With Weak Injective Hopf Algebra Structure

Constructing new Hopf algebras from known Hopf algebras is one of the most important problems in Hopf algebra theory . We can give different structures on a tensorproduct H Q to make it a bialgebra or Hopf algebra. In [1] Radford constructed aHopf algebra A X H with the smash algebra structure and the smash coproduct coalgebra structure , and pointed out that if a bialgebra B has a Hopf subalgebra H and a projection ! : B--> H , then there must be a subalgebra A of B such that B @ A x H is a bialgebra isomorphism. Schauengurg[2] considered more general case and found necessary and sufficient conditions for the cosmash product coalgebra H Q to be a Hopf algebra with a Hopf subalgebra H and a weak projection !:H Q-->H.In this paper , we will consider smash product algebras. We will derive necessaryand sufficient conditions for a smash product algebra A#H to be a bialgebra with aquotient bialgebra H and a weak injection i:H-->A#H. Furthermore, we induce necessary and sufficient conditions for A# H to be a Hopf algebra when H is a Hopf algebra.In section 2 , the starting point is to describe maps F : A#H --> B# H in terms oftheir 'constituents' F(1):A-->B and F(2) :H-->B, then we give the equivalentconditions that F(1) and F(2) satisfy. In section 3 , we will describe H - bicomodule algebra structures on A# H . Since these are given by maps A#H --> H A# H , and since H A# H is itself a smash product, all we have to do is how to use the characterization of maps between smash products obtained in section 2, and give the constituents of the comodule axioms. In section 4 , we first show that for twobicomodule algebra A#H and B#H, the cotensor product (A#H)H(B#H) isstill a smash product (A B)# H. Then we describe the structure on A B and givethe corresponding equivalent conditions.In section 5 , we will complete the description of the bialgebra structures we have given, and give the equivalent conditions for such bialgebras to be Hopf algebras .The Hopf structure on A#H can be described by four maps a: A -->> H A,:H-->H A, A:A-->AA and :H--> A A .We will show that this struc-ture includes biproduct, bicrossed product and bicrossed coproduct as special cases by restricting maps or A to some special maps.