);B = ¦(A) and j : B ! A be the inclusion. Suppose that ¦ is an algebra map. We show that (D;B)is an admissible pair and B ¿¦ j A À¼ i D is an admissible mapping system and that the generalized biproduct bialgebra B £LH D is isomorphic to A as bialgebras.