BIPRODUCT BIALGEBRAS WITH A PROJECTION ONTO A HOPF ALGEBRA

Abstract

Let (D, B) be an admissible pair. Then recall that B ×L H D (cid:192)πD D are bialgebra maps satisfying πD ◦ iD = I. We have solved a iD converse in case D is a Hopf algebra. Let D be a Hopf algebra with antipode sD and be a left H-comodule algebra and a left H-module coalgebra over a field k. Let A be a bialgebra over k. Suppose A (cid:192)π i D are bialgebra maps satisfying π ◦ i = ID. Set Π = ID ∗ (i ◦ sD ◦ π), B = Π(A) and j : B → A be the inclusion. Suppose that Π is an algebra map. We show that (D, B) is an admissible pair and B (cid:191)Π i D is an admissible mapping system and that the generalized biproduct bialgebra B ×L H D is isomorphic to A as bialgebras.

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