Twisting products of Hopf algebras and quantum double

Abstract

Let σ be a skew pairing on the pair (B,H) of Hopf algebras, and A a left (B, H) bicomodule algebra. A new algebraA σ , called the twisting product ofA, is obtained by alternating the multiplication ofA using σ and the coactions onA byB andH. σ induces a skew pairing $$\hat \sigma $$ on (B⊗B ccp,H⊗H cp), and the regular comodule structures ofB andH induce a left (B⊗Bccp, H⊗Hcp) bicomodule algebra structure onH⊗H, and the associated twisting product (B⊗H)σ- is a Hopf algebra, with the tensor coalgebra structure; moreover,A σ remains a left (B⊗H)σ- comodule algebra. In particular, a description of the Drinfeld double is obtained from the twisting point of view. In addition, smash products appear as special cases. Dually, the construction of twisting coproducts is introduced by using copairings, and the Drinfeld quantum codouble and some smash coproducts are described.