Duality Theorem And Cross Product Hopf Algebras In Braided Tensor Categories

The author discusses the duality theorem of infinite dimensional Hopf algebras in braided tensor categories in the first chapter, and shows the theorem by the use of braiding diagram. The main results are calculated as follows:Proposition 2.1 Let (A , m ,η) be an algebra, then ( A0 , △ , ε ) is a coalgebra with comultiplication △ - m and counitε =η .Proposition 2.2 Let (B, m, η , △, ε ) be a bialgebra, then (B0 , △, ε , m, η ) is also a bialgebra. In addition, if B=H is a Hopf algebra with antipodes, then H0 is a Hopf algebra with antipodes S .Proposition 2.8 Let H be a Hopf algebra and U a subHopfalgebra of H0 such that both H and U have bijective antipodes, and assume that U satisfies the RL-condition with respect to H. Let A be a U-comodule algebra, so that A is an H-module algebra as above. Let U act on AH by acting trivially on A and via - on H. then (AH)U= A (HU).The author gives the necessary and sufficient condition for the crossproduct bialgebra D= A1.2×2.1H to be a Hopf algebra. The resultsare got as follows:Lemma3.4 If both A and H have antipodes and(M1)-(M3), (CM1)-(CM3), (B1)-(B4),(CB2) , (CB4) hold, then D=Aα×βH has an antipode SD:Corollary3.5 If (A, H, α , β , φ ,ψ )is a Hopf datum and both A and H have antipodes, then the double bicrossproduct D=Aα×βH has an antipode.Proposition3.6 The cross product bialgebra D=A1.2×2.1H is a Hopf algebra when A and H have antipodes.