The Structure Of The Braid Of Category _hm Hopf Algebra

Since Drinfeld introduced the concepts of quasi-triangular bialgebras to solve the Yang-Baxter equation, many people have been studying its properties and obtained many significant results. It was known that there are many significant results for the antipode. Upon the background above, in this paper we mainly investigate the antipodes and constructions of Hopf algebras in braided category H MFirstly, suppose that (H,R) is a quasi-triangular bialgebra, we will introduce some concepts of the twisted H-module algebras and H-module coalgebras. We obtain that if A is H-module algebras then algebras (RA)op,(R’A)op,R(A(?)A)and R’(A(?)A) are H-module algebras, and if C is a H-module coalgebra then (RC)cop,(R’C)cop,R(C(?)C) and R’(C(?)C) are H-module coalgebras.Secondly, we will prove that the antipode s of a Hopf algebra B in category H M gives an left H-module algebra homomorphism s:Bâ†'(RB)op and a H-module coalgebra homomorphism s:(R’B)copâ†'B, see Theorem4.2. Afterwards, we also obtain that if s is bijiective then the inverse s-1:Bâ†'(R’B)op is a H-module algebra isomorphism, and s-1:(RB)copâ†'B is a H-module coalgebra isomorphism, see Corollary4.4.Lastly, by using the above properties, we construct some new Hopf algebras in H M from a given Hopf algebra in H M, see Theorem5.4, Theorem5.5, Theorem5.6and Theorem5.7.