PROPERTIES OF GENERALIZED BIPRODUCT HOPF ALGEBRAS

Abstract

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 ∈ Homk(B, B) is an inverse of idB then B ×L H d) = Σ(1B ×L H 1D). We show that the mapping system B (cid:191)ΠB D (where jB and iD are the canonical inclusions, jB ΠB and πD are the canonical coalgebra projections) characterizes B ×L H D. These generalize the corresponding results in [6].

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