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 ${}_H^H{\mathcal YD}$. We describe the bialgebras (R{#}H)^{op} and (R{#}H)^o explicitly as bi-products R^{\UOP}{#}H^{op} and R^{\UO}{#}H^o respectively where $R^{\UOP}$ is a bialgebra in ${}^{H^{op}}_{H^{op}}{\mathcal YD}$ and $R^{\UO}$ is a bialgebra in ${}^{H^o}_{H^o}{\mathcal YD}$. We use our results to describe two-cocycle twist bialgebra structures on the tensor product of bi-products.