PROPERTIES OF GENERALIZED BIPRODUCT HOPF ALGEBRAS

The biproduct bialgebra has been generalized to generalized biproduct bialgebra B£L H D in [5]. Let (D;B) be an admissible pair and let D be a bialgebra. We show that if generalized biproduct bialgebra B £L H D is a Hopf algebra with antipode s, then D is a Hopf algebra and the identity idB has an inverse in the convolution algebra Homk(B;B): We show that if D is a Hopf algebra with antipode sD and sB 2 Homk(B;B) is an inverse of idB then B£L HD is a Hopf algebra with antipode s described by s(b£LH d) =§(1B £LH sD(b¡1 ¢ d))(sB(b0) £LH 1D): We show that the mapping system B ¿¦BjBB £LH D À¼DiDD (where jB and iD are the canonical inclusions, ¦B and ¼D are the canonical coalgebra projections) characterizes B £LH D:These generalize the corresponding results in [6].