Twisting products of Hopf algebras and quantum double

Let sigma be a skew pairing on the pair (B, H) of Hopf algebras, and A a le ft (B, H) bicomodule algebra. A new algebra A(sigma), called the twisting p roduct of A, is obtained by alternating the multiplication of A using sigma and the coactions on A by B and H. sigma induces a skew pairing <(sigma)ov er cap> (BxB(oop) HxH(op)), and the regular comodule structures of B and H induce a left (BxB(oop), HxH(op)) bicomodule algebra structure on BxH, and the associated twisting product (BxH)<(sigma)over bar> is a Hoof algebra, w ith the tensor coalgebra structure; moreover, A(sigma) remains a left (BxH) <(sigma)over bar> comodule algebra. In particular, a description of the Dri nfeld double is obtained from the twisting point of view. In addition, smas h products appear as special cases. Dually, the construction of twisting co products is introduced by using copairings, and the Drinfeld quantum codoub le and some smash coproducts are described.