Biproducts and Two-cocycle Twists of Hopf Algebras

Let H be a Hopf algebra with bijective antipode over a field k and suppose that R#H is a bi-product. Then R is a bialgebra in the Yetter–Drinfel’d category HYD. We describe the bialgebras (R#H) and (R#H) explicitly as bi-products R#Hop and R#H respectively where R is a bialgebra in H op HopYD and R o is a bialgebra in H o HoYD. We use our results to describe two-cocycle twist bialgebra structures on the tensor product of bi-products.