BIPRODUCT BIALGEBRAS WITH A PROJECTION ONTO A HOPF ALGEBRA

Abstract

Let (D,B) be an admissible pair. Then recall that <TEX>$B\;{\times}^L_HD^{{\rightarrow}{\pi}_D}_{{\leftarrow}i_D}\;D$</TEX> are bialgebra maps satisfying <TEX>${\pi}_D{\circ}i_D=I$</TEX>. We have solved a converse in case D is a Hopf algebra. Let D be a Hopf algebra with antipode <TEX>$S_D$</TEX> and be a left H-comodule algebra and a left H-module coalgebra over a field <TEX>$k$</TEX>. Let A be a bialgebra over <TEX>$k$</TEX>. Suppose <TEX>$A^{{\rightarrow}{\pi}}_{{\leftarrow}i}D$</TEX> are bialgebra maps satisfying <TEX>${\pi}{\circ}i=I_D$</TEX>. Set <TEX>${\Pi}=I_D*(i{\circ}s_D{\circ}{\pi}),B=\Pi(A)$</TEX> and <TEX>$j:B{\rightarrow}A$</TEX> be the inclusion. Suppose that <TEX>${\Pi}$</TEX> is an algebra map. We show that (D,B) is an admissible pair and <TEX>$B^{\leftarrow{\Pi}}_{\rightarrow{j}}A^{\rightarrow{\pi}}_{\leftarrow{i}}D$</TEX> is an admissible mapping system and that the generalized biproduct bialgebra <TEX>$B{\times}^L_HD$</TEX> is isomorphic to A as bialgebras.